diff --git a/-9AyT4oBgHgl3EQfdffn/content/tmp_files/2301.00305v1.pdf.txt b/-9AyT4oBgHgl3EQfdffn/content/tmp_files/2301.00305v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cd21e965ce7990c0e22a8f4feffbc92d878aae01 --- /dev/null +++ b/-9AyT4oBgHgl3EQfdffn/content/tmp_files/2301.00305v1.pdf.txt @@ -0,0 +1,8207 @@ +arXiv:2301.00305v1 [math.CT] 31 Dec 2022 +UNIVERSITY OF CALGARY +The functorial semantics of Lie theory +by +Benjamin MacAdam +A THESIS +SUBMITTED TO THE FACULTY OF GRADUATE STUDIES +IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE +DEGREE OF DOCTOR OF PHILOSOPHY +GRADUATE PROGRAM IN COMPUTER SCIENCE +CALGARY, ALBERTA +JUNE, 2022 +© Benjamin MacAdam 2022 + +Abstract +Ehresmann’s introduction of differentiable groupoids in the 1950s may be seen as a starting point for +two diverging lines of research, many-object Lie theory (the study of Lie algebroids and Lie groupoids) +and sketch theory. This thesis uses tangent categories to build a bridge between these two lines of +research, providing a structural account of Lie algebroids and the Lie functor. +To accomplish this, we develop the theory of involution algebroids, which are a tangent-categorical +sketch of Lie algebroids. We show that the category of Lie algebroids is precisely the category of invo- +lution algebroids in smooth manifolds, and that the category of Weil algebras is precisely the classi- +fying category of an involution algebroid. This exhibits the category of Lie algebroids as a tangent- +categorical functor category, and the Lie functor via precomposition with a functor +∂ : Weil1 → �Gpd, +bringing Lie algebroids and the Lie functor into the realm of functorial semantics. +ii + +Preface +This thesis is the original work of the author. +This thesis project arose from attempts to broadly understand differential geometry, and in partic- +ular the differential geometry of mechanics. First steps were taken with Jonathan Gallagher in under- +standing how the enriched perspective on category theory introduced in Garner (2018) might relate to +differential geometric structures. The basic structures in this thesis, however, came out of discussions +and collaboration with Matthew Burke in the Lie theory context. +Chapter 1 is an introduction to the theory of tangent categories, and contains no new results. Chap- +ter 2 began as a joint project with Matthew, who ultimately had to leave the project due to time con- +straints, although by that time we had already found the basic structure of the proof that differential +bundles are vector bundles in the category of smooth manifolds (proved in Theorem 2.5.1). The cur- +rent structure of the chapter, particularly with the emphasis on associative coalgebras of the weak tan- +gent comonad (T,ℓ) and the tight connection to Grabowski’s previous work on the Euler Vector Field +construction (Grabowski and Rotkiewicz (2009)), is original work. The basic results in that chapter ap- +pear in MacAdam (2021); however, the results have been streamlined (there was originally a notion of +a strong differential bundle, in Section 2.4 it is observed that all differential bundles are strong), and +there are some new observations about linear connections (Theorem 2.6.6). +Chapter3 isa rewrite ofa preprintwritten with Matthew on involution algebroids(Burke and MacAdam +(2019)), while Section 3.4 is due to conversations with Richard Garner (we expect to release a new pa- +per on involution algebroids based on these results as a joint work). The original idea of augmenting +an anchored bundle with an involution map is entirely due to Matthew, and the isomorphism on ob- +jects between involution and Lie algebroids follows calculations shared by Richard Garner (Proposition +3.4.12). My own contribution in this section comes from connecting this to the work of Martínez (2001), +as well as giving the bijection on morphisms for involution and Lie algebroids (Theorem 3.5.1). +The first three chapters provide the background required for the deeper results in Chapter 4. In +the work of Weinstein (1996); Martínez (2001); de León et al. (2005) on Lie algebroids, it was observed +that the “prolongation” of a Lie algebroid acted like a tangent bundle. Proposition 4.5.1 makes this +intuition precise by showing the prolongation is a second tangent structure on the category of Lie al- +gebroids. The main theorem in this chapter (Theorem 4.4.8) shows that involution algebroids in a +tangent category � are equivalent to tangent functors (A,α) from Weil1 to � so that the functor A pre- +serves transverse limits and the natural transformation α is T -cartesian (Definition 4.4.1). These are +entirely new observations about Lie algebroids and are original work of the author. +Finally, Chapter 5 puts the first four chapters into the language of enriched category theory, us- +ing Garner’s enriched perspective on tangent categories (Garner (2018)). The first original results in +this chapter demonstrate that differential bundles and anchored bundles are models of � -sketches +(Propositions 5.2.6 and 5.2.10), where � is the site of enrichment for tangent categories. Next, the +enriched theories framework of Bourke and Garner (2019) is used to prove Theorem 5.4.14, that invo- +lution algebroids are models of a nervous theory, which is the enriched version of Theorem 4.4.8. The +thesis concludes with the Lie Realization, Theorem 5.5.13, which is a new characterization of Lie differ- +iii + +entiation and introduces an entirely new way to construct adjunctions between categories of “smooth +groupoids” and categories of “Lie algebroids” using purely enriched-categorical methods (this is in +contrast to the geometric approach used in Crainic and Fernandes (2003) and the homotopy theoretic +approach in Sullivan (1977)). +This thesis touches on relatively advanced topics in two areas of math: differential geometry (Lie +algebroids) and enriched category theory (enriched nerve constructions). I have striven to keep it as +self-contained as possible, introducing the category of smooth manifolds and tangent categories and +including an appendix with the basics of enriched category theory and locally presentable category the- +ory. Material on foundational category theory (which is to say, anything that can be found in MacLane +(1988)) and basic differential calculus (see any calculus textbook) is used without citation or introduc- +tion; this includes limit, adjunction, monads and monadicity theorems, and the calculus of Kan exten- +sions and coends. Some facts about horizontal composition spans are used in Chapter 4, but nothing +that goes beyond the basic definition. +iv + +Acknowledgments +I would first like to thank my supervisor, Robin Cockett, for guiding this project and for his investment +of time in reading various preprints and walking through proofs with me. I also owe a special debt of +thankstoMatthew Burke, whowasa close collaboratoron thisprojectand spearheaded the application +of tangent categories to Lie theory, and to Jonathan Gallagher for extensive discussions about tangent +categories. I am also grateful to Rory Lucyshyn-Wright for spending a summer teaching me enriched +categorytheory, and toKristine Bauerforhelpful conversationsoverthe years. I would alsolike tothank +all of the members of the Peripatetic Seminar and others over the years for friendly discussion and +feedback: Chad Nester, Prashant Kumar, J.S. Lemay, Priyaa Srinivasan, Cole Comfort, Daniel Satanove, +Rachel Hardeman, and Geoff Vooys. +Finally, I am grateful to my parents and family, and to my wife Niloofar for her support as I wrote +this thesis. +v + +For my wife, Niloofar. +vi + +Notation +We start with a table of symbols: +Notation +�,�,... +A (usually tangent) category, treated as a general context for mathematics, de- +noted using mathbb +� ,� ,... +A small category treated as a mathematical object, denoted by mathcal +A,B,... +Objects in a category and also functors. written using capital letters +f ,φ +Morphisms in a category and natural transformations: lower-case Roman and +Greek letters +f ◦ g +The composition of two maps g : A → B, f : B → C (applicative notation) +πi +The projection from the i t h component of an n-fold pullback A0q0×q1...qn−1×qnAn +or a product +�n Ai +F.G +Composition of two functors, F : � → �,G : � → � (applicative notation) +φ.G +Whiskering of a natural transformation +⊗ +Tensor product in a monoidal category +⊠ +A restricted notion of span composition, introduced in Definition 4.3.3 +T,p,0,+,ℓ,c +The data for an arbitrary tangent category, introduced in Definition 1.3.2 +D,⊙,0,!,δ +The data for an infinitesimal object, introduced in Definition 1.3.7 +T n +Iterated application of an endofunctor T +Tn +The n-fold pullback power of p : T → id for a tangent category, T p ×p ...p ×p T +vii + +We also provide a table of categories: +Notation +SMan +The category of smooth manifolds +�ℓ +The category of lifts in a tangent category �, introduced in Definition 2.2.3 +NonSing(�) +The category of non-singular lifts in a tangent category �, introduced in Defini- +tion 2.3.1 +PDiff(�) +The category of pre-differential bundles in a tangent category �, introduced in +Definition 2.4.1(i) +Diff(�) +The category of differential bundles in a tangent category �, introduced in Defi- +nition 2.4.1(ii) +LieAlgd +The category of Lie algebroids, introduced in Section 3.1 +InvAlgd(�) +The category of anchored bundles in a tangent category �, introduced in Defini- +tion 3.2.1 +Anc(A) +The category of anchored bundles in a tangent category �, introduced in Defini- +tion 3.2.1 +Anc� (A) +The category of involution algebroids with chosen prolongations in a tangent cat- +egory � +Weil1 +The category of Weil algebras, introduced in Section 4.1 +W +Notation for the Weil algebra �[x]/x 2 +� +The category of transverse-limit-preserving functors Weil1 → Set, introduced in +Definition 5.1.1 +Weiln +1 +The category of Weil algebras with width n, used in Definition 5.2.8 +Weil∗ +1 +The full subcategory of Weil1 spanned by {�,W } +� +The classifying � -category of an anchored bundles and all of its prolongations, +introduced in Definition 5.4.12 +viii + +Table of Contents +Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +ii +Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +iii +Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +v +Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +vi +Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +vii +Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii +1 +Tangent categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +1.1 +Differential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +1.2 +The category of smooth manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +1.3 +Tangent structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +1.4 +Local coordinates in a tangent category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +1.5 +Submersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +2 +Differential bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +2.1 +Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +2.2 +Lifts for the tangent weak comonad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +2.3 +Non-singular lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +38 +2.4 +Differential bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +2.5 +The isomorphism of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +44 +2.6 +Connections on a differential bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +45 +3 +Involution algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +48 +3.1 +Lie algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +50 +3.2 +Anchored bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +55 +3.3 +Involution algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +63 +3.4 +Connections on an involution algebroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +69 +3.5 +The isomorphism of Lie and involution algebroid categories . . . . . . . . . . . . . . . . . . . . +76 +4 +The Weil nerve of an algebroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +78 +4.1 +Weil algebras and tangent structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +80 +4.2 +Tangent structures as monoidal actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +83 +4.3 +The Weil nerve of an involution algebroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +87 +4.4 +Identifying involution algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +97 +4.5 +The prolongation tangent structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 +5 +The infinitesimal nerve and its realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 +5.1 +Tangent categories via enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 +5.2 +Differential and anchored bundles as enriched structures . . . . . . . . . . . . . . . . . . . . . . 113 +5.3 +Enriched nerve constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 +5.4 +Nervous monads and algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 +5.5 +The infinitesimal approximation of a groupoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 +6 +Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 +6.1 +Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 +6.2 +Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 +ix + +Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 +Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 +x + +Introduction +The study of Lie groupoids and Lie algebroids goes back to Charles Ehresmann and his student Jean +Pradines in the late 1950s, building upon Sophus Lie’s original research into the application of groups of +smooth symmetries to solving ordinary differential equations (Lie (1893)). Motivated by partial differ- +ential equations, Ehresmann (1959) introduced the notion of a differentiable groupoid, which models +the internal symmetries of a smooth manifold, in contrast to the external symmetries given by a Lie +group (a group object in the category of smooth manifolds) or Lie group action. Pradines (1967) ex- +tended the Lie functor (which sends Lie groups and Lie group actions to Lie algebras and Lie algebra +actions) from external to internal symmetries and introduced the notion of a Lie algebroid. In doing +so, he identified some shortcomings in Ehresmann’s original definition of differentiable groupoids, in- +troducing the modern notion of a Lie groupoid. +Ehresmann’s investigations into differentiable groupoids initiated one of the major differential ge- +ometry research programmes of the second half of the twentieth century, the study of Lie groupoids +and Lie algebroids (which one may refer to as many-object Lie theory to distinguish it from the “single- +object” Lie groups and Lie algebras that had classically been studied). Research into many-object Lie +theory in the 80s and 90s focused on extending Lie’s second theorem and the Cartan–Lie theorem, +which in modern terms state that Lie algebras form a coreflective subcategory of Lie groups; that is, the +Lie functorhasa fullyfaithful leftadjoint(Mackenzie and Xu (2000); Moerdijk and Mrˇcun (2002); Nistor +(2000)). The leftadjointisoften called Lie integration, and in theirfamouspaperCrainic and Fernandes +(2003) found the exactconditionsgoverningwhethera Lie algebroid integratestoa Lie groupoid. Weinstein +(1996) initiated a line of research into classical mechanics on Lie algebroids and groupoids, extending +Poincarè’s development of mechanics on a space with a Lie group action and the Euler–Poincarè equa- +tions (Poincaré (1901); see Marle (2013) for a modern treatment); this has been further developed by +Eduardo Martinez and his collaborators (de León et al. (2005); Martínez (2001); Martínez (2018); Fusca +(2018)). +However, Ehresmann’s work in differentiable groupoids signalled a change in focus for his own re- +search, as he increasingly focused within the then-new area of category theory. Over the course of +the 60s and 70s Ehresmann published a string of influential papers in the nascent area of functorial +semantics, developing the formalism of sketch theory. Sketch theory has proven to be highly influen- +tial in mathematical logic, and has been active for some 40 years, being extended to syntax/semantics +adjunctions Gabriel and Ulmer(2006), enriched category theory Kelly (1982) and generalized limit doc- +trines Adámek et al. (2002). The influence of sketch theory can still be seen on Lie theory in the work +of Kirill Mackenzie and his collaborators to develop the theory of double Lie algebroids, Lie-algebroid +groupoids, double-vector bundles, and other tensor-product theories (Mackenzie (1992, 2011)). +This thesis aims to provide a structural account of the Lie functor from Lie groupoids to Lie al- +gebroids, using tangent categories Cockett and Cruttwell (2014) to unify Ehresmann’s many-object Lie +theory and sketch theory. Tangent categories provide a syntactic description of tangent structure based +on Kock and Lawvere’s synthetic differential geometry (Kock (2006), Lawvere (1979)) and the Weil func- +tor formalism (a comprehensive account may be found in Kolár et al. (1993), while the explicit link to +1 + +abstract tangent structure is found in Leung (2017), another line of research in differential geometry +that has run parallel to modern Lie theory (albeit with some exchange, e.g. Kolár (2007)). Recent work +has recast tangent categories as a class of enriched categories (Garner (2018)), making it possible for +modern techniques from sketch theory and functorial semantics to be applied to differential geome- +try. In doing so, we demonstrate that the language of tangent categories sheds light on the study of +classical mechanics on Lie algebroids and groupoids, as well as Mackenzie’s investigations into “Ehres- +mann doubles” of vector bundles and Lie algebroids. +Overview +The first three chapters of this thesis build on previous work in the tangent category literature (for +example, Cockett and Cruttwell (2017, 2018); Lucyshyn-Wright (2018)), providing tangent-categorical +sketches of differential-geometric structures. These structures follow Ehresmann’s original notion of +a sketch quite closely; they are specified as graphs (a collection of objects and arrows in the category) +with a set of diagrams that must commute and cones that must be universal, except that the data may +now include the tangent functor T and the tangent natural transformations p, 0, +, ℓ, and c . Each +sketch is accompanied by a proof that its category of models in smooth manifolds (which we shall +often write SMan) is precisely the category of geometric structures it seeks to model. +The first chapter reviews the basic theory of tangent categories, paying particular attention to the +category of smooth manifolds. The first examples of “sketches” from the tangent categories literature +are covered, namely differential objects and affine connections, which model vector spaces and con- +nections respectively (Cockett and Cruttwell (2017, 2018)). The chapter concludes with a study of tan- +gent submersions, which model submersions from classical differential geometry and are a useful ex- +ample of the sort of work that occurs in Chapters 2 and 3. +The second and third chapters develop tangent categorical sketches for vector bundles and Lie +algebroids respectively. Chapter 2 extends the observation due to Grabowski and Rotkiewicz (2009) +that the category of vector bundles is a full subcategory of multiplicative monoid actions by the non- +negative reals �+, and then applies the Euler vector field construction (Definition 2.1.8). The tangent +categorical sketch for a vector bundle, called a differential bundle, is then developed based on a mor- +phism λ : E → T E , and an isomorphism of categories between differential bundles in SMan and the +category of smooth vector bundles is proved. Chapter 3 introduces involution algebroids, which re- +place the bracket of a Lie algebroid with an involution map +σ : A̺×T πT A → A̺×T πT A +(where ̺ : A → T M is the anchor of the Lie algebroid). Using Martinez’s presentation of the structure +equations for a Lie algebroid (Martínez (2001)), we are once again able to prove an isomorphism of +categories, this time that the category of Lie algebroids is isomorphic to that of involution algebroids +in SMan. This provides the initial bridge between differential geometric structures and tangent cate- +gorical sketches, making it possible to apply more sophisticated techniques in Chapters 4 and 5. +The fourth chapter constructs a syntactic tangent category for Lie algebroids, and demonstrates +that Lie algebroids are precisely generalized tangent bundles. We call this result the Weil nerve, as it +follows the same structure as Grothendieck’s original nerve theorem Segal (1974), in this case using the +categories presentation of tangent categories due to Leung (2017). This result has useful implications +for the study of generalized mechanics and geometric structures on Lie algebroids, as it introduces a +novel tangent structure on the category of Lie algebroids. This novel tangent structure corresponds to +Poincarè/Weinstein/Martinez’s characterization of classical mechanics on a Lie algebroid. +The fifth and final chapter introduces the enriched categories perspective on tangent categories +from Garner (2018), so that the work in Chapters 2 and 3 may be rephrased using the enriched sketches +2 + +of Kelly (2005). We construct a functor from the syntactic category of involution algebroids to the syn- +tactic category of a groupoid-in-a-tangent-category, thus giving a presentation of the Lie functor in +the spirit of Ehresmann’s sketch theory. The syntactic version of the Lie functor is built by construct- +ing another novel tangent structure on the category of Lie groupoids (or more generally, groupoids in +a tangent category), which also agrees with previous investigations into classical mechanics on a Lie +groupoid. As a final result, we demonstrate that in a locally presentable tangent category we may use a +left Kan extension to construct a left adjoint to the Lie functor, which we call the the Lie realization. +3 + +Chapter 1 +Tangent categories +Tangent categories are an example of convergent evolution in mathematics, in which two unrelated +lines of research with very different aims have arrived at a common endpoint, in this case the same +formal setting for abstract differential geometry. The older line of research has its roots in differential +geometry proper and, in particular, Weil’s algebraic characterization of the tangent bundle of a smooth +manifold (Weil (1953)). Weil’s work motivated Kock and Lawvere’s development of synthetic differen- +tial geometry, presented in the book of the same name Lawvere (1979); Kock (2006) as well as the Weil +functor formalism of Kolár et al. (1993). The second, more recent line of research has its foundations +in theoretical computer science following the publication of Linear Logic by Girard (1987). Ehrhard +and Regnier noticed that some models of linear logic have a notion of the “Taylor series approxima- +tion” of a proof; this led to the development of differential linear logic by Ehrhard and Regnier (2003). +Blute, Cockett, and Seely studied the categorical semantics of models of linear logic equipped with the +derivative operation - that is, they identified those categories whose internal language are were models +of differential linear logic - developing a categorical theory of differentiation in Blute et al. (2006, 2009). +Tangent categories arise naturally in each line of research: on the first path with the distillation +of synthetic differential geometry into abstract tangent functors in Rosický (1984), and more recently +when Cockett and Cruttwell refined abstract tangent functors following their investigations into the +manifold categoriesofcartesian differential restriction categoriesin Cockett et al.(2011); Cockett and Cruttwell +(2014). In a sense, they are categories that axiomatize Weil’s characterization of the tangent bundle as +an endofunctor in a way that captures the combinatorics of higher-order derivatives when looking at +a certain class of internal commutative monoids (Cockett and Seely (2011)), as will be made precise +in Chapter 4. Tangent categories also pull Kock and Lawvere’s synthetic differential geometry into the +framework of enriched category theory, which is explored in Chapter 5. +Instances of tangent structure abound throughout mathematics and computer science. For exam- +ple, many categories of geometric spaces have natural tangent structure, such as the category of conve- +nient manifolds (the categoryofmanifoldsmodelled locallybyconvenient vectorspaces Kriegl and Michor +(1997)) and the category of schemes (see point (ii) in Example 2 of Garner (2018)). An example from +mathematical logic is the category of Köthe sequence spaces( Ehrhard (2002)), and categorical models +ofthe differential lambda-calculus(Cockett and Gallagher(2019)). More recently, tangentand differen- +tial categorieshave found applicationsin differentiable programmingand machine learning(Wilson and Zanasi +(2021)), and to understanding Johnson and McCarthy’s functor calculus Bauer et al. (2018). +This thesis studies differential geometric structures using the language of tangent categories, fol- +lowing the tradition of synthetic differential geometry. As such, this chapter will develop tangent cate- +gories with a focus on the category of smooth manifolds. Extending the study of these formal structures +in the context of novel tangent categories is a significant endeavour and should be treated as a direc- +4 + +tion for future research. The first section introduces Cartesian differential categories as the categorical +axiomatization of multivariable calculus. The second section introduces the category of smooth man- +ifolds and two characterizations of its tangent bundle (kinematic versus operational), while the third +section identifies the structure of the kinematic tangent bundle that characterizes abstract tangent +structures. The fourth section presents a pair of structures that allow for “local-coordinate calcula- +tions” in the tangent category of differential objects and connections. The final section introduces +tangent submersions. A submersion is a differentiable map between differentiable manifolds whose +differential is everywhere surjective; as a preview of the work in Chapters 2 and 3, this section shows +that in the category of smooth manifolds, a tangent submersion is precisely a submersion. Section 1.5 +first appeared in MacAdam (2021), and is the only original work in this chapter. +1.1 +Differential calculus +As with most treatments of synthetic differential geometry, e.g. Kock (2006), it makes sense to begin +with the differential calculus - in this case, an introduction to the categorical theory of differentiation. +Categorical differentiation has recently gained quite a bit of attention due to its relationship with ma- +chine learning Cockett et al. (2020), and applications to homotopy theory Bauer et al. (2018). This sec- +tion will just consider the basic structures introduced in Blute et al. (2009), and the canonical example +of a cartesian differential category (the category of finite-dimensional real vector spaces and smooth +maps between them). +Definition 1.1.1. [Definition 1.2.1 Blute et al. (2009)] A cartesian left additive category is a cartesian +category1 � so that: +(i) Each hom-set �(A,B) is a commutative monoid with addition +AB and zero map 0AB : A → B (the +subscript AB will be suppressed when the context is clear). +(ii) The composition operation ◦ preserves addition on the left: +(g + h) ◦ f = g ◦ f + h ◦ f +(iii) Projection is an additive map (preserves addition): +πi ◦ (f + g ) = (πi ◦ g ◦ f ) + (πi ◦ g ) +Where πi denotes the projection from the i t h component of a product or pullback. j +There are various examples of cartesian left additive categories - they all fit the same pattern of a +category where each object is equipped with a non-natural, but coherent, choice of linear structure: +Example 1.1.2. +(i) Any category with biproducts is a cartesian left additive category where every map is additive. +(ii) The category of cartesian spacesCartSp, whose objects are finite-dimensional real vector spaces and +morphisms are smooth maps between them, is a cartesian left additive category." Clearly smooth +maps from A → B are closed under addition, projection is an additive map, and (g + h) ◦ f = +g ◦ f + h ◦ f . +1We use the standard notation where 1 is the terminal object, × is product, and πi is the i t h projection. +5 + +(iii) The category of topological vector spaces and continuous morphisms is a cartesian left additive +category. +In fact, cartesian left additive categories may equivalently be described as cartesian categories +where each object has a coherent choice of commutative monoid structure. +Proposition 1.1.3 (Blute et al. (2009)). The following are equivalent: +(i) � is a cartesian left additive category +(ii) � is a cartesian category so that each object has a chosen commutative monoid structure (A,+A,0A) +where the following coherence holds: +(A × B)2 +A × B +A2 × B 2 ++A×B +τ ++A×+B +(where τ = ((π0 ◦ π0,π0 ◦ π1),(π1 ◦ π0,π1 ◦ π1))). +(iii) There is a category with biproducts �+ and a bijective-on-objects subcategory inclusion i : �+ �→ � +that creates products. +Cartesian left additive categories provide an appropriate to define a differentiation operation. Re- +call that the usual derivative of a map f : � → � from elementary calculus can be written +∂ f +∂ x : � → �. +More generally, for a map f : �n → �, one writes the Jacobian of f at x: +J [f ] : �n → (�n ⊸ �m) := + + +∂ f1 +∂ x1 +... +∂ f1 +∂ xn +... +... +... +∂ f1 +∂ x1 +... +∂ f1 +∂ xn + + +The Jacobian, however, requires some notion of a “matrix”2 representing a linear map from �n to �m— +not every category that has a notion of differentiation supports that operation. Instead, the directional +derivative: +D [f ](x,v) := lim +d→0 +f (x + t · v) +t +gives an appropriately general notion of differentiation that extends to categories where the space of +linear maps A → B is not representable by an object in the category 3. A cartesian differential category +axiomatizes the directional derivative as a combinator on a cartesian left-additive category. +2This would be called an internal hom in the categorical logic literature. +3In the case of automatic differentiation, it is also worth noticing that computing the directional derivative of a map +�n → �m has complexity 2� (f ), while forming the Jacobian has complexity n� (f ), so the directional derivative is a more +appropriate primitive for purely practical computational reasons (see Section 5 of Hoffmann (2016) for a discussion of the +computational complexity of forward-mode automatic differentiation). +6 + +Definition 1.1.4. [Definition 2.1.1 in Blute et al. (2009)] A cartesian differential category is a cartesian +left additive category equipped with a combinator (e.g. a function on hom-sets) +A +f−→ B +A × A −−→ +D[f ] B +satisfying the following axioms: +[CD.1] Additive: +D [f + g ] = D[f ]+ D[g ] +D[0] = 0 +[CD.2] Additive in the second variable: +D[f ]◦ (g ,h + k) = D[f ]◦ (g ,h) + D[f ]◦ (g ,k) +D[f ]◦ (g ,0) = 0 +[CD.3] Projection is linear: +D [πi ] = πi ◦ π1 +D[id ] = π1 +[CD.4] Pairing: +D[(f ,g )] = (D[f ],D[g ]) +[CD.5] Chain rule: +D[g ◦ f ] = D[g ]◦ (π0,D[f ]) +[CD.6] Linear in the second variable: +D[D [f ]]◦ ((a,0),(0,d )) = D[f ]◦ (a,d ) +[CD.7] Symmetry of partial differentiation: +D [D[f ]]◦ ((a,b ),(c ,d )) = D[D [f ]]◦ ((a,c ),(b,d )) +Example 1.1.5. The category of cartesian spaces (Example 1.1.2(ii)), CartSp, is the canonical cartesian +differential category. Let f : �n → �m, and consider its Jacobian at x ∈ �n, J [f ](x) ∈ �n×m. Define the +differential combinator: +D [f ]◦ (u,v) = J [f ](v) · u = lim +t →0 +f (x + t · v) +t +. +In Bauer et al. (2018), the authors construct a cartesian differential category based on the Abelian functor +calculus of Johnson and McCarthy (1998). +In Wilson and Zanasi (2021), the authors consider a cartesian differential category whose objects are �2- +modules to apply gradient-based methods to learn the parameters of of Boolean circuits. +Every cartesian differential category comes with a notion of linearity. This notion of linearity is +strictly stronger than additivity - there do exist examples of non-linear additive maps. +Definition 1.1.6 (Definition 2.2.1 of Blute et al. (2009)). A map f : A → B is linear whenever D[f ] ◦ +(0AB,id ) = f . +We denote the category of linear maps in a cartesian differential category � as Lin(�). The category +Lin(�) will have biproducts, and will be a cartesian differential subcategory of �. +7 + +Lemma1.1.7(Corollary2.2.3in Blute et al. (2009)). Let � be a cartesian differential category, and denote +its category of linear maps as Lin(�). +(i) Linear maps preserve addition. +(ii) The category Lin(�) is a bijective-on-objects subcategory of � with biproducts, and the inclusion +Lin(�) �→ � creates products. +(iii) Every category with biproducts is a cartesian differential category, where +D [f ] = f ◦ π1, +and this differential structure makes the inclusion Lin(�) �→ � preserve the left additive structure +and differential combinator (that is, it is a cartesian differential functor). +1.2 +The category of smooth manifolds +Tangent categories axiomatize a more general structure than differential calculus, one in which spaces +are only “locally linear.” The category of smooth manifolds gives the historically canonical example, +and a good portion of this thesis relates to structures internal to that category, so it seems worthwhile to +set a working definition for that context. We follow Tu (2011), and allow for disconnected components +of a manifold to have different dimensions. +Definition 1.2.1. [Definitions 5.5–5.7 in Tu (2011)] A chart on a topological space M is pair (Ui,φi : +Ui �→ �n), where Ui is an open subset Ui ⊆ M and φi : Ui → �n is a local homeomorphism. An atlas is +a collection of charts {(Ui,φi : Ui → �n)|i ∈ I } (where n is fixed for each connected component of M ) so +that for each i, j ∈ I , the transition function ψi,j that completes the diagram +φ−1 +i (Ui ∩Uj) +�n +Ui ∩Uj +Uj +φi +⊆ +φj +ψi,j +is a smooth map. A smooth manifold is a topological space equipped with a (maximal4) atlas. A mor- +phism of smooth manifolds is a topological map f : M → N that is locally smooth - for each chart pair of +charts (Ui,φi) on M and (Vj ,θj), see Figure 1.1 for an illustration. The map f is smooth whenever each +map fi,j that completes the diagram is smooth +Ui +Vj +M +N +f +fi,j +The category of smooth manifolds, SMan, is the category of smooth manifolds and their morphisms. +Remark 1.2.2. In Chapter 2, some results will implicitly use partition of unity arguments, which require +that the underlying topological space for a manifold is Hausdorff and has a countable basis (i.e. it is +second-countable). We will avoid any direct reference to these properties, and so we omit them from the +definition of a smooth manifold. +4With respect to subset inclusion. +8 + +φi +φ−1 +i +φ−1 +j +φj +M +Ui +Uj +φi(Ui) +�m +ψi j +φj(Uj) +�m +Figure 1.1: +Overlapping charts in an atlas (Credit for this tex code belongs to user Cragfelt +https://tex.stackexchange.com/a/388493/101171.) +Example 1.2.3. +(i) Each vector space �n, for n ∈ �, has a canonical smooth manifold structure whose atlas is a single +chart (the identity map �n → �n). +(ii) Most geometric shapes that do not have any singularities or sharp edges can be equipped with an +atlas without any issue. For example, consider the circle: {(cos(x),sin(x)) : x ∈ [−π,π)} +For any appropriately small ε > 0, there are two charts from Iε = (−ε,π+ ε); +φ0(t ) = (cos(t ),sin(t )), φ1(t ) = (cos(t − π),sin(t − π)) +making the circle a smooth manifold. +Category theory has not seen as many applications in differential geometry as it has in topology +or algebra, likely because the category of smooth manifolds (Definition 1.2.1) is somewhat poorly be- +haved. The two main reasons that the category of smooth manifolds is “inconvenient” are as follows: +• The set of smooth maps between two manifolds M ,N fails, in general, to form a smooth mani- +fold; thus, the category is not cartesian closed. +• The category of manifolds does not have quotients or arbitrary fibre products. +9 + +However, the category of smooth manifolds admits some limits, for example finite products. +Proposition 1.2.4 (1.12 Kolár et al. (1993)). The category SMan of smooth manifolds has finite products. +Proof. Given two manifolds M ,N , take the product of their underlying topological spaces together +with the product charts +(φi × ψj ) :Ui × Vj → �n × �m. +The category of smooth manifolds—using our definition where disconnected components of a +manifold may have different dimensions—does have a class of (co)limits used throughout this paper, +namely idempotent splittings. +Definition 1.2.5 (Borceux and Dejean (1986)). An idempotent is an endomorphism e : E → E so that +e ◦e = e . The splitting of an idempotent is given by a pair of maps e = s ◦r so that r ◦s = id . The existence +of a splitting of an idempotent e is equivalent to asking that the following pair of parallel arrows has a +(co)equalizer: +E +E +e +The idempotent splitting � of a category (also known as the Cauchy completion of the category), is the +full subcategory of presheaves [�op ,Set] that are retracts of representable functors. Any functor into a +category with idempotent splittings will factor through this inclusion of categories, so it is the free co- +completion of � under idempotent splittings. +Proposition 1.2.6 (Lawvere (1989)). The category of smooth manifolds is the idempotent splitting of +the category whose objects are open subsets of Cartesian spaces and whose morphisms are smooth maps +f :U → V . +An idempotent in the idempotent splitting of a category is also an idempotent in the base category, +and thus admits a splitting. +Corollary 1.2.7. The category of smooth manifolds is closed to idempotent splittings: for every map +e : M → M so that e = e ◦ e there exists a pair of maps r :Q → M ,s : M →Q so that e = s ◦ r,r ◦ s = id . +The main construction of interest on the category of smooth manifolds, for tangent categories at +least, is the tangent bundle. Given a smooth map f : M → N , restrict it to a morphism between coor- +dinate patches, so it may be regarded as a map f |U :U → V , where U ⊆ �m,V ⊆ �n. This gives a local +derivative operation (remembering that πi denotes the i t h projection from a product +�n Ai )5 +D[f |U ] :U × �m → �n, +(f |U ◦ π0,D[f |U ]) :U × �m → V × �n. +The tangent bundle makes this construction global; that is, there is a functor T : SMan → SMan giving +an assignment +T.M +T.f +−→ T.N +that agrees with the local derivative on coordinate patches of M ,N . +5The wording here was originally muddled, it has since been corrected. +10 + +Definition 1.2.8 (Kolár et al. (1993)). Write the algebra of smooth functions on a manifold as C ∞(M ) := +SMan(M ,�). The set SMan(�,M )/ ∼= of tangent vectors on a smooth manifold M comprises the curves +� → M subject to the equivalence relation that for a pair of curves φ,θ : � → M , φ ∼= θ if and only if +φ(0) = θ(0) and for every f ∈ C ∞(M ), +∂ f ◦ φ +∂ x +(0) = ∂ f ◦ θ +∂ x +(0). +The set SMan(�,M )/ ∼= has a naturally determined smooth manifold structure which we call the tangent +bundle over M , T M . +Example 1.2.9. +(i) For a vector space, the space of linear paths crossing through a point v ∈ V is isomorphic to V , so +T V ∼= V × V . +(ii) The tangent bundle above the circle is diffeomorphic to the cylinder. This is follows from the classi- +cal result that a tangent vector on the circle must be perpendicular to its position vector. +The tangent bundle lifts the “local derivative” into a globally defined construction, so the tangent +bundle construction is functorial. +Proposition 1.2.10. The tangent bundle is a product-preserving endofunctor on the category of smooth +manifolds. +Proof. Functoriality follows by showing that a morphism of smooth manifolds preserves the equiva- +lence relation on curves that defines a tangent vector: +∀g ∈ C ∞(M ), ∂ g ◦ φ +∂ x +(0) = ∂ g ◦ θ +∂ x +(0) +Note that if f : N → M ∈ C ∞(N ), so that g ◦ f ∈ C ∞(M ), the chain rule ensures that +∀g ∈ C ∞(N ), ∂ f ◦ g ◦ φ +∂ x +(0) = ∂ f ◦ g ◦ θ +∂ x +(0) +To show that T is product-preserving, it suffices to show that the equivalence classes of curves are +stable under pairing. First, note that for any M and φ ∼= θ : � → M , +(φ,id ) ∼= (θ,id ) : � → M × � +Given a pair of curves θM ,ψM : � → M , where θM ∼= ψM and similarly θN ∼= ψN for N , this implies that +f ◦ (φM ,φN ) += f ◦ (φM × id ) ◦ (id ,φN ) += f ◦ (φM × id ) ◦ (id ,θN ) += f ◦ (id ,θN ) ◦ (φM × id ) += f ◦ (id ,θN ) ◦ (θM × id ) += f ◦ (θM × θN ) +so (φM ,φN ) ∼= (θM ,θN ). +11 + +The scalar action by � on tangent vectors and a partially defined addition additionally give the +tangent bundle the structure of a fibered �-module (that is, an �-module in the slice categorySMan/M +whose objects are morphisms into M , f : X → M , and morphisms are commuting triangles). +Proposition 1.2.11. The tangent bundle over M is an �-module in SMan/M , as follows: let γ,ω be +tangent vectors on M , and define +• p : T M → M ;p(γ) = γ(0). +• 0 : M → T M ;0(m) = [r �→ m] (the constant map � → M sending all r ∈ � to m ∈ M ) +• ·p : T M × � → T M ;γ ·p r = [x �→ γ(r · x)] +• + : T M p ×p T M → T M := [γ],[ω] �→ [γ + ω] (where addition around γ(0) = ω(0) is defined using +local coordinates). +The second derivative is involved in the more nuanced axioms for a cartesian differential category, +namely linearity in the vector argument and the symmetry of mixed partial derivatives. First, set f |U +to be the restriction of f : M → N to a map between local coordinate patches U ⊆ M ,V ⊆ N , and then +define +f0 = f |U ◦ π0 ◦ π1, +, +and +f1 = D[f ]◦ (π0 ◦ π0,π1 ◦ π0), +f2 = D[f ]◦ (π0 ◦ π0,π0 ◦ π1) . +The axioms [C D C .6],[C D C .7] then give: +(U × �n) × (�n × �n) +(V × �m) × (�m × �m) +T 2M +T 2N +T M +T.N +(U × �n) +(V × �m) +T 2.f +T.f +((f0,f1),(f2,D[D[f |U ]])) +(f ,D[f |U ]) +ℓ +ℓ +((π0,0),(0,π1)) +((π0,0),(0,π1)) +(U × �n) × (�n × �n) +(V × �m) × (�m × �m) +T 2M +T 2N +T 2M +T 2.N +(U × �n) × (�n × �n) +(V × �m) × (�m × �m) +c +((π0◦π0,π0◦π1),(π1◦π0,π1◦π1) +T 2.f +T 2.f +c +((π0◦π0,π0◦π1),(π1◦π0,π1◦π1) +((f0,f1),(f2,D[D[f |U ]]) +((f0,f2),(f1,D[D[f |U ]]) +The two natural transformations ℓ and c —the vertical lift and canonical flip—capture these coher- +ences. Locally, ℓ is the map inserting zeros into the second and third coordinates, while c flips the +second and third arguments, leading to the coherences established in the next proposition. To capture +12 + +these coherences on the tangent bundle, first note that a tangent vector on T M is equivalent to an +equivalence class of surfaces on M , +φ ∼= θ : �2 → M ⇐⇒ φ(0,0) = θ(0,0) +and ∀f ∈ C ∞(M ), ∂ f ◦ φ +∂ xi +(0,0) = ∂ f ◦ θ +∂ xi +(0,0),i = 0,1 +Proposition 1.2.12 (Cockett and Cruttwell (2014)). There are two natural transformations +ℓ : T M → T 2M ;ℓ([γ]) = [γ ◦ (π0 ·� π1)] +c : T 2M → T 2M ;c ([γ]) = [γ ◦ (π1,π0)] +satisfying the following coherences: +(i) ℓ.T ◦ ℓ = T.ℓ◦ ℓ6 +(ii) The following maps are morphisms of fibred �-modules. +T M +T 2M +M +T M +ℓ +p +p.T +0 +T M +T 2M +M +T M +ℓ +p +T.p +0 +(iii) c ◦ c = id +(iv) T.c ◦ c .T ◦ T.c = c .T ◦ T.c ◦ c .T +(v) c ◦ ℓ = ℓ +(vi) T.c ◦ c ◦ T.ℓ = ℓ.T ◦ c . +Observation 1.2.13. Equation (iv) is known as the Yang–Baxter equation. It is one of the coherences for +a symmetric monoidal category, and states that the twisting operation between two variables is coherent. +We may regard the category of endofunctors on a category as a strict monoidal category and use string +diagram notation (see e.g. Selinger (2010)). Interpreting the map c as twisting two strings, the coherence +becomes += +The projection p : T M → M is locally trivial: for each connected component of M that is modeled +on �n, each point m lies in an open subset Um so that p −1(Um) ∼=Um ×�n. This local triviality property +leads to the following universality condition. +Proposition 1.2.14. The following diagram is an equalizer: +T M +T 2M +T M +p +T.p +0◦p◦p +ℓ +6Recall that we are using the 2-categorical notation described in the front-matter +13 + +Therefore the diagram in the following corollary is a pullback: +Corollary 1.2.15. Write the map µ : T M p ×p T M → T 2M to be T. + ◦(ℓ× 0). The following diagram is a +pullback: +T M p ×p T M +T 2M +M +T M +0 +T.p +µ +⌟ +The five maps (p,0,+,c ,ℓ), along with their coherences and universal properties, characterize the +kinematic tangent bundle, axiomatized as a tangent structure in the next section. However, there is an +equivalent characterization of the tangent bundle for a finite-dimensional smooth manifold that will +be important throughout this thesis: the operational tangent bundle. We first need to define the mod- +ule of vector fields on a manifold, where a vector field is essentially an ordinary differential equation +defined on a manifold rather than a cartesian space. +Definition 1.2.16 (3.1,3.3of Kolár et al. (1993)). A vector field on a manifold M is a section X : M → T M +of the projection p : T M → M so that p ◦ X = idM . The set of vector fields on a manifold M is written +χ(M ) and carries a C ∞(M )-module structure using the fibered �-module structure on p : T M → M : +X +χ(M ) Y := +.M ◦ (X ,Y ), +0χ(M ) := 0.M , +f ·χ(M ) X (m) := f (m) ·T M X (m) +where X ,Y ∈ χ(M ), f ∈ C ∞(M ). +The module χ(M ) has an important universal property as a C ∞(M )-module—it is precisely the +module of derivations of C ∞(M ): +χ(M ) = {X : C ∞(M ) → C ∞(M ) : ∀f ,g ∈ C ∞(M ),X (f ·g ) = X (f ) ·g + f · X (g )} +Proposition 1.2.17 (3.4 of Kolár et al. (1993)). There is an isomorphism of C ∞(M ) modules: +Der(C ∞(M )) ∼= χ(M ). +Finally, we observe that there is a C ∞(M )-Lie algebra structure on χ(M ), with two equivalent def- +initions. First, there is the kinematic definition of the bracket, which is induced using the universality +of the vertical lift. Given vector fields X ,Y on M , note that +X = p.T ◦ (T.Y ◦ X −T M c ◦ T.X ◦ Y ), +0 = T.p ◦ (T.Y ◦ X −T M c ◦ T.X ◦ Y ) +so by Corollary 1.2.15, there is a unique map [X ,Y ] : M → T M so that +(T.Y ◦ X −T M c ◦ T.X ◦ Y ) = µ([X ,Y ],X ). +(1.1) +Similarly, there is a bracket defined on Der(M ) using the anticommutator of derivations: +[X ,Y ](f ) = X (Y (f )) − Y (X (f )). +(1.2) +These brackets are equivalent for finite-dimensional smooth manifolds. +Proposition1.2.18. [Mackenzie(2013)]Recall that a Lie algebra overa ring R isan R-module A equipped +with a bilinear map +[−.−] : A ⊗ A → A +that is alternating and satisfies the Jacobi identity: +[X ,[Y ,Z ]]+ [Z ,[X ,Y ]]+ [Y ,[Z ,X ]] = 0. +The two brackets on χ(M ) (viewed as a Lie algebra) from Equations 1.1 and 1.2 coincide. +14 + +1.3 +Tangent structures +This section develops the categorical framework to study more general categories of smooth manifolds +by axiomatizing the tangent bundle, tangent categories, which first appeared in Rosický (1984). An ar- +bitrary tangent category is significantly more general than smooth manifolds and captures examplesof +categorieswith “tangentbundle” from computerscience and logic, asdeveloped in Cockett and Cruttwell +(2014). First, observe that tangent categories forget the base ring from the previous section and only +consider the fibered commutative monoid structure of the tangent bundle. +Definition 1.3.1. An additive bundle in a category � is a triple E +q−→ M ,+ : E q×qE → E ,ξ : M → E which +gives (q,+,ξ) the structure of a commutative monoid in the slice category �/M . If (q,ξ,+),(q ′,ξ′,+′) are +both additive bundles, a bundle morphism +E +E ′ +M +M ′ +q +f +q +f0 +is additive if f ◦+ = +′ ◦(f ◦π0, f ◦π1) and f ◦ξ = ξ′ ◦ f0. The category of additive bundles in a a category +� is given by additive bundles in � and additive bundle morphisms. +We will often write pullback powers of an additive bundle E q ×q ...q ×q E as En, and use infix no- +tation to write addition so + ◦ (a,b ) becomes a + b . In the category of smooth manifolds, the tangent +bundle functor gives a well-behaved functorial vector bundle. A tangent category has a functorial ad- +ditive bundle and axiomatizes the coherences and universal properties of the tangent bundle from the +category of smooth manifolds. +Definition 1.3.2 (Rosický(1984); Cockett and Cruttwell (2014)). A tangent structure consists of a functor +T : � → � equipped with natural transformations +p : T ⇒ id , 0 : id ⇒ T,+ : T p ×p T ⇒ T +ℓ : T ⇒ T.T +c : T.T ⇒ T.T +satisfying the following axioms; we call a category equipped with a tangent structure a tangent category. +[TC.1] Additive bundle axioms: +(i) Pullback powers of p exist and are preserved by T ; write these Tn. +(ii) Each triple (p.M : T M → M ,0.M : M → T M ,+.M : T2M → T M ) is an additive bundle. +T +T +T2 +T +id +id +id ++ +p◦πi +p +p +0 +(1.3) +[TC.2] Symmetry axioms: +(i) Involution: +T.T +T.T +T.T +c +c +(1.4) +15 + +(ii) Yang–Baxter: c .T ◦ T.c ◦ c .T = T.c ◦ c .T ◦ T.c +T.T.T +T.T.T +T.T.T +T.T.T +T.T.T +T.T.T +T.c +c .T +T.c +c .T +c .T +T.c +(1.5) +(iii) Naturality equations: +T.T +T.T +T +T +T.T.T +T.T.T +T.T.T +T.T +T.T +T.T2 +T2T +T.T +T.T +T.T +T.T +T +T +T.ℓ +c +c .T +T.c +ℓ.T +(c ◦T.π0,c ◦T.π1) +T.+ ++.T +c +c +p.T +T.p +0.T +c +T.0 +(1.6) +[TC.3] Lift axioms: +(i) Naturality with addition: +T +T.T +T.T +T.T +T2 +T.T2 +id +T +T +T +T +T.T +(ℓ◦π0,ℓ◦π1) ++ ++ +ℓ +T.ℓ +0 +0 +0.T +ℓ +p +0 +T.p +(1.7) +(ii) Coassociativity: +T +T.T +T.T +T.T.T +ℓ +ℓ +T.ℓ +ℓ.T +(1.8) +(iii) Symmetric co-multiplication: +T +T.T +T.T +ℓ +c +ℓ +(1.9) +(iv) Universality: for µ := T. + ◦(0◦ π0,ℓ◦ π1), the following diagram is a pullback for all X : +T2X +T 2X +X +T X +µ +p◦πi +T.p +0 +16 + +Definition1.3.3(Cockett and Cruttwell (2014)). Atangent category ismonoidal whenever� isa monoidal +category, T is a monoidal functor, and +,p,c ,ℓ are monoidal natural transformations. A tangent cate- +gory is cartesian whenever � is cartesian and is a strict monoidal tangent category for products. +Notation 1.3.4. Throughout this thesis, two key pieces of notation apply: +• Tn denotes pullback powers of p : T ⇒ id (and more generally En for q : E → M ); iterated powers +of T are written T n. +• There will often be long strings of Tn pullbacks and functors F : A → B, so a 2-categorical notation +where functor composition is written with a period, Tn.F.T ′ +m, will often be adopted. While the +natural transformation c at Tn.Tm.M under the image of the functor T would often be written +T (cTn.Tm.M ), in this thesis it will be written as +T.c .Tn.Tm.M : T.T 2.Tn.Tm ⇒ T.T 2.Tn.Tm +while the composition of 2-cells will be written using ◦ in applicative order, rather than the dia- +grammatic order typically used in the tangent category literature. That is, the composition +A +g−→ B +f−→ C +would be written f ◦ g rather than g f . +Example 1.3.5. Applying the results from section 1.2, the category of smooth manifolds is a tangent +category. Recall that if M is a smooth manifold, there is a coordinate patch m ∈ U �→ M around each +point m ∈ M so that U ∼=U ′ ⊆ �n. fibre above U , p −1(U ), is locally isomorphic to U ′ ×�n and similarly +(p ◦ p)−1(U ) ∼=U ′ × (�n)3, so that: +p :U × �n → U +(m, x) �→ m +0 :U →U × �n +m �→ (m,0) ++ :U × �n × R n →U × �n +(m, x, y ) �→ (m, x + y ) +ℓ :U × �n →U × �n × �n × �n +(m, x) �→ (m,0,0, x) +c :U × �n × �n × �n →U × �n × �n × �n +(m, x, y,z) �→ (m, y, x,z) +The study of tangent categories is closely related to Lawvere’ssynthetic differential geometry, first in- +troduced in Lawvere (1979) and laterdeveloped in Kock (2006); Lavendhomme(1996); Moerdijk and Reyes +(1991). The setting of synthetic differential geometry is a topos � (for our purposes, we need only a com- +plete, cartesian closed category) equipped with a chosen ring object R that satisfies the Kock-Lawvere +axiom: given the object of nilpotent elements in R, D = [d : R|d 2 = 0], the following map is an isomor- +phism: +α : R × R → [D,R]; α(a,b ) = (d �→ a + d b ). +One can find a class of objects that form a tangent category: the infinitesimally linear objects. +Definition 1.3.6. Let � be a model of synthetic differential geometry where the ring object is (R,·,1,+,0). +An object M in a model of synthetic differential geometry is infinitesimally linear when it satisfies the +following axioms. +(i) The following natural morphism must be an isomorphism: +[D (2),M ] ∼= [D,M ]0∗×0∗[D,M ]; where D(2) := [(d0,d1) ∈ D 2 : di ·d j = 0]. +17 + +(ii) M satisfies property W (credited by Kock (2006) to Gavin Wraith), namely that the following dia- +gram is a ternary equalizer: +[D,M ] +[D × D,M ] +[D,M ] +[0◦!×D,M ] +[(0×0)◦!,M ] +[D×0◦!,M ] +[·,M ] +(We will often use M as shorthand for idM in diagrams). +An infinitesimal object generalizes the object D in the category of infinitesimally linear objects in +a model of synthetic differential geometry. The definition adopted in this thesis is a strict generaliza- +tion of the definition given in Cockett and Cruttwell (2014): here, we work with a symmetric monoidal +closed category rather than a cartesian closed category in order to capture some examples from logic. +Definition 1.3.7 (Definition 5.6 Cockett and Cruttwell (2014)). An infinitesimal object in a symmetric +monoidal category +(�,⊗,I ,α,ρ,σ) +is a tuple (⊙ : D ⊗ D → D,0 : I → D,ε : D → I ,δ : D → D(2)) (where D(n) denotes pushout powers of 0) +so that: +[IO.1] Pushout powers D(n) of 0 : I → D exist, and ε ◦ 0 = idI . +[IO.2] ⊙ is a commutative semigroup with zero, so that the following diagrams commute: +D ⊗ D +D ⊗ D +D ⊗ D ⊗ D +D ⊗ D +D +D ⊗ D +D +D +D +D +⊙ +σ +⊙ +⊙⊗D +⊙ +D⊗⊙ +⊙ +(D,0) +⊙ +0D ◦ε +The third diagram shows that 0 is an absorbing element rather than a unit (think of 0 as being in +the commutative semigroup on the set [0,1) given by multiplication). +[IO.3] The map δ : D → D(2) makes (0 : I → D,δ,ε) into a commutative comonoid in the coslice I /�: +D +D(2) +D +D(2) +D +D (2) +D +D (2) +D(3) +D(2) +δ +δ +D+I δ +δ+I D +δ +δ +(ι1|ι0) +(0◦ε+I D) +δ +[IO.4] The following diagram commutes (⊙ is coadditive): +D ⊗ D +D +D ⊗ D(2) +D(2) +1×δ +(ι0◦⊙|ι1◦⊙) +δ +⊙ +The notation (f |g ) : A + B → C for pushouts/coproducts is dual to pairing (f ′,g ′) : C → A × B for +pullbacks/products, just as ιi is dual to projection. Therefore, (f |g ) ◦ ι0 = f just as π0 ◦ (f ,g ) = f . +18 + +[IO.5] The following diagram is a coequalizer: +D ⊗ I +D ⊗ D +D(2) +0◦ε⊗0 +D⊗0 +(⊙|I ε⊗D )◦(δ⊗id) +There are two tangent structures associated to an infinitesimal object in a symmetric monoidal +category �. The first relies on the exponentiability of the infinitesimal object, while the second tangent +structure is on the opposite category of �. The enriched perspective on tangent structure in Section +5.1 will clarify the relationship between these two tangent structures. +Proposition 1.3.8. Let (�,⊗,I ,α,ρ,σ) be a symmetric monoidal category, and +⊙ : D ⊗ D → D,0 : I → D,ε : D → I ,δ : D → D(2) +define an infinitesimal object in �. +(i) There is a tangent structure on �op , where T = D ⊗ (−). +(ii) If � is also a symmetric monoidal closed category, then it is a tangent category with T = [D,−]. +Proof. +(i) The opposite category of � is a symmetric monoidal category: +(�op,⊗,I ,α−1,ρ−1,σ). +The tangent functor is D ⊗ (−), and the projection is +p = D ⊗ (−) +0op +−→ I ⊗ (−) +ρ−1 +−→ (−). +The zero map is given by +(−) +ρ−→ I ⊗ (−) +εop +−→ D ⊗ (−). +Addition in �op is given by co-addition in �: +D(2) ⊗ (−) +δop ⊗(−) +−−−−→ D ⊗ (−). +The lift is given by the semigroup structure (along with the monoidal coherences): +D ⊗ (−) +⊙⊗(−) +−−−→ (D ⊗ D) ⊗ (−) +α−1 +−→ D ⊗ (D ⊗ (−)). +The flip is also given by monoidal coherences, combined with the symmetry on the monoidal +category: +D ⊗ (D ⊗ (−)) +α−→ (D ⊗ D) ◦ (−) +σ⊗(−) +−−−→ (D ⊗ D ) ⊗ (−) +α−1 +−→ D ⊗ (D ⊗ (−)). +(ii) First, we identify the following natural isomorphisms: +u : id ⇒ [I ,−], +b : [A,[B,−]] ⇒ [A ⊗ B,−]. +19 + +• The tangent functor is [D,−], +and the triple (0 +: +I +→ +D,δ +: +D +→ +D(2), +ε : D → I ) gives the additive bundle structure +p : [D,−] +0∗ +−→ [I ,−] +u−→ id , +0 : id +u−→ [I ,−] +[ε,−] +−−→ [D,−] ++ : [D (2),−] +δ∗ +−→ [D,−]. +Note that by the continuity of [−,M ] : �op → �, we have [D(2),M ] ∼= [D,M ]p ×p [D,M ]. +• The lift is given by +ℓ : [D,−] +⊙∗ +−→ [D ⊗ D,−] +b −1 +−→ [D,[D,−]]. +• The canonical flip is given by +c : [D,[D,−]] +b−→ [D ⊗ D,−] +σ∗ +−→ [D ⊗ D,−] +b −1 +−→ [D,[D,−]]. +The coherences and couniversality properties of an infinitesimal object, along with the continuity +of +[−,M ] : �op → � +then induce the coherences and universality properties for the tangent bundle. This completes +the proof. +The tangent structure on �op induced by the infinitesimal object is the dual tangent structure on +�. This will be revisited in Chapters 4 and 5 when looking at tangent categories from the enriched +perspective. +Returning to synthetic differential geometry, the co-universality conditions on an infinitesimal ob- +ject corresponds to infinitesimal linearity (Definition 1.3.6). In some sense, the category of infinitesi- +mally linear objects is the largest subcategory of � for which D is an infinitesimal object. +Corollary 1.3.9. In a model of synthetic differential geometry (�,R), the object D = [d ∈ R|d 2 = 0] is an +infinitesimal object in the category of infinitesimally linear objects in �. +This thesis makes use of the 2-category of tangent categories (Section 2.3 of Cockett and Cruttwell +(2014)), which formalizes the notion of a morphism of tangent structure and 2-cells between them. +This 2-categorical framework is a departure from the classical theory of synthetic differential geometry, +where the literature only really addresses morphisms of tangent structure in the form of fully faithful +embeddings SMan �→ Microl(�). +Definition 1.3.10. Let (�,�),(�,�′) be a pair of tangent categories. A pair (F : � → �,α : F.T ⇒ T ′.F ) is +a tangent functor if the following diagrams commute: +F +F.T +F.T2 +F.T +F.T +F +T.F +T2.F +T.F +T.F +F.T +T.F +F.T 2 +T.F.T +T 2.F +F.T 2 +T.F.T +T 2.F +F.T 2 +T.F.T +T 2.F +F.+ +α2 ++.F +α +α +F.p +p.F +0.F +F.0 +α +ℓ.F +F.ℓ +α +α.T +T.α +α.T +T.α +α.T +T.α +F.c +c .F +20 + +A tangent functor is strong whenever α is a natural isomorphism; for a sub-(tangent category) inclusion, +α = id . A tangent functor (F,α) between cartesian tangent categories is a cartesian tangent functor if F +is an isomonoidal functor and α is a monoidal natural transformation. +Example 1.3.11. +(i) The coherences on the canonical flip c guarantee that (T : � → �,c : T.T ⇒ T.T ) is a strong tangent +endofunctor on any tangent category. +(ii) Given a pair of tangent functors (A,α) : � → �,(B,β) : � → �, the composition +(B.A : � → �,β.A ◦ B.α : B.A.T C +B.T D .A +T E .B.A +B.α +β.A +) +is a tangent functor. +(iii) A model of synthetic differential geometry (�,R) is well-adapted whenever there is a fully faithful, +strict tangent functor fromSMan to the category of microlinear spaces of �,SMan �→ Microl(�). The +original development of well-adapted models for synthetic differential geometry may be found in +Dubuc (1981), and the reader may check Bunge et al. (2018) for a recent account of the construction +of such models, or section 3 of Kock (2006). +Definition1.3.12. Atangentnatural transformation γ between tangent functors(F,α),(G ,β) isa natural +transformation so that the following diagram commutes: +F.T (A) +G .T (A) +T.F (A) +T.G (A) +γT A +αA +βT A +T γA +If F,G are cartesian tangent functors, then γ is cartesian whenever it is an isomonoidal natural transfor- +mation. +Definition 1.3.13. We will call the 2-category of tangent categories, tangent functors, and tangent nat- +ural transformations TangCat. The 2-category of cartesian tangent categories is CartTangCat. +1.4 +Local coordinates in a tangent category +This section develops some structures to facilitate reasoning about higher tangent bundles, using “lo- +cal coordinates.” In the case of a cartesian differential category, T n(A) = +� +2n A, whereas for an open +subset U ⊆ �m the tangent bundle decomposes as T nU ∼= U × (�m)2n−1. This section introduces two +structures that allow for these arguments in an arbitrary tangent category: differential objects and con- +nections. +Definition 1.4.1 (Cockett and Cruttwell (2018)). A differential object7 in a cartesian tangent category +is a commutative monoid (A,+A,0A) such that there is a section-retract pair A +λ−→ T A +ˆp−→ A which exhibits +T (A) as a biproduct in the category of commutative monoids: +A ⊕ A ∼= T A. +Concretely, a differential object is a commutative monoid equipped with λ : A → T A, ˆp : T A → A, ˆp ◦λ = +id so that the following axioms hold: +7Not to be confused with 4.1 from Barr (2002). +21 + +DO.1 Coherence between + and λ, ˆp: +T (A × A) +T A +A × A +A +T A × T A +T A × T A +A × A +A +T (A × A) +T A +ˆp +T.+A +(T π0,T π1) +ˆp× ˆp ++A ++A +λ×λ +(T π0,T π1)−1 +T.+A +λ +DO.2 Coherence between 0A and 0.A: +A +1 +A +1 +T A +A +T A +A +0 +! +0A +ˆp +λ +p +! +0A +DO.3 Coherence between +A and +.A: +T2A +A × A +A × A +T2A +T A +A +A +T A +ˆp× ˆp ++A ++ +ˆp +λ×λ ++A +λ ++ +DO.4 Coherence between λ and ˆp with ℓ: +T A +T 2A +A +T A +A +T A +T A +T 2A +ℓ +T. ˆp +ˆp +ˆp +λ +λ +ℓ +T.λ +A map f : A → B between differential objects (A,λA, ˆpA,+A,0B) → (B,λB, ˆpB,+B,0B) is linear whenever +f preserves the lifts and projections +(T.f ◦ λA = λB ◦ f ) and ( ˆpB ◦ T.f = f ◦ ˆpA). +Following the work in Section 3 of Cockett and Cruttwell (2018), it is sufficient to check that f pre- +serves λ or ˆp (each condition implies the other). +Example 1.4.2. +(i) In the category of smooth manifolds, the real numbers are a differential object, as T.� = �[x]/x 2, +so the lift map in this case is +λ(a) = 0+ a · x, +ˆp(a + b · x) = b. +More generally, finite-dimensional real vector spaces, with their canonical smooth manifold struc- +ture, are exactly differential objects in the category of smooth manifolds. The lift map is defined +using T � ∼= �[x]/x 2, so that +V +(0,1�) +−−−→ T (V × �) +T ·V +−→ T V. +This is equivalent to the isomorphism T.V = R[x]/x 2 ⊗ V . +22 + +(ii) Every object in a cartesian differential category has a canonical differential object structure, as +T A := A × A. +Classically, the tangent space above each point of a smooth manifold is a vector space. We have a +similar result for differential objects from Cockett and Cruttwell (2018). +Lemma 1.4.3. Suppose we have the following pullback in a cartesian tangent category �, and all powers +of T preserve it. +E +T M +1 +M +ι +! +p +m +There is a unique differential object structure (E ,λ, ˆp) so that ℓ◦ ι = T.ι ◦ λ.8 +There are two natural classes of morphisms between differential objects, linear and “smooth” (that +is, arbitrary morphisms). +Definition 1.4.4 (Cockett and Cruttwell (2018)). Let � be a cartesian tangent category. We define the +following categories: +(i) Diff(�) is the category of differential objects and arbitrary morphisms, so for any differential objects +A,B, we have Diff(�)(A,B) := �(A,B). +(ii) DLin(�) is the category of differential objects and linear morphisms. +The category of differential objects and smooth maps is a cartesian differential category, exhibiting +differential calculus as a specialized logic in a tangent category. +Proposition 1.4.5 (Section 3.5 of Cockett and Cruttwell (2018)). Let � be a cartesian tangent category. +Then: +(i) Diff(�) is a cartesian differential category, where we define the differential combinator D to be +A +f−→ B +A × A +λ×0 +−−→ T (A × A) +T.+A +−−→ T (A) +T (f ) +−−→ T (B) +ˆpB +−→ B +(where λA,+A are the lift and addition for the differential object structure on A, and ˆpB is the pro- +jection map on the differential object structure on B). +(ii) There is an equality of categories, Lin(Diff(�)) = DLin(�), meaning that a morphism between differ- +ential objects in the cartesian tangent category � is linear if and only if it is linear in the cartesian +differential category Diff(�). +Recall that if M is an open subset of �n, the second tangent bundle of an open subsetU ⊆ �n splits +as +T 2(U ) = T (U × �n) = (U × �n) × (�n × �n) = T3U . +Every smooth manifold admits such a decomposition on its second tangent bundle; these are known +as a affine connections9 and provide a way to reason about an object as though it has local coordinates +in an arbitrary tangent category. +8Some diagrammatic notation had creeped into the original draft here, I have fixed it. +9The prefix “affine” differentiates these from more general connections on differential bundles, which are introduced in +Section 2.6. +23 + +Definition 1.4.6 (Cockett and Cruttwell (2017)). In a tangent category �, define the following: +(i) An affine vertical connection is a map κ : T 2M → T M so that +a) κ is a vertical descent, namely a section of the vertical lift ℓ : T M → T 2M , so κ ◦ ℓ = id ; +b) κ is compatible with both lifts on T 2M : T.κ ◦ ℓ.T = T.κ ◦ T.ℓ = ℓ◦ κ. +(ii) An affine horizontal connection is a map ∇ : T2 → T 2M so that +a) ∇ is a horizontal lift, namely a section to the horizontal descent (p.T,T.p) : T 2 ⇒ T2, so that +(p.T,T.p) ◦ ∇ = id ; +b) ∇ +is +compatible +with +the +linear +structures +on +T 2,T2:T.∇ +◦ +(ℓ +× +0) += ℓ◦ ∇and T.∇(0× ℓ) = T.ℓ◦ ∇. +(iii) An affine connection is a pair (κ,∇) comprising a vertical and horizontal connection on M satisfy- +ing the compatibility conditions: +a) T. + ◦(+.T ◦ (ℓ◦ κ,T.0◦ p.T ),∇(p.T,T.p)) = idT 2M , +b) κ ◦ ∇ = 0◦ p ◦ πi. +An affine connection is torsion-free if κ ◦ c = κ. +The data ofa vertical connection issufficienttodefine a full connection, asobserved in Lucyshyn-Wright +(2018). +Lemma1.4.7(Lucyshyn-Wright (2018)). Afull connection isequivalent to a vertical connection in which +the following diagram is a fiber product: +T M +T 2M +T M +M +T M +p +p +p +p.T +T.p +κ +Example 1.4.8. +(i) Every differential object in a tangent category has a canonical vertical connection given by (p ◦ +p.T, ˆp ◦ T. ˆp). A morphism of differential objects will preserve this vertical connection. +(ii) Every smooth manifold has a non-natural choice of Riemannian metric. By the fundamental the- +orem of Riemannian geometry, there is a torsion-free connection associated with the metric. (See +any standard reference on Riemannian geometry, e.g. do Carmo (1992).) +Connections allow for arguments on higher powers of the tangent bundle to be pushed down to +pullback powers of T . +Observation 1.4.9. Suppose M ,N each have connections (κ−,∇−). The map T 2.f : T 2M → T 2N can be +written using local coordinates � +T 2.f : T3M → T3N as +T 2M +T 2N +T3M +T3N +T.+◦(+.T ◦(ℓ◦π2,T.0◦π0),∇(π0,π1)) +T 2.f +(p.T,T.p,κN ) +24 + +where � +T 2.f is given by +(T.f ◦ π0,T.f ◦ π1,T.f ◦ π2 +N ∇[f ](π0,π1)) +with ∇[f ] := κN ◦ T 2.f ◦ ∇M . +Note that in the case that f preserves the connections, ∇[f ] = 0, as +κN ◦ T 2.f ◦ ∇M = T f ◦ κM ◦ ∇M = T f ◦ 0◦ p = 0◦ f ◦ p. +1.5 +Submersions +The category of smooth manifolds is incomplete: there are cospans10 X +f−→ M +g←− Y for which the +pullback fails to exist. Following Thom (1954), the pullback of a cospan exists and is preserved by T +(i.e. it is a T-limit) whenever for each point f (x) = g (y ), the direct sum of the images of Tx f and Ty g is +the full vector space Tf (x)M , such cospans are called transverse. Submersions, then, form a convenient +class of maps, as any cospan where one map is a submersion will be transverse. More precisely: +Definition +1.5.1. +If +A +and +B +are +smooth +manifolds, +a +smooth +function +f +: +A → B is a submersion if and only if the derivative D f |a of f at every point a ∈ A is a surjective lin- +ear map. +With this definition we have the following result: +Proposition 1.5.2. In the category of smooth manifolds, let the class of submersions be denoted by � . +(i) Submersions are closed under the tangent functor: f ∈ � ⇒ T.f ∈ � . +(ii) Submersions are closed to pullback along arbitrary maps: +X +g ∗X +X +N +M +N +M +g +f ∈� +g ∗ f ∈� +g +f +¯g +⌟ +This will often be referred to as T -stability under reindexing, as it induces a functor between slice +categories: +g ∗ : Submersions/M → Submersions/N . +The properties of the class of submersions in the category of smooth manifolds were studied in +Cockett and Cruttwell (2018) and axiomatized as a tangent display system. +Definition 1.5.3. A tangent display system in a tangent category � is a class of maps � in � that is +• stable under the tangent functor, d ∈ � ⇒ T.d ∈ �, +• T -stable under reindexing (as in Proposition 1.5.2). +We call any tangent display system that is closed to retracts in the arrow category a retractive display +system. If for all M , pM ∈ �, we call � a proper (retractive) display system. +10Following a general conventionincategory theory,where the prefix "co"-X means anX inthe opposite category,a cospan +in � is a span in the dual category of �. +25 + +This section will show that the submersions in the category of smooth manifolds give a retractive +display system, yielding a general construction of retractive display systems from display systems. +The definition of a submersion may be rephrased as follows: f is a submersion if and only if for all +a ∈ A and all v ∈ T (B) such that f a = pv, there exists a w ∈ T (A) such that T.f ◦w = v. This is a weakly +universal cone over A +f−→ B +p←− T B: there exists at least one morphism into it for any other cone over +the diagram. +Definition 1.5.4. A commuting square is a weak pullback if for any x : X → A and y : X → B so that +f x = g y , there exists a map X → W making the following diagram commute: +X +W +A +B +C +x +y +∃ +a +b +f +g +If the above diagram is a weak pullback for each T n, then it is a weak T -pullback. +Lemma 1.5.5. Should the pullback of A +f−→ C +g←− B exist, Definition 1.5.4 is equivalent to asking that the +induced map (a,b ) : W → A f ×g B be a split epimorphism. +Proof. Let r be a retract of (a,b ) : W → A f ×g B. For any X +(x,y ) +−−→ A f ×g B, the map r ◦ (x, y ) exhibits the +diagram as a weak pullback. For the converse, the unique map (a,b ) : W → A f ×g B will be a section of +any map A f ×g B → W induced by weak univerality. +We now restate the submersion property for a map f using global elements (for all a ∈ A and all +v ∈ T (B) such that f a = pv, there exists a w ∈ T (A) such that T (f )w = v) using generalized elements. +Definition 1.5.6. An arrow f : A → B in a tangent category is a tangent submersion if and only if the +naturality diagram +T A +T B +A +B +T f +p +p +f +is a weak T -pullback. +Following Lemma 1.5.5, in the case that the pullback exists this is equivalent to asking for a section +h : A f ×p T B → T A of the horizontal descent (p,T f ) : T A → A f ×p T B (this section is sometimes called +a horizontal lift in differential geometry literature Cordero et al. (1989)). In smooth manifolds, the T - +pullback along the projection p : T ⇒ id always exists, so to prove that every submersion is a tangent +submersion it suffices to show the existence of a horizontal lift. +Proposition 1.5.7. In the category of smooth manifolds, the tangent submersions are precisely the sub- +mersions (Definition 1.5.1) +Proof. There is an explicit construction of a horizontal lift for a classical smooth submersion in VII.1 +of Cushman and Bates (2015). +26 + +It is possible to show that the T -stability properties for submersions in the category of smooth +manifolds follow from the general theory of weak pullbacks. We begin by showing that weak pullbacks +satisfy a weakened version of the pullback lemma and then show that the retract of a weak pullback is +a weak pullback (the second lemma is Lemma 2.1 of Adámek et al. (2010)). +Lemma 1.5.8 (Pullback lemma). Consider the diagram +• +• +• +• +• +• +(A) +f +g (B) +(i) If f ,g are jointly monic and (A) + (B) is a weak pullback, then (A) is a weak pullback. +(ii) If (A),(B) are weak pullbacks then (A) + (B) is a weak pullback. +Proof. +(i) If (A)+(B) is a weak pullback, a map can be induced for a cone over (A) by concatenating it with (B); +the jointly monic condition on f ,g guarantees that the map induced for (A) + (B) will commute +for (A). +(ii) Given a cone for (A) + (B) induce a map for (B), which then induces a cone for (A). +Lemma 1.5.9. (Weak) pullbacks are closed to retracts. +Proof. Suppose that S ′ is a weak pullback, and S is a retract of it in the category of commuting squares. +Consider the following diagram (suppressing the subscripts for s,r ): +Z +A +A′ +A +B +B ′ +B +C +C ′ +C +D +D ′ +D +x +y +s +x ′ +y ′ +r +x +y +w +s +w ′ +r +w +z +s +z ′ +r +z +s +r +Given a cone for S, there is a corresponding cone for S ′ which induces a map Z → A′ and postcompo- +sition with rA gives the desired map into A. +Using these lemmas, it is straightforward to prove that the following T -stability properties hold for +tangent submersions. +Lemma 1.5.10. In any tangent category �, +(a) tangent submersions are closed to composition; +(b) tangent submersions are closed to retracts; +(c) any T -pullback of a tangent submersion is a tangent submersion. +27 + +Proof. (a) follows from Lemma 1.5.8 while (b) follows from Lemma 1.5.9. It remains to prove (c). Con- +sider a T -pullback, where u is a tangent submersion: +A +M +B +N +f +v +u +g +By naturality, the outer paths of the following two diagrams are equal: +T A +T M +M +T B +T N +N +T f +T v +p +T u +u +T g +p += +T A +A +M +T B +B +N +p +T v +f +v +u +p +g +Note that the left diagram is a weak pullback by composition. Therefore the outer perimeter of the right +diagram is a weak pullback, and the right square is a pullback, so the left square is a weak pullback by +Lemma 1.5.8, as desired. +The class of tangent submersions is closed to retracts in the arrow category and is conditionally T - +stable under reindexing (if the T -pullback of a tangent submersion exists, it is a tangent submersion). +This stability property leads to the following result: +Proposition 1.5.11. Let � be a tangent category that allows for reindexing of the class of tangent sub- +mersions �. Then the class of tangent submersions is a display system. +Proof. Any class of maps that is closed to reindexing is a tangent display system, and the class of sub- +mersions is closed to retracts in the arrow category. +Corollary 1.5.12. The class of submersions in the category of smooth manifolds is a proper retractive +display system. +28 + +Chapter 2 +Differential bundles +Our principal aim in this thesis is to provide an abstract tangent-categorical axiomatization for Lie +algebroids. To accomplish this, we must provide an axiomatization for Lie algebroids which is essen- +tially algebraic (in the sense of Freyd and Kelly (1972)). However, in the category of smooth manifolds, +Lie algebroids are defined in terms of vector bundles and these are prima facie a highly non-algebraic +notion. +In addition to algebraic axioms which make it an �-module in the slice over its base M , a vector +bundle q : E → M satisfies a crucial topological requirement: it must be locally trivial. This means +that the projection q : E → M must be locally isomorphic to a projection π0 : U × �n → U for some +open subset U of M and natural number n. It is this property that permits calculations using local +coordinates, an approach deeply enshrined in the culture of differential geometry. +Cockett and Cruttwell (2017) introduced the algebraicnotion ofa differential bundle. Evidence that +differential bundles are the appropriate generalization of vector bundles was provided by showing how +classical results for vector bundles could be generalized to differential bundles in any tangent category +Cockett and Cruttwell (2017, 2018). However, the precise relationship in the category of smooth man- +ifolds between vector bundles and differential bundles was left open. The main result of this chap- +ter (see MacAdam (2021)) is that vector bundles and differential bundles coincide in the category of +smooth manifolds. +The axiomatization of differential bundles focuses on another important property of vector bun- +dles: given a vector bundle q : E → M and a vector v in the fibre Ex above x ∈ M , the tangent space +Tv(Ex) can be naturally identified with Ex . This gives a lift map λ : E → T E which can be axiomatized. +While the lift map had long been noted in the differential geometry literature in the guise of the Euler +vector field (see 6.11 of Kolár et al. (1993) and also Section 1 of Michor (1996) which explicitly uses the +term "lift"), it had not been adopted as the basis of an abstract axiomatization. +More recently, in the differential geometryliterature, Grabowski and Rotkiewicz (2009) and Bursztyn et al. +(2016) realized that the multiplicative �+-action on the total space E determines the vector bundle +structure of q : E → M , and conversely such a multiplicative action determines a vector bundle pre- +cisely when its Euler vector field (Definition 2.1.8) satisfies an additional “non-singular” property. This +chapter extends these more recent observations on vector bundles to differential bundles. +The chapter begins by reviewing vector bundles, describing the Euler vector field construction that +sends a vector bundle q : E → M to a "lift" map λ : E → T E from Grabowski and Rotkiewicz (2009), +whereas the rest of the chapter contains new results developed in collaboration with Matthew Burke. +The second section establishes that these lifts are associative coalgebras for the weak comonad (T,ℓ), +and that there is a fully faithful functor from vector bundles into the category of lifts for smooth mani- +folds. The third section identifies the universal property satisfied by the lift (or equivalently Euler vector +29 + +field), while the fourth section shows that a non-singular lift corresponds precisely to a differential bun- +dle. The fifth section proves the main theorem of the chapter: vector bundles are precisely differential +bundles for smooth manifolds. The final section contains some remarks on extending affine connec- +tions to arbitrary differential bundles, which will be useful in Chapter 3. +2.1 +Vector bundles +A vector bundle over a manifold M axiomatizes the notion of a smoothly varying family of vector spaces +indexed by the points m ∈ M . The driving example is that of the tangent bundle over a smooth man- +ifold M , where the fibre above each point m ∈ M is the tangent space TmM . The manifold structure +guarantees that the projection is locally trivial: given a chart U �→ M , the next pullback splits as a +product: +U × �n +T M +U +M +p +⌟ +The local triviality of the tangent bundle is essential for various constructions and is part of the defini- +tion of a vector bundle. +Definition 2.1.1. A vector bundle is a tuple +(q : E → M ,ξ : M → E ,+ : E q ×q E → E ,· : � × E → E ) +of morphisms in SMan so that +(i) the tuple (q,ξ,+,·) defines an �-module in SMan/M ; +(ii) the map q : E → M is locally trivial. +The fibred �-module structure means E is a family of vector spaces indexed by M , {Em|m ∈ M }. +It is important to note that the local triviality axiom guarantees that the projection of a vector bun- +dle is a submersion (Definition 1.5.1); thus pullback powers of q : E → M exist and are preserved by +the tangent functor. +Example 2.1.2. Consider the cylinder, defined as the subset of �3 spanned by C = {(x, y,z)|x 2 + y 2 = +1,z ∈ �}: +� +Above each point i ∈ S 1 = {(x, y )|x 2 + y 2 = 1} the fibre over i is �. For each point i, we can choose a +sufficiently small ε and take the open set +Ui = {(x, y ) ∈ S 1|(ix − x)2 + (iy − y )2 ≤ ε}, +which may be flattened to (−(1+ ε),1+ ε) × �. +30 + +The sections of a vector bundle also give rise to a C ∞(M )-module, generalizing that aspect of the +tangent bundle’s fibred �-module structure. +Lemma 2.1.3. Given a vector bundle q : E → M , write the set of sections of q as Γ(q); as, for example, +Γ(p.M ) = χ(M ) (recall the notation from Definition 1.2.16). The set Γ(q) has a C ∞(M )-module structure +in much the same way as χ(M ): +X +Γ(q) Y := + ◦ (X ,Y ), 0Γ(q) := ξ, (f ·Γ(q) X )(m) := f (m) · X (m) +There are also a variety of general constructions that yield vector bundles. +Example 2.1.4. +(i) The tangent bundle is a vector bundle: the construction in Section 1.2 makes it clear that the pro- +jection p : T M → M is a locally trivial, fibred �-module over the base space M . +(ii) A trivial vector bundle over M with fibres in V is the product M × V . In particular, every vector +space is a trivial vector bundle above the one-point space {∗}. +(iii) Each TkM will be locally trivial; locally it lookslike the k-fold product of the tangent space p −1 +k (U ) ∼= +U × (�n)k for an n-dimensional manifold M . More generally, one can take the fibrewise pullback +Ek = E q ×q E q ×q ...q ×q E and discover a vector bundle over M . +(iv) The cotangent bundle of M , T ∗M , has the dual vector space of TmM above each point M : T ∗ +m(M ) = +(TmM )∗. This space can be appropriately topologized to be smooth, and a set of sections of Γ(T ∗M ) +is isomorphic to the set of morphisms T M → � that are linear in each fibre. This construction may +be applied to any vector bundle and is called the dual vector bundle. +(v) Consider the space Λn(E ), the alternating tensor product of E ∗. The set of sections of this vector +bundle is equivalent to the alternating n-linear morphism En → �; when restricted to the tangent +bundle, this is the space of differential n-forms. +There are two constructions on vector bundles that will be necessary to prove the main theorem of +this section. +Proposition 2.1.5. Let (q : E → M ,ξ,+q,·q) be a vector bundle. +(i) For any map f : N → M , the T -reindexing of q by f is a vector bundle: +f ∗E +E +N +M +q +f +f ∗q +¯f +⌟ +(2.1) +(ii) Any retract of q in the space of arrows is a vector bundle; that is, given +F +E +F +N +M +N +q +r ′ +r +π +s +s ′ +π +(2.2) +if there is a vector bundle structure on q, then there is a vector bundle structure on π. +31 + +The category of vector bundles has “locally linear” bundle morphisms as its maps. +Definition 2.1.6. A morphism of vector bundles between q : E → M and π : F → N is a commuting +square +E +F +M +N +q +f +v +π +that is fibrewise linear, so that above each fibre +f |m : Em → Fv(m) +isa linearmorphism of vectorspaces. Thismay equivalently be stated asa morphism of fibred �-modules, +so that the following diagrams commute: +E +F +E +F +M +N +M +N +E2 +F2 +� × E +� × F +E +F +E +F +q +f +v +π +ξ +ζ +f +v ++ ++ +f +f2 +·E +�×f +·F +f +Example 2.1.7. +(i) For the pullback vector bundle in Diagram 2.1, the pair ( ¯f , f ) is a linear bundle morphism. +(ii) For the section/retract vector bundle structure from Diagram 2.2, the section and retract are linear +morphisms. Note that this is exactly the splitting of a linear idempotent on q : E → M . +The lift on the tangent bundle was defined in Section 1.2 as +[γ]∼ �→ [γ ◦ ·�]∼. +Instead, consider the action of � on a tangent vector: +([γ]∼,r ) �→ [γ ◦ (r · x)]∼. +Note that T.· gives the equation +T. ·◦([ω ◦ (x, y )],[(a,b ) �→ a + b · x]) = (ω ◦ (a · x,a · b · y )); +so the lift map ℓ can be rederived as follows: +T M +T M × � +T (T M × �) +T 2M +[γ] +([γ],1) +([γ ◦ π0],[r �→ 1• r ]) +[γ ◦ •�] +(id,1�) +0×λ +This general construction is known as the Euler vector field of a multiplicative action by �+. +32 + +Definition 2.1.8. Consider a multiplicative monoid action a : �+ × E → E . The Euler vector field1 of +the action is the morphism λ : E → T E constructed as follows: +λ := E +(id,1�◦!) +−−−−→ E × � +0×λ +−−→ T E × T � ∼= T (E × �) +T.a +−→ T E . +Local triviality for the tangent bundle is encoded by the universality of the vertical lift condition. A +similar universality condition holds for vector bundles. +Proposition 2.1.9. Let q : E → M be a vector bundle with corresponding Euler vector field λ. Then the +following diagram is a T -pullback: +E +T E +M +E × T M +(ξ,0) +q +λ +(p,T.q) +Exploiting the fact that fibred �-modules have subtraction, the following result holds. +Corollary 2.1.10. The following two diagrams are T-equalizers: +E2 +T E +T M +T M p ×q E +T E +E +µE :=0◦π0+T q λ◦π1 +νE :=T.ξ◦π0+p λ◦π1 +T.q +T.q◦0◦p +p +p◦T.ξ◦T.q +(recall that E2 is the pullback of a submersion along a submersion and is therefore guaranteed to exist +and be preserved by the tangent functor). +Proof. Given v : X → T E so that +T.q ◦ v = T.q ◦ 0◦ p ◦ v +then +p ◦ (v −T.q 0◦ p ◦ v) = p ◦ v −q p ◦ v = ξ ◦ q ◦ v. +So there is a unique v ′ so that +λ ◦ v ′ = (v −T.q 0◦ p ◦ v) +meaning that +v = 0◦ p ◦ v +T.q λ ◦ v ′ = µ(0◦ p ◦ v,v ′) +as required. The projection q is a submersion, so the pullback E2 is preserved by the tangent functor, +as is the pullback in Proposition 2.1.9, and the same calculation may be applied for each T n. The proof +for ν follows by the same argument. +Recall that the class of submersions forms a retractive display system in the category of smooth +manifolds (Definition 1.5.3), so they are stable under reindexing and closed to retracts. We may now +infer the following: +Corollary 2.1.11. The projection for a vector bundle is a submersion. +Proof. This follows from the fact that π : T E → E is a submersion, so that q ◦ π0 : E q ×p T M → M is a +submersion, so the map q : E → M is a retract of the projection p.E : T E → E in the arrow category. +1Somewhat confusingly, the Euler vector field is almost never a vector field. +33 + +Preservation of the Euler vector field is also sufficient to guarantee that a morphism f : E → F +determines a vector bundle morphism. +Proposition 2.1.12. Let q : E → M ,π : F → N be a pair of vector bundles with Euler vector fields λE ,λF . +Then a bundle morphism (f ,v) : q → π is a vector bundle morphism if and only if +λF ◦ f = T.f ◦ λE +Proof. Note that the νF map from Corollary 2.1.10 is monic, and if f preserves the lift, it preserves ν: +νF ◦ (T.v, f ) = + ◦ (T.ζ ◦ T.v,λF ◦ f ) = + ◦ (T.f ◦ T.xi,T.f ◦ λE ) = T.f ◦ νE . +Next, observe that T.v × f is the unique map making the following diagram commute: +M +E +T M p ×q E +T E +T N p ×q F +T F +N +F +∃! +ν +ν +T.f +p +p +f +q◦π1 +π◦π1 +ζ +ξ +⌟ +⌟ +Now ν is a vector bundle morphism and monic, and T.f is a vector bundle morphism, so it follows +that T.v × f is a vector bundle morphism and hence f is also a vector bundle morphism. The reverse +implication is immediate. +2.2 +Lifts for the tangent weak comonad +The lift ℓ : T ⇒ T 2 gives rise to a weak comonad. Weak comonads were introduced in Wisbauer (2013) +and have a natural notion of an associative algebra2. An associative coalgebra of the weak comonad +(T,ℓ) is called a lift; this chapter will demonstrate that lifts provide the essential structure necessary to +formulate vector bundles. +Definition 2.2.1. A weak comonad on a category � is an endofunctor S : � → � equipped with a coas- +sociative map δ : S ⇒ S.S: +S +S.S +S.S +S.S.S +δ +δ +S.δ +δ.S +An associative algebra of a weak comonad is an object E equipped with a map λ : E → SE so that +E +S.E +S.E +S.S.E +λ +λ +δ.E +S.λ +2This is not strictly true. Wisbauer has a more nuanced hierarchy of almost-monads, and in his language (T,ℓ) would be +an endofunctor with an associative product. +34 + +A morphism of these algebras is a map f : (E ,λ) ⇒ (D,γ) so that +E +D +S.E +S.D +f +S.f +λ +γ +Recall that for a full (co)monad, there is an adjunction between the base category and the category +of (co)algebras. That result is weakened in this case: +Lemma 2.2.2. For every weak comonad on a category �, there is a free coalgebra functor +F : � → CoAlg(�);E �→ (S.E ,δ : S.E → S.S.E ) +an underlying object functor +U : CoAlg(�) → �;(E ,λ) �→ E +and a natural transformation +λ : id ⇒ F.U ; +E +S.E +S.E +S.S.E +λ +S.λ +λ +δ +Definition 2.2.3. A lift in a tangent category � is an associative coalgebra of (T,ℓ), namely, a pair (E ,λ : +E → T E ) so that the following diagram commutes: +E +T E +T E +T 2E +λ +λ +ℓ.E +T.λ +A morphism of lifts is a coalgebra morphism. The category of lifts and lift morphisms in a tangent cate- +gory � is written Lift(�). +Note that the tangent bundle is not, in general, a comonad: while the tangent projection has the +correct type for a counit, p : T ⇒ id , it does not satisfy p ◦ℓ = id . In fact, if p were a counit, this would +force id = p ◦ ℓ = 0 ◦ p so that 0 = p −1, thus if (T,ℓ,p) is a comonad then T is naturally isomorphic to +the identity functor. +Example 2.2.4. +(i) For every object M in a tangent category �, the pair (T M ,ℓ : T M → T 2M ) is a lift, called the free +lift on M . +(ii) Every object M in a tangent category has a trivial lift, 0 : M → T M , where T.0◦ 0 = ℓ◦ 0. +(iii) Every differential object has a lift λ; the coherence is equivalent to axiom [D0.3] in Definition 1.4.1. +(iv) The Euler vector field of a multiplicative �+-action h : R + ×E → E in SMan is a lift. Recall that the +Euler vector field over the scalar action sM : T M × � → � induces the vertical lift on a manifold: +ℓ = T.sM ◦ (0,λ� ◦ 1�◦!). +35 + +In the category of smooth manifolds, T � ∼= �[x]/x 2, where λ(r ) = [x �→ r ·x] corresponds to the map +λ′(r ) = 0+r ·x. Similarly, there is an isomorphism �[x, y ]/(x 2, y 2) so that for the maps 0.Tand T.0, +0.T (a + r · x) ∼= a + r · x + 0y + 0x y, +T.0(a + r · x) ∼= a + 0x + r · y + 0x y. +Since we know that (0 + x)(0 + y ) = (0 + x y ) in �[x, y ]/(x 2, y 2) (following 1.4.2), we can use these +isomorphisms to see that +ℓ◦ λ� ◦ 1�◦! = (0.T ◦ λ� ◦ 1�◦!) ·T 2.� (T.0◦ λ� ◦ 1�◦!). +Consider a monoid action (�+,h) on a manifold E . The Euler vector field of this action, +λ : E +(id,1�◦!) +−−−−→ E × � +(0,λ�) +−−−→ T E × T � +T.h +−→ T E +will define an algebra if the induced scalar action on T E commutes with the natural scalar action: +T.λ ◦ λ = T 2.h ◦ (T.0◦ λ,T.λ ◦ 0◦ 1�◦!) += T 2.h ◦ (T.0◦ T.h ◦ (0,λ ◦ 1�◦!),0◦ λ ◦ 1�◦!) += T 2.h ◦ (T 2.h ◦ (T 0◦ 0,T 0◦ λ ◦ 1�◦!),0◦ λ ◦ 1�◦!) += T 2.h ◦ (T.0◦ 0,(T.0◦ λ ◦ 1�◦!) ·T 2.� (0.T ◦ λ ◦ 1�◦!)) += T 2.h ◦ (ℓ◦ 0,ℓ◦ λ ◦ 1�◦!) += ℓ◦ T.h ◦ (0,λ ◦ 1�◦!) += ℓ◦ λ. +Observation 2.2.5. Recall that by Proposition 2.1.12, morphisms preserve a monoid action if and only +if they preserve the associated Euler vector field of the action. This means the Euler vector field construc- +tion gives a fully faithful functor from monoid actions to lifts in the category of smooth manifolds, and +therefore from the category of vector bundles to the category of lifts in SMan. +The following proposition gives a pair of constructions on lifts—closure under the tangent functor +and finite T -limits—that will be useful in this section. +Lemma 2.2.6. Let � be a tangent category. +(i) The tangent functor lifts to an endofunctor on the category of lifts in �. +(ii) Given a diagram +D : � → Lift(�) +in the category of lifts of �, if the T -limit ofU .D exists in �, then limU .D has a natural lift λ′ asso- +ciated to it so that (limU .D,λ′) is the limit of D in Lifts(�). (That is, T -limits of lifts are computed +pointwise in the base category.) +Proof. +(i) Simply check that +T.(c ◦ T.λ) ◦ c ◦ T.λ = T.c ◦ T 2.λ ◦ c ◦ T.λ = T.c ◦ c .T ◦ T 2.λ += T.c ◦ c .T ◦ T.ℓ◦ T.λ = ℓ.T ◦ c ◦ T.λ. +36 + +(ii) Concretely, a tangent terminal object will have a lift: +(1,1 +0−→ T.1 ∼= 1). +Given (E ,λ) and (F,l ), if the tangent product E × F exists there is a lift +(E × F,E × F +λ×l +−−→ T E × T F ∼= T (E × F )). +Given the T -equalizer of a fork f ,g : (E ,λ) → (F,l ), the equalizer has a lift induced as follows: +C +E +F +T.C +T.E +T.F +f +g +k +λ +k +λ +T.f +T.g +l +Proposition 2.2.7. The category of lifts is a tangent category. +Proof. The tangent functor sends +f : (E ,λ) → (F,l ) +T.f : (T E ,c ◦ T.λ) → (T F,c ◦ T.l ). +To see that this is still an algebra morphism, compute +c ◦ T.l ◦ T.f = c ◦ T 2.f ◦ T.λ = T 2.f ◦ c ◦ T.λ +The structure maps are the structure maps on the underlying object of the lift; the universality condi- +tions follow by Proposition 2.2.6. +The following idempotent is key in the theory of lifts and will be used in defining non-singular +lifts (Definition 2.3.1), and its splitting will present the projection and zero-section of a vector bundle +(Definition 2.4.1). +Proposition 2.2.8. The category of lifts in a tangent category � has a natural idempotent: +e : id ⇒ id ;e(E ,λ) : (E ,λ) +p◦λ +−−→ (E ,λ). +Proof. First, we see that e = p ◦ λ is an idempotent: +p ◦ λ ◦ p ◦ λ = p ◦ p.T ◦ T.λ ◦ λ = p ◦ p.T ◦ ℓ◦ λ = p ◦ 0◦ p ◦ λ = p ◦ λ. +Moreover, every f : (E ,λ) → (F,l ) preserves the idempotent: +f ◦ p ◦ λ = p ◦ T.f ◦ λ = p ◦ l ◦ f . +Finally, note that the idempotent is a lift morphism.: +T.λ ◦ λ = ℓ◦ λ = c ◦ ℓ◦ λ = c ◦ T.λ ◦ λ +which implies that +λ ◦ e = λ ◦ p ◦ λ = p ◦ T.λ ◦ λ = p ◦ c ◦ T.λ ◦ λ = T.p ◦ T.λ ◦ λ = T.e ◦ λ. +37 + +2.3 +Non-singular lifts +Grabowski and Rotkiewicz (2009) introduced the notion of a non-singular lift as a means to axioma- +tize the Euler vector field of a vector bundle’s multiplicative �+-action. While it is not immediately +clear that our definition is the same as Grabowski’s, the results of Section 2.5 will justify the use of this +language as they are necessarily the same. +Definition 2.3.1. A lift (E ,λ) in a tangent category � is non-singular whenever the following diagram is +a T -equalizer: +E +T E +T E +λ +T.e +e .E +where e .E = p ◦ ℓ (the idempotent associated to the free lift on E ) and T.e = T.p ◦ T.λ (the image of the +idempotent associated to (E ,λ) under the tangent functor). The category of non-singular lifts is written +NonSing(�). +The most prominent class of examples is given by the Euler vector field of the �-action on a vector +bundle. +Proposition 2.3.2. The Euler vector field of a vector bundle is a non-singular lift. +Proof. Let (q : E → M ,+,ξ,·) be a vector bundle with Euler vector field λ. By Proposition 2.1.9, the +diagram +T M +E +T E +E +λ +T.q +p◦T.ξ◦T.q +p +T.q◦0◦p +is a T -limit. The T -universality of this diagram will hold if and only if the diagram is universal after +each parallel pair of arrows is post-composed by a T -monic. A section is a T -monic, so the previous +diagram is T -universal if and only if the following diagram is T -universal: +E +T E +E +T E +T M +T E +p +ξ◦q◦p +T.q +0◦q◦p +λ +0 +T.ξ +Now simplify this diagram using the fact that T.(ξ ◦ q) = T.e ,0◦ p = e .E : +T E +E +T E +T E +e .T +e .E ◦T.e +T.e +e .E ◦T.e +λ +38 + +Note that the the two pairs of parallel arrows have a common arrow, implying that they may be pulled +together into a single ternary equalizer. All that remains to check, then, is that for any x : X → T E , +(e .e ◦ x = T.e ◦ x = e .E ◦ x) ⇐⇒ (T.e ◦ x = e .E ◦ x). +The forward implication is trivial, so it remains to prove the reverse. Suppose T.e ◦ x = e .E ◦ x; then +e .e ◦ x = e .E ◦ T.e ◦ x = e .E ◦ e .E ◦ x = e .E ◦ x +giving the result, namely that the diagram +E +T E +T E +λ +T.e +e .E +is a T -equalizer. +Every map f : E → F gives a map of free coalgebras T.f : (T E ,ℓ) → (T F,ℓ) and the idempotent e is +a coalgebra morphism by Proposition 2.2.8, so the following is immediate: +Proposition 2.3.3. A non-singular lift is an equalizer in the category of lifts: +(E ,λ) +(T E ,ℓ) +(T E ,ℓ) +λ +T.e +e .E +Observe that the category of non-singular lifts is closed under finite limits in the category of lifts. +Proposition 2.3.4. The category of non-singular lifts in � is closed under T -limits: +(i) The tangent functoron liftspreservesnon-singularlifts, so that if (E ,λ) isnon-singularthen (T E ,c ◦ +T.λ) is non-singular. +(ii) The trivial lift on an object, 0 : M → T M , is non-singular. +(iii) T -products of non-singular lifts are non-singular lifts. +(iv) T -equalizers of non-singular lifts are non-singular lifts. +Proof. +(i) This follows from the fact that the non-singularity condition is a T -limit. +(ii) The zero map splits the idempotent 0◦ p, so it is the equalizer of 0◦ p,p ◦ 0 = id . +(iii) This follows by stability of limits under products. +(iv) The following diagram commutes by naturality: +C +E +F +T.C +T.E +T.F +T.C +T.E +T.F +λ +λ +f +T.f +T.g +e .E +T.e +e .F +T.e +T.f +T.g +g +λ +e .C +T.e +Each horizontal diagram is a T -equalizer, and the two columns on the right are T -equalizers, so +the column on the left is a T -equalizer. +39 + +Finally, when a tangent category has certain T -equalizers, there is an idempotent monad on the +category of lifts, whose algebras are non-singular lifts: +Theorem2.3.5. Let � be a tangent category with chosen T -equalizersof idempotents. Then the following +equalizer determines a left-exact idempotent monad on the category of lifts, whose algebras are non- +singular lifts. +(F,l ) +(T.E ,ℓ) +(T.E ,ℓ) +e .E +T.e +Proof. First, take the equalizer in Lift(�); the functor sends (E ,λ) to the chosen limit (F,l ).The unit of +the monad is the unique morphism from (E ,λ) to the equalizer (F,l ) induced by universality: +(F,l ) +(T E ,ℓ) +(T E ,ℓ) +(E ,λ) +λ +T.e +e .E +∃! +Note that non-singular lifts are closed under finite limits, so (F,l ) is a non-singular lift. If (E ,λ) is a +nonsingular lift then λ : (E ,λ) → (T E ,ℓ) equalizes the diagram, so there is a unique isomorphism +(F,l ) ∼= (E ,λ), making the multiplication of the monad a natural isomorphism (and thus yielding an +idempotent monad). The functor is defined as a T -limit and therefore preserves all T -limits of lifts, so +it is left-exact. +In the category of smooth manifolds, ℓ is the Euler vector field of an �+-action, this guarantees that +every λ is the Euler vector field of a multiplicative �+-action. We note the following corollary. +Corollary 2.3.6. In the category of multiplicative �+ actions in SMan, multiplication by 0 is equivalent +to the natural idempotent e in the fully faithful functor sending an �+-action to its Euler vector field. +By non-singularity, the following diagram is an equalizer, giving E a multiplicative action by �+ whose +Euler vector field is λ: +(E ,·E ) +(T E ,·T ) +(T E ,·T ) +T.e +e .T +λ +Moreover, λ is the Euler vector field of this lift, and the following diagram commutes: +E +T E +T E +�+ × E +�+ × T E +�+ × T E +T (�+ × E ) +T (�+ × T E ) +T (�+ × T E ) +T E +T T E +T T E +T (�+×e .E ) +T (�+×T.e ) +�+×λ +·p +T ·p +e .E +T.e +T λ +T ·E +(1�,id) +(1�,id) +(1�,id) +λ�×0 +λ�×0 +λ�×0 +e .E +T.e +�+×e .E +�+×T.e +λ +�+×λ +40 + +2.4 +Differential bundles +This section introduces (pre-)differential bundles, which provided the rest of the data for a vector bun- +dle: namely the projection, the zero section, and the addition map. The zero section and projection +data arise by splitting the natural idempotent e : id ⇒ id , and non-singularity will induce the addi- +tion map. Every differential bundle satisfies a pair of universality diagrams, linking this presentation +of differential bundles to the original definition in Cockett and Cruttwell (2018). +Definition 2.4.1. +(i) A pre-differential bundle is a lift λ : E → T E equipped with a chosen splitting of the natural idem- +potent e = p ◦ λ from Proposition 2.2.8. Pre-differential bundles are formally written (q : E → +M ,ξ,λ), where q : E → M is the retract, ξ : M → E the section, and λ : E → T E the lift (the types +are only necessary for the projection as the rest may be inferred, and will generally be suppressed to +save space). +(ii) A differential bundle is a pre-differential bundle (q : E → M ,ξ,λ) with the properties that λ is +non-singular and T -pullback powers of q exist. +Morphisms of (pre-)differential bundles are exactly morphisms of their underlying lifts. The categories +of (pre-)differential bundles are exactly (pre-)differential bundles and lift morphisms, and are written +Pre(�),DBun(�) respectively. +Recall that as e is a natural idempotent in the category of lifts, any lift morphism will preserve e +and will consequently preserve its idempotent splitting. Preserving the idempotent means that every +differential bundle morphism is a bundle morphism, where the base map is given by +E +F +M +N +q +f +π +m:=π◦f ◦ξ +We now look at the limits of (pre-)differential bundles. +Observation 2.4.2. +(i) The limit for a diagram of pre-differential bundles is the limit of the underlying lifts equipped with +a chosen splitting of p ◦ λ due to basic properties about idempotent splittings. +(ii) The limit for a diagram of differential bundles is the limit in the category of pre-differential bun- +dles (the lift will be universal by Proposition 2.3.4), so long as T -pullback powers of the resulting +projection exist. +Because there is a projection associated to a (pre-)differential bundle, the T -reindexing operation +described in Proposition 1.5.2 can now be applied. This gives a pullback differential bundle in a similar +way to Proposition 2.1.5. +Lemma 2.4.3 (Cockett and Cruttwell (2018)). Let (q : E → M ,ξ,λ) be a (pre-) differential bundle in a +tangent category �, and consider a T -pullback in �: +u∗E +E +N +M +u +q +u∗q +¯u +⌟ +41 + +Then the induced triple maps +T.u∗E +T E +N +u∗E +E +u∗E +E +N +M +N +M +T N +T M +u +q +u∗q +¯u +⌟ +T. ¯u +λ +0 +T.q +T.u +0 +T.u∗q +u∗λ +q +u +⌟ +ξ◦u +u∗ξ +induce a (pre-)differential bundle (u∗q : u∗E → N ,u∗ξ,u∗λ). If λ is non-singular and T -pullback pow- +ers of u∗q exist, then (u∗q,u∗ξ,u∗λ) is a differential bundle. +Proof. Note thata pre-differential bundle (q : E → M ,ξ,λ) in � mayalsobe regarded asa pre-differential +bundle (q : (E ,λ) → (M ,0),ξ,λ) in Lift(�). Take the following pullback in Lift(�): +(u∗E ,u∗λ) +(E ,λ) +(N ,0) +(M ,0) +u +q +⌟ +It follows by construction that ι ◦ u∗λ = λ ◦ ι. Thus the result holds. +Proposition 2.4.4. Let (q : E → M ,ξ,λ) be a non-singular pre-differential bundle in a tangent category +�, so that T -pullback powers of q exist. Then there is an additive bundle structure (q,ξ,+q) so that the +differential bundle morphisms are additive: +(λ,ξ) : (q,ξ,+q) → (p,0,+) +(λ,0) : (q,ξ,+q) → (T.q,T.ξ,T.+q). +Furthermore, every differential bundle morphism preserves addition. +Proof. Non-singularity forces the existence of an addition map: +E2 +T2E +T2E +E +T E +T E +∃!+q ++ +λ×λ +λ +e .E +T.e ++ +(e ×e ).E +T2.e +Note that this diagram commutes because T2E is a pre-differential bundle whose lift is ℓ×ℓ, and also + +is a linear morphism, so it commutes with the addition map e ◦a +e ◦b = e ◦(a ×b ). Post-composition +with λ ensuresthatξ isthe unitand thatassociativityholds. Adifferential bundle morphism will induce +a morphism of the equalizer diagrams that induce each addition map to preserve addition. +Recall that by Proposition 2.3.2, the Euler vector field for every vector bundle is a non-singular lift. +If T M q ×q E exists, the map +νE : T M p ×q E +T.ξ×λ +−−−→ T2E ++.E +−→ T E +may be formed. Similarly, using the additive bundle structure from Proposition 2.4.4, the µ map may +be formed: +µE : E q ×q E +0×λ +−−→ T (E q ×q E ) +T.+q +−−→ T E +Note that any differential bundle morphism will preserve µ and ν. +42 + +Lemma 2.4.5. Differential bundle maps preserve µ(x, y ) := 0◦ x +T.q λ◦ y and ν(v, y ) := T.ξ◦v +p λ◦ y . +Proof. The following diagram demonstrates that lift maps preserve ν: +E q ×p T M +F q ′×p T N +T.E +T.F +T.E +T.F +ν +ν +f ×T.m +T.f +T.g +e .e +T.e +e .e +T.e +T.f +T.g +g ×T.n +Similarly, this diagram demonstrates that lift maps preserve µ: +T E +T F +T E2 +T F2 +E2 +F2 +(T.f ×T.f ) +(λ×0) +(λ′×0) +(f ×f ) +T.+ +T.+′ +T f +Lemma 2.4.6. Consider the full subcategory of differential bundles in � whose objects are differential +bundles (E ,λ) so that the forks +E2 +T E +T M +T M p ×q E +T E +E +µE :=0◦π0+T q λ◦π1 +νE :=T.ξ◦π0+p λ◦π1 +T.q +T.q◦0◦p +p +p◦T.ξ◦T.q +(2.3) +are T -equalizers. This subcategory is closed under T -equalizers. +Proof. Start with the T -equalizer of lifts: +C +E +F +f +g +k +Observe that k is a lift map, so by Lemma 2.4.5, the following diagram commutes: +C q c ×p T P +E q ×p T M +F q ′×p T N +T.C +T.E +T.F +T.C +T.E +T.F +ν +ν +f ×T.m +T.f +T.g +e .e +T.e +e .e +T.e +T.f +T.g +k×T.k ′ +ν +e .e +T.e +T.k +T.k +g ×T.n +43 + +Next, since k is a morphism of pre-differential bundles, by Lemma 2.4.5 the following diagram com- +mutes: +C2 +E2 +F2 +T.C +T.E +T.F +T.C +T.E +T.F +µ +µ +f2 +T.f +T.g +e .e +T.e +e .e +T.e +T.f +T.g +g2 +µ +e .e +T.e +The top row follows because maps satisfying Rosicky’s universality condition preserve µ, and the bot- +tom row by the naturality of e .C and the fact that linear maps preserve the natural idempotent. Thus, if +E ,F satisfy the universality diagrams in Diagram 2.3, the equalizer C will as well, because T -equalizers +are closed to T -limits in the category of fork diagrams. +Theorem 2.4.7. For every differential bundle (q : E → M ,ξ,λ) in a tangent category, the diagram +E2 +T E +T M +µE +T.q +T.q◦0◦p +is a T -equalizer, and if T M p ×q E exists, then +E q ×p T M +T E +E +νE :=λ◦π0+p T.ξ◦π0 +p +p◦T.ξ◦T.q +is a T -equalizer. +Proof. The tangent bundle satisfies both universality conditions, and by Lemma 2.4.6every differential +bundle will satisfy these conditions by the non-singularity of the lift. +Corollary 2.4.8. In a complete tangent category, the category of differential bundles is precisely the cat- +egory of algebras of the monad in Theorem 2.3.5 on pre-differential bundles. +Remark 2.4.9. The universality conditions in Theorem 2.4.7 demonstrate that this definition of differ- +ential bundle agrees with that of Cockett and Cruttwell (2018). +2.5 +The isomorphism of categories +It is now straightforward to show that the main theorem of this chapter holds. First, observe that for +every differential bundle (q : E → M ,ξ,λ) in the category of smooth manifolds, the T -pullback E p ×q +T M exists , as p is a submersion. Note that the universality of the two diagrams is equivalent: +E q ×p T M +T E +E +νE :=λ◦π0+p T.ξ◦π0 +p +p◦T.ξ◦T.q +E q ×p T M +T E +M +E +νE +λ◦π0+p T.ξ◦π1 +p +ξ +⌟ +Thus the following holds. +44 + +Theorem 2.5.1. There is an isomorphism of categories between vector bundles and differential bundles +in smooth manifolds. +Proof. Note that Proposition 2.3.2 gives a fully faithful functor from the category of vector bundles to +differential bundles of smooth manifolds, as the lift associated to the vector bundle is non-singular and +the projection and zero section give the rest of the structure of a differential bundle: as remarked after +Definition 2.1.1, local triviality guarantees that the projection is a submersion, so pullback powers of +the projection exist, yielding a differential bundle. +To see there is an isomorphism on objects, recall that every differential bundle is a fibred �-module +by Corollary 2.3.6, and that this identification is a bijective mapping (it recovers the original �-action +from the Euler vector field of the �-action, and vice versa). Last, note that the universality condition +E q ×p T M +T E +M +E +νE +λ◦π0+p T.ξ◦π1 +p +ξ +⌟ +along with Proposition 2.1.5 ensures the local triviality of q, so that the unique fibred �-module struc- +ture associated to a differential bundle is indeed a vector bundle. +2.6 +Connections on a differential bundle +The connections discussed in this section generalize the notion of an affine connection to a differential +bundle, giving a “local coordinates” presentation for T E similar to the presentation of T 2M as T3M +induced by an affine connection. Chapter 3 makes extensive use of connections on vector bundles, so +it is useful to set out the basic definitions before embarking on algebroid theory. +Definition 2.6.1 (Cockett and Cruttwell (2017)). Let (q : E → M ,ξ,λ) be a differential bundle in a tan- +gent category �. +• A vertical connection is a map κ : T E → E so that +(i) κ is a vertical descent, and hence a retract of the lift; thus, κ ◦ λ = id ; +(ii) κ is compatible with both differential bundle structures on T E , so that the maps +κ : (T E ,ℓ) → (E ,λ) +κ : (T E ,c ◦ T.λ) → (E ,λ) +are lift morphisms. +• A horizontal connection is a map ∇ : E q ×p T M → T E so that +(i) ∇ is a horizontal lift, and so is a retract of (p.E ,T.q) : T E → E q×pT M 3; thus, (p,T.q)◦∇ = id ; +(ii) ∇ is compatible with each pair of lifts, so that the maps +∇ : (E q ×p T M ,λ × ξ) → (T E ,c ◦ T.λ) +∇ : (E q ×p T M ,0× ℓ) → (T E ,ℓ) +are lift morphisms. +3That (p.E ,T.q) land in the pullback E q ×p T M is a consequence of naturality, as p ◦ T.q = q ◦ p. +45 + +• A full connection is a pair (κ,∇) that satisfies the following compatibility relations: +(i) κ ◦ ∇ = ξ ◦ q ◦ π0, +(ii) ∇(p,T.q) +T.q µ(p,κ) = id , so that there is an isomorphism E q ×p T M p ×q E ∼= T E . +A notion that will be useful when dealing with classical differential geometry is that of a covariant +derivative, whose definition is equivalent to that of a vertical connection in the category of smooth +manifolds. +Definition 2.6.2. Let (q : E → M ,ξ,λ,κ) be a vertical connection.4 The covariant derivative associated +to (κ,∇) is the map +∇(−)[=] : Γ(π) × Γ(p) → Γ(π);(A,X ) �→ (κ ◦ T A ◦ X ). +Lucyshyn-Wright (2018) drastically simplified the notion of a full connection by showing that it is +exactly a vertical connection satisfying a universal property. +Proposition 2.6.3 (Lucyshyn-Wright (2018)). A connection on a differential bundle is equivalently spec- +ified by a vertical connection +κ : T E → E +so that the following diagram exhibits T E as a biproduct in the category of differential bundles over M : +E +T E +T M +M +E +p +T q +κ +q +q +p +There is also a notion of flatness for connections that extends to vertical connections on a differen- +tial bundle. +Definition 2.6.4. A connection κ on a differential bundle (q : E → M ,ξ,λ) is flat whenever +κ ◦ T.κ ◦ c = κ ◦ T.κ. +We can see that connections are closed under similar constructions to differential bundles, in par- +ticular idempotent splittings and the reindexing construction from Lemma 2.4.3. +Lemma 2.6.5. Let (q : E → M ,ξ,λ) be a differential bundle equipped with a vertical connection. +(i) Any linear retract of (q,ξ,λ) will have a vertical connection. +(ii) The pullback differential bundle induced by pulling back q along f : N → M will have a vertical +connection. +If the vertical connection is flat or is part of a full connection (that is, it satisfies the universality condition +in Proposition 2.6.3), then the induced connection will be as well. +Proof. The universal property induces the vertical connection in each case. The construction will pre- +serve flatness as it is an equational condition, and it preserves effectiveness by the commutativity of +limits. +4The definition of a covariant derivative only uses a vertical connection. +46 + +There is no reason for every differential bundle in a tangent category to have a connection (for +example, the tangent bundle in the free tangent category Weil1 in Chapter 4 is a differential bundle +that does not have a connection). However, if the total space of a differential bundle has an affine +connection, this induces a compatible connection on the differential bundle and its base space. +Theorem 2.6.6. Let (q : E → M ,ξ,λ) be a differential bundle in a tangent category in �, where E has a +(flat) vertical connection. Then the total space M and the differential bundle (q,ξ,λ) each have a (flat) +vertical connection. If the connection is full (so there is a compatible horizontal connection), then the +induced connections are likewise full. +Proof. By the strong universality condition for differential bundles, the differential bundle (q ◦ π0 : +E q×pT M → M ,(ξ,0),λ×ℓ) is the pullback differential bundle of p : T E → E along ξ : M → E . This gives +(q◦π0,(ξ,0),(λ,ℓ)),a (flat, effective) vertical connection byLemma 2.6.5, and soityields(q : E → M ,ξ,λ) +and (p : T M → M ,0,ℓ) as (flat, effective) vertical connections by the idempotent splitting property. +Every smooth manifold has an affine connection, thus inducing a connection on any vector bundle. +Corollary 2.6.7. Every vector bundle has a connection. +47 + +Chapter 3 +Involution algebroids +This chapter accomplishes the first major goal of this thesis by providing a tangent-categorical axiom- +atization of Lie algebroids, namely involution algebroids. Much like vector bundles, Lie algebroids are +a highly non-algebraic notion (in the sense of Freyd and Kelly (1972)), being vector bundles equipped +with a Lie algebra structure on the set of sections of the projection. Furthermore, the bracket on sec- +tions must satisfy a product rule with respect to �-valued functions on the base space (the Leibniz +law), introducing another piece of non-algebraic structure to the definition. The tangent-categorical +definition of Lie algebroids will treat the tangent bundle as the “prototypical Lie algebroid” in which +the vertical lift ℓ : T ⇒ T 2 identifies the vector bundle structure and the canonical flip c : T 2 ⇒ T 2 plays +the role of the Lie bracket. +Lie algebroidsare the natural many-objectanalogue toLie algebras, in the same waythatLie groupoids +are the many-object analogue of Lie groups. In the single-object case, a Lie group is classically thought +of as a space of symmetries for some smooth manifold (one often identifies a group action G ×M → M ), +and a Lie algebra may similarly be thought of as a space of derivations (often identified as a sub-Lie- +algebra of χ(M ) for a manifold M ). The extension of groups to groupoids is natural; in fact, Brandt’s +introduction of groupoids in Brandt (1927) predates MacLane and Eilenberg’s invention of category +theory in Eilenberg and MacLane (1945) by nearly two decades. The translation of Lie algebras to the +many-object case is not as straightforward. The first step is to replace the vector space underlying a Lie +algebra with a vector bundle (π : A → M ,ξ,λ). The idea is to axiomatize this vector bundle so that each +section in Γ(π) corresponds to a derivation on C ∞(M ). The anti-commutator operation on derivations +from Proposition 1.2.18 suggests there should be a Lie bracket [−,−] : Γ(π) ⊗ Γ(π) → Γ(π) (similar to the +partially-defined multiplication for a groupoid), while the correspondence with derivations on C ∞(M ) +suggests there be a vector bundle morphism ̺ : A → T M satisfying the Leibniz law: +[X , f · Y ] = f ·[X ,Y ]+ [X , f ]· Y ; +[X , f ] := ˆp ◦ T.f ◦ ̺ ◦ X 1 +(3.1) +(the full definition of Lie algebroids may be found in 3.1.1). This “operational” definition of Lie alge- +broids makes it difficult to describe their morphisms, and furthermore it essentially fails to be an alge- +braic structure in the classical sense, as it axiomatizes structure on the set of sections of a map rather +than a morphism in the category itself. +Involution algebroids were introduced to provide a tangent-categorical presentation of Lie alge- +broids, similar to the relationship between differential bundles and vector bundles. Chapter 2 focused +on the Euler vector field construction on a vector bundle, showing that this induced a fully-faithful +functor from vector bundles to associative coalgebras (lifts) of the weak comonad (T,ℓ), and identified +vector bundles with a subcategory of Lift(SMan) satisfying a universal property. The corresponding +1Recall the notation from Lemma 2.1.3. +48 + +construction for Lie algebroids, then, is the canonical involution, which was identified by Eduardo +Martinez and his collaborators (a clearly written exposition may be found in Section 4 of de León et al. +(2005)). Given a Lie algebroid (π : A → M ,̺ : A → T M ,[−,−] : Γ(π) ⊗ Γ(π) → Γ(π)), its canonical involu- +tion is a map +σ : A̺×T πT A → A̺×T πT A. +Using this σ map, there is a straightforward characterization of Lie algebroid morphisms: a Lie alge- +broid morphism is precisely a vector bundle morphism (f ,m) : A → B that preserves the anchor and +involution maps: +A +B +A̺×T πT A +B ̺B ×T.πB T B +T M +T N +A̺×T πT A +B ̺B ×T.πB T B +̺A +̺B +f +T.m +σB +f ×T.f +σA +f ×T.f +Furthermore, it is implicit in Martinez’s work (Martínez (2001)) that σ satisfies axioms corresponding +to the Lie algebroid axioms. Thus, involutivity corresponds to antisymmetry of the Lie bracket +σ ◦ σ = id ⇐⇒ [X ,Y ]+ [Y ,X ] = 0, +while the Leibniz law holds if and only if the ̺ map sends the algebroid involution to the canonical flip +on M , +T.̺ ◦ π1 ◦ σ = c ◦ T.π◦ ̺ ⇐⇒ ∀f ∈ C ∞(M ),[X , f · Y ] = f ·[X ,Y ]+ [X , f ]· Y +(using the same definition as before for [X , f ]). +The idea of an involution algebroid, then, is to axiomatize the canonical involution directly, just as +differential bundles axiomatize the Euler vector field of a differential bundle. An involution algebroid +is a differential bundle equipped with a pair of structure maps +̺ : A → T M , σ : A̺×T πT A → A̺×T πT A +satisfying a collection of axioms. Some of them are straightforward translations of the structure equa- +tions for Lie algebroids given in Martínez (2001), for instance +T.̺ ◦ λ = ℓ◦ ̺, σ ◦ σ = id , T.̺ ◦ π1 ◦ σ = c ◦ T.̺ ◦ π1. +However, this requires a new coherence between the Euler vector field of the underlying vector bundle +and the involution map: +σ ◦ (ξ ◦ π,λ) = (ξ ◦ π,λ). +The most striking new fact about this coherence is that the Jacobi identity on the bracket [−,−] corre- +sponds to the Yang–Baxter equation on σ: +A̺×T πT AT.̺×T 2.πT 2A +A̺×T πT AT.̺×T 2.πT 2A +A̺×T πT AT.̺×T 2.πT 2A +A̺×T πT AT.̺×T 2.πT 2A +A̺×T πT AT.̺×T 2.πT 2A +A̺×T πT AT.̺×T 2.πT 2A +σ×c .A +σ×c .A +1×T.σ +σ×c .A +1×T.σ +This is both surprising (it is a new characterization of a central object of study in differential geom- +etry and mathematical physics) and yet in a way expected (the work in Cockett and Cruttwell (2015), +49 + +Mackenzie (2013) indicates that the Lie algebra structure on the set of vector fields over a manifold +follows from the Yang–Baxter equation on c ). This vector-field-free presentation of the Jacobi identity +allows for a structural approach to Lie algebroids that drives the work presented in Chapters 4 and 5. +As with Chapter 2, the first section is expository, and is concerned with introducing the category of +Lie algebroids. The second section introduces anchored bundles, together with the space of prolonga- +tions of an anchored bundle. The relationship between anchored bundles and involution algebroids +is equivalent to that between reflexive graphs and groupoids (the subject of Chapter 5), the space of +prolongations of an anchored bundle being equivalent to the set of composable arrows for a groupoid. +This section is mostly a translation of Martinez’s prolongation construction to a general tangent cate- +gory. The rest of the chapter contains new results, developed in collaboration with Matthew Burke and +Richard Garner. +Section 3 introduces involution algebroids, which are anchored bundles equipped with an involu- +tion map on their space of prolongations. Section 4 considers an anchored bundle in a tangent cate- +gory with negatives that is equipped with a connection. The connection gives an involution algebroid +a “local coordinates” presentation (in the sense of Section 1.4) that is equivalent to the local character- +ization of Lie algebroids from Section 1. The final section of this chapter establishes the main result: +the category of Lie algebroids is isomorphic to that of involution algebroids in smooth manifolds. +3.1 +Lie algebroids +This section reviews the basic theory of Lie algebroids: their definition and that of their morphisms, +along with some introductory examples. The classical definition will not appear elsewhere in this +chapter, however, as we quickly introduce Martinez’s structure equations for a Lie algebroid (Martínez +(2001)), then translate them into tangent-categorical terms using a connection. +Definition 3.1.1. A Lie algebroid is a vector bundle π : A → M equipped with an anchor ̺ : A → T M +and a bracket [−,−] : Γ(π) ⊗ Γ(π) → Γ(π) satisfying the following axioms: +• bilinear: [a X1 + b X2,Y ] = a[X1,Y ]+ b [X2,Y ] and [X ,a Y1 + b Y2] = a[X ,Y1]+ b [X ,Y2] +• anti-symmetric: [X ,Y ]+ [Y ,X ] = 0 +• Jacobi: [X ,[Y ,Z ]] = [[X ,Y ],Z ]+ [Y ,[X ,Z ]] +• Leibniz: [X , f · Y ] = f ·[X ,Y ]+ [X , f ]· Y +(where [X , f ] is defined as in Equation 3.1). +Example 3.1.2. +(i) The canonical example of a Lie algebroid is, of course, the tangent bundle using the operational +tangent bundle from Definition 1.2.16. +(ii) ALie algebra isa Lie algebroid overthe terminal object: fora groupG , the bundle of source-constant +tangent vectors is the usual Lie functor from Lie groups to Lie algebras, because a groupoid is a one- +object group. +50 + +(iii) The bundle of source-constant tangent vectors s,t : G → M of a Lie groupoid forms a Lie algebroid. +This bundle is defined by the pullback +A +T G +T M +M +T M ×G +(T.s,p) +(0,e ) +π +T.t +̺ +⌟ +where the projection is π and the target is given by ̺ in the diagram. There is an injective �-module +morphism from sections of π, Γ(π) to vector fields on G , χ(G ), and the Lie bracket on G is closed +over the image of this lift, putting a Lie bracket on Γ(π). In particular, we can see that T M is the +bundle of source-constant tangent vectors for the pair groupoid on a manifold M : +T M +T (M × M ) +M +(M × M ) × T M +(p,T.π0) +(∆,0) +p +(id,0◦p) +⌟ +Given (u,v) : X → T (M × M ) above (m,m) : X → T M with u = 0 ◦ m, it follows that u = 0 ◦ p ◦ u, +so v is the unique map induced into T M . +(iv) Every group is a groupoid over a single object. The Lie algebroid associated with a group G , then, is +the usual Lie algebra. +(v) From the Hamiltonian formalism of mechanics, every Poisson manifold has an associated Lie alge- +broid. A Poisson manifold is a manifold M equipped with a Poisson algebra structure on C ∞(M ), +namely a Lie algebra that is also a derivation: +[f ·g ,h] = f ·[g ,h]+ [f ,h]·g +where · is the multiplication in the algebra C ∞(M ) as in Lemma 2.1.3. The cotangent bundle over +a Poisson manifold M is canonically a Lie algebroid, called a Poisson Lie algebroid Courant (1994). +(vi) Any Lie algebra bundle—that is, a vectorbundle equipped with a Lie bracket on itsspace of sections— +is a Lie algebroid, with anchor map ξ ◦ π. +Morphisms of Lie algebroids are notoriously difficult to work with, and have an involved definition. +Definition 3.1.3. Let A,B be a pair of anchored bundles over M ,N , and Φ : A → B an anchored bundle +morphism over a map φ : M → N . A Φ-decomposition of X ∈ Γ(πA) is a set of Xi ∈ Γ(πB) and fi ∈ C ∞(M ) +so that +Φ ◦ X = +� +i +fi · Xi ◦ φ. +An anchor-preserving vector bundle morphism Φ is a Lie algebroid morphism if and only if for any X ,Y ∈ +Γ(πA) and Φ-decompositions {Xi , fi},{Yj,g j} of X ,Y , the following equation holds: +Φ ◦ [X ,Y ] = +� +fi ·g j ·([Xi ,Yj]◦ φ) + +� +[X , fi ]·(Xi ◦ φ) − +� +[Y ,g j](Yj ◦ φ). +51 + +The equation defininga Lie algebroid morphism holdsindependentlyofthe choice ofΦ-decomposition +(see Higgins and Mackenzie (1990) for a proof). +Example 3.1.4. +(i) If A,B are Lie algebroids over a base manifold M , Φ : A → B is a Lie algebroid morphism if and only +if it preserves the anchor and Lie bracket. +(ii) Given two Lie algebroid morphisms f : A → B,g : B → C , their composition g ◦ f is also a Lie +algebroid morphism. +(iii) A Poisson Sigma model (see Bojowald et al. (2005)) is a morphism of Lie algebroids +φ : T Σ → T ∗M +for which Σ is a 2-dimensional manifold and T Σ denotes its tangent Lie algebroid, while M is a +Poisson manifold and T ∗M denotes the Lie algebroid structure on its cotangent bundle . +Lie algebroids are a natural generalization of Lie algebras to the “multi-object” setting, but they +are ill-suited for a functorial presentation of the theory. A step in this direction is to consider the +coordinate-based presentation of the Lie bracket and its coherences due to Martínez (2001). Let A +be an anchored bundle over M equipped with a bilinear bracket on its space of sections, and choose a +pair of bases for Γ(π) and χ(M ): for Γ(π) write {eα}, and for χ(M ) write { ∂ +∂ x i }. The anchor and bracket +then have a presentation in local coordinates: +̺(eα) = +� +i +̺i +α +∂ +∂ x i +[eα,eβ] = +� +γ +C γ +αβ eγ +(from here on out, we use Einstein summation notation to simplify our calculations, so instead write +̺(eα) = ̺i +α +∂ +∂ x i +[eα,eβ] = C γ +αβ eγ +with +� +suppressed). The following characterization of the Lie algebroid axioms uses Martinez’s struc- +ture equations. +Proposition 3.1.5. [Martínez (2001)] An anchored bundle A over M equipped with a bracket is a Lie +algebroid if and only if ̺ and [−,−] satisfy the following structure equations: +(i) Alternating: +C ν +αβ + C ν +βα = 0 +(ii) Leibniz: +̺ j +α +∂ ̺i +β +∂ x j = ̺i +γC γ +αβ + ̺ j +β +∂ ̺i +α +∂ x j +(iii) Bianchi: +0 = ̺i +α +∂ C ν +βγ +∂ x i + ̺i +β +∂ C ν +γα +∂ x i + ̺i +γ +∂ C ν +αβ +∂ x i + C µ +βγC ν +αµ + C µ +γαC ν +βµ + C µ +αβC ν +γµ +52 + +This proposition is a straightforward translation of the Lie algebroid axioms into local coordinates +using a covariant derivative. First, recall that by the smooth Serre–Swan theorem (11.33 Nestruev +(2003)), the bilinearity of the bracket +[−,−] : Γ(π) × Γ(π) → Γ(π) +as a morphism of C ∞(M )-modules guarantees that it corresponds to a bilinear morphism A2 → A of +vector bundles, meaning that there exists a globally defined bilinear map A2 → A that is equal to the +Lie bracket when applied to sections of the projection. We record this as a lemma: +Lemma 3.1.6. For every Lie algebroid � , there is a bilinear morphism +〈−,−〉 : A2 → A +so that for any sections X ,Y ∈ Γ(π), 〈X ,Y 〉 = [X ,Y ]. +There are two structure maps derived from 〈−,−〉 that encode the coherences of a Lie algebroid. +The first map measures the extent to which the anchor maps fail to preserve the chosen connections +on the vector bundle π : A → M and the tangent bundle p : T M → M . +Definition 3.1.7. Let A be a Lie algebroid, and for a chosen horizontal connection ∇ on A and vertical +connection κ′ on T M , set +{v, x}κ′,∇ := T M p ×q A +∇ +−→ T A +T.̺ +−→ T 2M +κ′ +−→ T M . +When the choice of connection is evident by context, we will suppress the subscript. +Observation 3.1.8. The above parentheses bracket corresponds to the symbol +{−,−} = ̺ j +β +∂ ̺i +α +∂ x j . +The Leibniz coherence may be rewritten as follows: +Lemma 3.1.9. Let A be an anchored bundle with a bilinear bracket (inducing an involution σ). Choose +connections (κ,∇),(κ′,∇′) on A and T M respectively. The bracket and anchor map satisfy the Leibniz +law if and only if +̺ ◦ 〈x, y 〉(κ,∇) + {̺x, y }(κ′,∇) = {̺y, x}(κ′,∇). +Proof. The condition is equivalent to the identity +̺ j +α +∂ ̺i +β +∂ x j = ̺i +γC γ +αβ + ̺ j +β +∂ ̺i +α +∂ x j . +The Bianchi axiom measures the failure of the Jacobi identity in local coordinates, and states that +it must be corrected for by the curvature of the brackets. We see that +C ν +αµC µ +βγ = [eα,[eβ,eγ]] +while +∂ C ν +βγ +∂ x i eν = κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ ( ∂ +∂ x i ,eβ,eγ) +53 + +determines a trilinear map +{ ∂ +∂ x i ,eβ,eγ}(κ,∇) := κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ ( ∂ +∂ x i ,eβ,eγ). +This is the second derived map used in the structure equations for a Lie algebroid. +Definition 3.1.10. Let (π : A → M ,ξ,λ,̺) be an anchored bundle equipped with a bilinear map +〈−,−〉 : A2 → A. +The derived ternary bracket {−,−,−} : T M p ×πAπ×πA → A is defined as +{v, x, y }(κ,∇) := T M p ×πAπ×πA +∇[A2] +−−−→ T A2 +T.〈−,−〉 +−−−−→ T A +κ−→ A +where ∇[A2] is the pairing (∇(π0,π1),∇(π0,π2)). +Observation 3.1.11. The ternary bracket corresponds to the following symbol: +{−,−,−}(κ,∇) := ̺i +α +∂ C ν +βγ +∂ x i . +Lemma 3.1.12. Let A be an anchored bundle over M with a bilinear bracket and denote the induced +involution by σ. Choose a pair of connections and write the derived maps 〈−,−〉,{−,−,−}. Then +(i) the bracket is antisymmetric if and only if its globalization is; that is, 〈eα,eβ〉 + 〈eβ,eα〉 = 0; +(ii) the bracket satisfies the Jacobi identity if and only if it is alternating in the last two arguments and +0 = +� +γ∈Cy(3) +〈xγ0,〈xγ1, xγ2〉〉 + +� +γ∈Cy(3) +{̺xγ0, xγ1, xγ2}. +Proof. +(i) This is equivalent to C ν +αβ + C ν +βα = 0. +(ii) This is equivalent to +0 = C µ +βγC ν +αµ + C µ +γαC ν +βµ + C µ +αβC ν +γµ + ̺i +α +∂ C ν +βγ +∂ x i + ̺i +β +∂ C ν +γα +∂ x i + ̺i +γ +∂ C ν +αβ +∂ x i . +Proposition 3.1.13. A Lie algebroid is exactly an anchored vector bundle (π : A → M ,ξ,λ,̺) with a +bilinear, alternating map +〈−,−〉 : A2 → A; +〈x, y 〉 + 〈y, x〉 = 0 +so that for any connection (∇,κ) on A, the maps +{v, x}(κ,∇) := κ′ ◦ T.̺ ◦ ∇(v, x), +{v, x, y }(κ,∇) := κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ (v, x, y ) +(3.2) +satisfy the equations +(i) ̺ ◦ 〈x, y 〉 + {̺x, y }(κ′,∇) = {̺y, x}(κ′,∇), +54 + +(ii) +� +γ∈Cy(3)〈xγ0,〈xγ1, xγ2〉〉 + +� +γ∈Cy(3){̺xγ0, xγ1, xγ2}(κ,∇) = 0. +There is also a local coordinates presentation of morphisms as in Section 2 of Martínez (2018). An +anchored bundle morphism A → B is a Lie algebroid morphism whenever +f β +γ Aγ +αδ + ̺i +δ +∂ f β +α +∂ x i = B β +θσ f θ +α f σ +δ + ̺i +α +∂ f β +δ +∂ x i . +The A and B arguments are understood as the brackets, so this condition can be rewritten as +f β +γ Aγ +αδ = f ◦ 〈α,δ〉, +B β +θσ f θ +α f σ +δ = 〈f ◦ α, f ◦ δ〉. +Set the following notation for maps between vector bundles with connection: +f : A → B +(κA,∇A,A,λA) +(κB,∇B,B,λB) +∇[f ] : A2 → B := κB ◦ T.f ◦ ∇A +(3.3) +The ̺ terms are understood to be the torsion, so that +̺i +δ +∂ f β +α +∂ x i = κ ◦ T.f ◦ ∇(̺eδ,eα) = ∇[f ](̺eδ,eα), +̺i +α +∂ f β +δ +∂ x i = κ ◦ T.f ◦ ∇(̺α,δ) = ∇[f ](̺eα,eδ), +using the notation set up in Equation 3.3. The notion of a Lie algebroid morphism, then, has the fol- +lowing presentation: +Proposition 3.1.14 (Martínez (2018)). Let (π : A → M ,̺A,〈−,−〉A),(q : B → N ,̺B,〈−,−〉B) be a pair of +Lie algebroids with chosen connections (κ−,∇−). An anchor-preserving vector bundle morphism f : A → +B is a Lie algebroid morphism if and only if +f ◦ 〈eα,eδ〉 + ∇[f ]◦ (̺eδ,eα) = 〈f ◦ α, f ◦ δ〉 + ∇[f ]◦ (̺eα,eδ). +3.2 +Anchored bundles +Anchored bundlesare toLie algebroidswhatreflexive graphsare togroupoids. Each theoryhas(mostly) +the same structure, but while a reflexive graph is missing a groupoid’s composition operation, an an- +chored bundle lacks a Lie algebroid’s bracket operation. This section reviews the basic theory of an- +chored bundles and their prolongations (see Mackenzie (2005) for more details). +Definition3.2.1. An anchored bundle in a tangent category isa differential bundle (A +π−→ M ,ξ,λ) equipped +with a linear morphism +A +T M +M +̺ +π +p +A morphism of anchored bundles is a linear bundle morphism (f ,v) that preserves the anchors +T A +T B +A +B +A +B +T M +T N +λ +l +f +T.f +ρ +̺ +f +T.v +The category of anchored bundles and anchored bundle morphisms in a tangent category � is written +Anc(�), and a generic anchored bundle is written (A +π−→ M ,ξ,λ,̺). +55 + +There are two pullbacks that are associated with every anchored bundle. These play the role of the +spaces of composable arrows G2 := G t ×s G ,G3 := G t ×sG t ×sG for a reflexive graph s,t : G → M . +Definition 3.2.2. Let (A +π−→ M ,ξ,λ,̺) be an anchored bundle. Its first and second prolongations are +given by the limits +� (A) +T A +� 2(A) +T 2A +A +T M +T A +T 2M +A +T M +̺ +T.π +⌟ +T 2.π +T.̺ +T.π +̺ +⌟ +(The notation for the fibre product is slightly non-standard, as it is not technically a pullback.) Through- +out this chapter, it will always be assumed that the first and second prolongations of an anchored bundle +exist (although no choice of prolongation is explicitly made). +Remark 3.2.3. It is not strictly necessary that the prolongations of an anchored bundle exist; this con- +dition is primarily a matter of convenience when discussing involution algebroids. Every result in this +section, that does not explicitly mention prolongations, holds for an anchored bundle independently of +their existence. +Example 3.2.4. +(i) For any object M , id : T M → T M is an anchor for the tangent differential bundle, and every +f : M → N yields a morphism of anchored bundles. Moreover, for any anchored bundle over M the +anchor is itself a morphism of anchored bundles (A,π,ξ,λ,̺) → (T M ,p,0,ℓ,id ). This induces a +fully faithful functor: +� �→ Anc(�). +This inclusion has a left adjoint, which sends an anchored bundle (π : A → M ,ξ,λ,̺) to its base +space M (the unit is the anchor map ̺), so that � is a reflective subcategory of Anc(�). +(ii) For any differential bundle (A,π,ξ,λ), the map 0 ◦ π : A → T M is an anchor and every morphism +f : A → B of differential bundles again yields a morphism of anchored bundles. The naturality of +0 ensures that every differential bundle morphism will preserve this trivial anchor map, giving a +fully faithful functor +DBun(�) �→ Anc(�). +This functor has a right adjoint that replaces the anchor map with the trivial anchor map +(π : A → M ,ξ,λ,̺) �→ (π : A → M ,ξ,λ,0◦ π) +where the counit is given by the natural idempotent e = ξ ◦ π : (A,λ) → (A,λ). It is trivial to check +that the anchor map is preserved by the bundle morphism (e ,id ): +̺ ◦ ξ ◦ π = 0◦ T.id ◦ π +and so the following diagram commutes: +T M +T M +A +A +0◦π +p◦λ +̺ +56 + +This means that differential bundles are a coreflective subcategory of anchored bundles. +(iii) Any reflexive graph (s,t : C → M ,e : M → C ) in a tangent category has an anchored bundle (when +sufficient limits exist), constructed as +C ∂ +T.C +T.C +e .T1 +T.e s +(where e s : C → C = e ◦ s). Construct a lift on C ∂ : +T.C ∂ +T 2C1 +T 2C1 +C ∂ +T.C1 +T.C1 +e .C1 +T.e s +ℓ +ℓ +T.e .C1 +T.T.e s +λ +This lift will be non-singular by the commutativity of T -limits. The pre-differential bundle data is +given by the projection +C ∂ �→ T.C +p.e s +−−→ M . +The section is induced by +M +(0.e ) +−−→ T.C +while post-composition with T.t gives the anchor map: +C ∂ �→ T.C +T.t +−→ T.C . +The diagram is a pullback by composition, and the outer perimeter defines the pullback � (A). Note +that any reflexive graph morphism will give rise to an anchored bundle morphism by naturality, +making the construction of an anchored bundle from a reflexive graph functorial. +(iv) For any object M in a tangent category, cM : T 2M → T 2M is an anchor on (T.p,T.0,c ◦ T.ℓ), and +every map f : M → N gives a morphism of anchored bundles. +(v) For any anchored bundle (q : E → M ,ξ,λ,̺), the differential bundle (T.q,T.ξ, +c ◦ T.λ) has an anchor +̺T : T E +T.̺ +−→ T 2M +c−→ T 2M . +The first prolongation of an anchored bundle � (A) behaves similarly to the second tangent bundle, +except that it does not have a canonical flip. In the definition of an affine connection, the tangent +bundle played a similar role to the “arities” of a theory. There is a lift map that makes this connection +stronger: +Definition 3.2.5. Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category �. We define a +generalized lift: +ˆλ : A → A̺×T πT A := (ξ ◦ π,λ). +This generalized lift satisfies the same coherences as the lift on the second tangent bundle: +Proposition 3.2.6. Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category �. It follows that +(i) (Coassociativity of ˆλ) (ˆλ × ℓ) ◦ ˆλ = (id × T.ˆλ) ◦ ˆλ; +57 + +(ii) (Universality) the following diagram is a T -pullback: +A2 +A̺×T πT A +M +A +ˆµ +π0 +π◦πi +ξ +⌟ +where ˆµ := (ξ ◦ π◦ π0,µ). +Proof. +(i) Compute +(ˆλ × ℓ) ◦ ˆλ = (ξ ◦ π◦ ξ ◦ π,λ ◦ ξ ◦ π,ℓ◦ λ) += (ξ ◦ π,T.(ξ ◦ π) ◦ λ,T.λ ◦ λ) += (π0,T.(ξ ◦ π) ◦ π1,T.λ ◦ π1) ◦ (ξ ◦ π,λ) += (id × T (ˆλ)) ◦ ˆλ. +(ii) Use the pullback lemma to observe that the following diagram is universal for any anchored bun- +dle: +A2 +� (A) +T A +M +A +T M +ξ +̺ +0 +T π +π0 +π◦π1 +ˆµ +π1 +µ +The top triangle of the diagram commutes by definition. The right square and outer perimeter +are pullbacks by definition, and the bottom triangle also commutes by definition. The pullback +lemma ensures that the left square is a pullback, so for every anchored bundle, the general lift is +universal for � (A). Now post-compose with the involution: +A2 +� (A) +� (A) +M +A +A +ξ +π0 +π◦π1 +ˆµ +id +pπ1 +σ +ˆν +It suffices to check that the top triangle commutes, so σ ◦ ˆµ = ν: +σ ◦ ˆµ ◦ (a,b ) = σ ◦ ((ξ ◦ π,0) ◦ a +π0 (ξ ◦ π,λ) ◦ b ) = (id ,T.ξ ◦ ̺ ◦ a) +pπ1 (ξ ◦ π,λ) ◦ b. +Thus, the lift (ξπ,λ) involution algebroid is universal for � (A). +Example 3.2.7. +(i) For T (M ) = T (M ), the space of prolongations is T (M )id ×T p T 2M = T 2M , and the second prolon- +gation is given by T 2M . +58 + +(ii) In a tangent category with a tangent display system, if π ∈ � then the prolongations for (π,ξ,λ,̺) +automatically exist. In particular, for every anchored bundle in the category of smooth manifolds, +all prolongations exist because the projection is a submersion (see Section 1.5). +(iii) For any differential bundle with the anchor 0 ◦ π, it follows that � (A) ∼= Aπ×π◦πi (A2) ∼= A3. The +universality of the vertical lift factors b into (a1,a2), as the following diagram is a pullback: +A3 +Aπ×πA +T A +A +M +T M +π0 +(π1,π2) +µ +ππi +T π +π +0 +(iv) Returning to the anchored bundle constructed from a graph, the space of prolongations � (A) em- +beds into the second tangent bundle of the space of composable arrows: +A̺×T πT A �→ T 2(G t ×sG ). +The category of anchored bundles is, in a sense, “tangent monadic” over the category of differential +bundles: the forgetful functor from anchored bundles to differential bundles “creates” T -limits and +the tangent structure (this is all made precise in Chapter 5 using an enriched perspective on tangent +categories). +Observation 3.2.8. A limit of anchored bundles is the limit of the underlying differential bundles. Be- +cause the anchor is preserved by every map in the diagram, this induces a natural transformation for +any D : � → Anc(C ): +� +Anc(�) +Anc +DBun(�) +D +S +U +̺ +Thus, limU .S.D computes the limit of the underlying objects, while limU .D computes the limit of the +underlying differential bundles in the diagram. The anchor/unit map, then, induces a differential bun- +dle map: +lim̺i : (limAi ,limλi ) → (T.(lim +i Mi),ℓ.(lim +i Mi)). +This is the limit in the category of anchored bundles (so long as pullback powers of the limit projection +and the two prolongations of the limit anchor bundle exist). +The tangent structure on lifts defined in Proposition 2.2.7 lifts to a tangent structure on anchored +bundles. +Lemma 3.2.9. The category of anchored bundles in a tangent category has a tangent structure that maps +objects as follows: +(π : A → M ,ξ,λ,̺) �→ (T.π : T A → T M ,T.ξ,c ◦ T.λ,c ◦ T.̺) +where the structure maps are all defined using the pointwise structure maps in �. +59 + +Proof. Given an anchored bundle (q : A → M ,ξ,λ,̺), there is an anchor on the differential bundle +(T q,T ξ,c ◦ T λ) given by c ◦ T ̺; the diagram commutes by naturality and the coherences on c ,ℓ. +T 2A +T 3M +T 3M +T A +T 2M +T 2M +T M +T M +T M +T q +T p +c ◦T λ +c ◦T ℓ +ℓ +c +p +T ̺ +T 2̺ +T c +The tangent structure maps and universality properties all follow from the forgetful property of the +functor from anchored bundles to differential bundles as a consequence of Observation 3.2.8. +For any anchored bundle A, there are two differential bundles associated to � (A). The first is the +usual pullback differential bundle given by pulling back T.π along ̺ as in Lemma 2.4.3: +A̺×T πT A +T A +A +T M +π1 +π0 +T π +̺ +This gives the differential bundle structure +(A̺×T πT A +π0 +−→ A,A +(id,T.ξ◦̺) +−−−−−−→ A̺×T πT A,A̺×T πT A +(0,c ◦T.λ) +−−−−−→ T (A̺×T πT A)). +Taking the fibre product in the category of anchored bundles as in Observation 3.2.8 yields the second +differential bundle structure: +(A,π,ξ,λ) +(̺,id) +−−−→ (T M ,p,0,ℓ) +(T π,π) +←−−− (T A,p,0,ℓ). +The two lifts behave similarly to the pair (T.ℓ,ℓ.T ) on the second tangent bundle, and (ℓ,λT ) on the +tangent bundle of a pre-differential bundle. +Lemma 3.2.10. Let � be a tangent category and Anc(�) the category of anchored bundles in �. If T - +pullback powers of p ◦π0,π1 : A̺×T πT A → A exist, then there are two differential bundles on � (A), with +lifts induced as +(� (A),λ × ℓ) +(T A,ℓ) +(� (A),0× (c ◦ T.λ)) +(T A,c ◦ T.λ) +(A,λ) +(T M ,ℓ) +(A,0) +(T M ,0) +̺ +T.π +⌟ +T.π +̺ +⌟ +with structure maps +1. (� (A) +p◦π1 +−−→ A,A +(ξπ,0) +−−−→ � (A),� (A) +λ×ℓ +−−→ T ◦ � (A)), +2. (� (A) +π0 +−→ A,A +(id,T.ξ◦̺) +−−−−−−→ � (A),� (A) +0×c ◦T.λ +−−−−→ T.� (A)). +Furthermore, the two lifts λ� and λ̺ commute: +c ◦ T.(λ × ℓ) ◦ (0× c ◦ T.λ) = T.(0× c ◦ T.λ) ◦ (λ × ℓ). +60 + +Proof. The two differential bundles exist as a consequence of Observation 2.4.2 and Lemma 2.4.3, re- +spectively. The commutativity of limits follows by postcomposition, as both c ◦T.λ,ℓ and λ,0 commute +by the differential bundle axioms. +The above lemma determines a functor, which we denote � , that sends an anchored bundle in � +to an anchored bundle in Lift(�) (equipped with the tangent structure from Proposition 2.2.7). +Proposition 3.2.11. There is a functor � from anchored bundles in � to anchored bundles in the cate- +gory of lifts Lift(�), that sends an anchored bundle (π : A → M ,ξ,λ,̺) to the tuple in Lift(�) +π� := (� (A),0× c ◦ T.λ) +p◦π1 +−−→ (A,0), +ξ� := (A,0) +(ξ◦π,0) +−−−→ (� (A),0× c ◦ T.λ) +λ� := (� (A),0× c ◦ T.λ) +(λ×ℓ) +−−→ (T (� (A)),c ◦ T (0× c ◦ T.λ), +̺� := (� (A),0× c ◦ T.λ) +π1 +−→ (T A,c ◦ T.λ) +(note that this functor lands in anchored bundles of non-singular lifts; see Definition 2.3.1). Morphisms +of anchored bundles +(f ,m) : (π : A → M ,ξ,λ,̺) → (q : E → N ,ζ,l ,δ) +are sent to +� (f ,m) := f Tm ×Tm T f : A̺×T πT A → E δ×T q T E +Proof. First, note that the tuple +(π� : � (A) → A,ξ� ,λ� ,̺� ) +is an anchored bundle, and that each morphism is a lift morphism: +• π� : (� (A),0× c ◦ T.λ) → (A,0) follows because +T.p ◦ T.π1 ◦ (λ × ℓ) = T.p ◦ ℓ◦ π1 = 0◦ p ◦ π1 +• ξ� : (A,0) → (� (A),0× c ◦ T.λ) follows since +(T.ξ ◦ T.π,T.0) ◦ 0 = (0◦ ξ ◦ π,T.0◦ 0) = (0◦ ξ ◦ π,c ◦ T.λ ◦ 0) = (0× c ◦ T.λ) ◦ (ξ ◦ π,0) +• λ� : (� (A),0× c ◦ T.λ) → (T.� (A),0× c ◦ T.(c ◦ T.λ)) follows by the commutativity of 0× c ◦ T.λ) +and λ × ℓ, and +• ̺� : (� (A),0× c ◦ T.λ) → (T A,c ◦ T.λ) is a lift by definition of ̺� = π1. +This gives an anchored bundle in the category of non-singular lifts in �. +Next, check that the mapping is functorial. To see that � (f ,m) preserves the lifts, note that (f ,m) +gives a morphism of diagrams for each pullback in Lift(�) defining the two lifts, so the induced map +� (f ,m) preserves each lift. To see that � (f ,m) preserves the anchor, check that +̺� B ◦ � (f ,m) = π1 ◦ (f × T.f ) = T.f ◦ π1 = T.f ◦ ̺� A. +61 + +Corollary 3.2.12. In a tangent category � where pullbacks along differential bundle projections exist, +such as a tangent category equipped with a proper retractive display system (Definition 1.5.3) like the +category of smooth manifolds, there is an endofunctor on the category of anchored bundles in �, +� ′ : Anc(�) → Anc(�) =U Lift.� . +Observation 3.2.13. T -Limits in Anc(Lift(�)) are computed as pointwise limits in � by Observations +3.2.8 and 2.4.2, and � is constructed as a limit, so it follows that � preserves T -limits. +Proposition 3.2.14. The prolongation endofunctor � ′ : Anc(�) → Anc(�) has natural transformations +• p ′ : � ′ ⇒ id +• 0′ : id ⇒ � ′ +• +′ : � ′p ′×p ′� ′ ⇒ � ′ +• ℓ′ : � ′ ⇒ � ′.� ′ +satisfying all of the axioms of a tangent category that do not incvolve the canonical flip (Definition 1.3.2). +(Sketch). Note that the full argument is given in Section 4.3, but it is not difficult to sketch it out here. +For the projection, zero map, and addition, use the differential bundle structure induced by Corollary +3.2.11. To see the lift axiom, note that up to a choice of pullback, we have: +� ′.� ′(π : A → M ,ξ,λ,̺) = + + + + + + + + + +(p ◦ π1,p ◦ π2) : +� 2(A) → � (A) +(ξ ◦ π◦ π0,0◦ π1,0◦ π2) : +� (A) → � 2(A) +(λ × ℓ× ℓ) : +� 2(A) → T.� 2(A) +(π1,π2) : +� 2(A) → T.� (A) +The “lift map” is then +ℓ′ : � ′ ⇒ � ′.� ′; � (A) +A×T.ˆλ +−−−→ � 2(A) +(where we recall that ˆλ = (ξ ◦ π,λ) : A → � (A)). +Remark 3.2.15. The structure described in Proposition 3.2.11 leads to the theory of double vector bun- +dles, developed by MacKenzie and his collaborators F. and Mackenzie (2019); Mackenzie (1992). A dou- +ble vector bundle is a commuting square +D +B +A +M +q D +A +q D +B +q A +q B +where each projection is a vector bundle projection, and “vertical” and “horizontal” orientations of the +square are each vector bundle homomorphisms. It was observed by Grabowski and Rotkiewicz (2009) +that this is equivalent to a pair of commuting �+-actions on the total space E ; following the development +in Chapter 2, this is a commuting pair of non-singular lifts on E , +λA,λB : E → T E , T.λB ◦ λA = c ◦ T.λA ◦ λB. +A proper exposition of the so-called “Ehresmann doubles” (Mackenzie (2011)) for the structures in Lie +theory would substantially expand the scope of this thesis, and so it has been relegated to the margins. +62 + +Finally, observe that a connection (see Section 2.6) on an anchored bundle’s underlying differential +bundle behaves similarly to an affine connection. +Lemma 3.2.16. Let (π : A → M ,ξ,λ,̺) be an anchored bundle with � (A) existing in a tangent category +�. If (π,ξ,λ) has a connection, then define +ˆκ := κπ1, +ˆ∇ := ∇(π0,̺π1) +and note that +(i) ˆκ : � (A) → A is a retract of (ξπ,λ), and ˆκ : (� (A),l ) → (A,λ) is linear for both l = (λ×ℓ),(0×c ◦T.λ); +(ii) ˆ∇ : A2 → � (A) is a section of (π0,p ◦ π1) and is bilinear; +(iii) there is an isomorphism � (A) ∼= A3: +∇(p ◦ π1,̺ ◦ π0) +(π,T.π) ˆµ(p ◦ π1, ˆκ) = id . +Example 3.2.17. Every vector bundle in the category of smooth manifolds has a connection; it follows +that Lemma 3.2.16 holds for every anchored bundle in the category of smooth manifolds. +Remark 3.2.18. The category of anchored bundles in a tangent category is almost a tangent category, +except that it lacks a symmetry map. The “differential objects” in such a category will act like a cartesian +differential category, except that the symmetry of mixed partial derivatives fails. There has been some +interest in settings for differentiable programming where the symmetry of mixed partial derivatives need +not hold (see Definition 3.4 along with the discussion at the end of Section 6 in Cruttwell et al. (2021)); +the category of anchored bundles in a tangent category appears to be a source of examples. +3.3 +Involution algebroids +Involution algebroids are a tangent-categorical axiomatization of Lie algebroids. Recall that a Lie al- +gebroid has almost all of the structure of the operational tangent bundle, in particular a Lie bracket +on sections that satisfies the Leibniz law so that it gives a directional derivative. In Proposition 3.2.14, +it was demonstrated that the category of anchored bundles have almost all of the same maps as the +tangent bundle, the only structure map missing being the canonical flip +c : T 2M ⇒ T 2M +which, we may recall from the chapter introduction (and Cockett and Cruttwell (2015), Mackenzie +(2013)), is used in constructing the Lie bracket of vector fields in a tangent category with negatives. +Thus, a natural next step is to add an involution map to an anchored bundle and require that it satis- +fies the same coherences as c from the tangent bundle. +Definition 3.3.1. An involution algebroid is an anchored bundle (A,π,ξ,λ,̺) equipped with a map +σ : � (A) → � (A) satisfying the following axioms: +(i) Involution: +� (A) +� (A) +� (A) +σ +σ +63 + +(ii) Double linearity: +T.� (A) +T.� (A) +� (A) +� (A) +σ +0×c ◦T.λ +T.σ +λ×ℓ +(iii) Symmetry of lift: +A +� (A) +� (A) +ˆλ=(ξ◦π,λ) +σ +ˆλ +(iv) Target: +� (A) +T A +T 2M +� (A) +T A +T 2M +σ +π1 +T.̺ +π1 +T.̺ +c +(v) Yang–Baxter: +� 2(A) +� 2(A) +� 2(A) +� 2(A) +� 2(A) +� 2(A) +σ×c +σ×c +σ×c +A×T.σ +A×T.σ +A×T.σ +A is an almost-involution algebroid if the involution does not satisfy the Yang–Baxter equation. A mor- +phism of involution algebroids is a morphism of anchored bundles, so that � (f )◦σA = σB ◦� (f ). (Note +that because σ is an isomorphism and σ = σ−1, +σ : (� (A),0× c ◦ T.λ) → (� (A),λ × ℓ) +is linear as well.) Write the category of involution algebroids and involution algebroid morphisms in � +is written Inv(�). +Observation 3.3.2. It is not immediately clear that the Yang–Baxter equation is well-typed. This follows +from the target axiom (iv) and the double linearity axiom (ii). Starting with (u,v,w ) : A̺×T πT AT.̺×T 2.π +T 2A, we see that σ × c is well-typed if and only if +T.̺ ◦ π1 ◦ σ(u,v) = T 2.π◦ c ◦ w = c ◦ T 2.π◦ w = c ◦ T.̺ ◦ v. +Similarly, 1 × T.σ is well-typed if T.̺ ◦ u = T.π ◦ π0 ◦ T.σ(v,w ); then use the double linearity axiom to +compute +T.π◦ π0 ◦ T.σ(v,w ) = T.π◦ T.p ◦ T.π1(v,w ) = T.π◦ T.p ◦ w += T.p ◦ T 2.π◦ w = T.p ◦ T.̺ ◦ v = T.π◦ v. +64 + +This perspective on Lie algebroids has already appearad in the work of Martinez and his collabora- +tors in de León et al. (2005), where a “canonical involution” was derived on space of prolongations of a +Lie algebroid using the formula +σ : � (A) → � (A);σ(x, y,z) = σ(y, x,z + 〈x, y 〉). +The structure of this map has been largely unexplored; helpfully, involution algebroids succeed in +reverse-engineering axioms for an involution map that will induce a Lie bracket on the sections of +the projection map. The bracket from the original Lie algebroid is induced using the same formula as +for the bracket of vector fields on the tangent bundle: +λ ◦ [X ,Y ]∗ = +� +(π1 ◦ σ ◦ (id ,T.X ◦ ̺) ◦ Y −p T.Y ◦ X ◦ ̺) −T.π 0Y +� +. +Furthermore, a morphism of anchored bundles is a Lie algebroid morphism if and only if it preserves +the derived involution map. +Example 3.3.3. +(i) For any M in �, (T M ,p,0,id ,c ) is an involution algebroid. Furthermore, for any involution alge- +broid anchored on M , (̺,id ) is a morphism of involution algebroids (by the target axiom). This +defines a fully faithful functor � �→ Inv(�). The same construction as the anchored bundle case in +Example 3.2.4(i) exhibits � as a reflective subcategory of Inv(�). +(ii) For any differential bundle, the trivial anchored bundle (π : A → M ,ξ,λ,0◦ π) has an involution +using the isomorphism � (A) ∼= A3, given by σ := (π1,π0,π2) (proving this map satisfies the involu- +tion algebroid axioms is just an exercise in combinatorics). It follows that every differential bundle +morphism gives rise to a morphism of these trivial involution algebroids. The same construction +from the anchored bundle case (Example 3.2.4(ii)) exhibits Diff(�) as a coreflective subcategory of +involution algebroids. +(iii) Consider a groupoid +s,t : G → M , e : M → G , (−)−1 : G → G , m : G2 → G . +The underlying reflexive graph has an associated anchored bundle, constructed in Example 3.2.4, +and the space of prolongations of this anchored bundle includes into (T 2.G2). Note that there is a +well-formed involution map: +σ(u,v) = c ◦ ((0◦ p ◦ v)−1;(T.0◦ u);v) +The direct proof of this involves a more conceptual construction, which is the focus of Section 5.5. +An involution algebroid resembles a generalized tangent bundle, and so the lift (ξ◦π,λ) : A → � (A) +satisfies the same universality conditions as ℓ : T ⇒ T 2. The double linearity condition is equivalent to +the naturality condition for c ,ℓ in Definition 1.3.2: +� (A) +� (A) +T 2 +T 2 +� 2(A) +� 2(A) +� 2(A) +T 3 +T 3 +T 3 +T.ℓ +c .T +T.c +c +ℓ.T +id×T.ˆλ +σ +σ×c +id×T.σ +ˆλ×ℓ +(3.4) +65 + +which, using string diagrams for monoidal categories (Selinger (2010)), is the equation += +where the circle denotes the lift and c is the crossing of two lines. +Proposition 3.3.4. Let (π : A → M ,ξ,λ,̺) be an anchored bundle, and suppose that +σ : � (A) → � (A) +satisfies the involution axiom (i) and the symmetry of lift axiom (iii),and furthermore that the involution +“exchanges” the idempotents associated to the two lifts (λ×ℓ) and (0×c ◦T.λ) on � (A) (Proposition 2.2.8) +σ ◦ ((p ◦ λ) × (p ◦ ℓ)) = (p ◦ 0) × (p ◦ c ◦ T.λ) = id × (T.p ◦ T.λ) +Then the double linearity axiom is equivalent to the left-hand commuting diagram in Diagram 3.4: +(ˆλ × ℓ) ◦ σ = (id × T.σ) ◦ (σ × c ) ◦ (id × T.ˆλ). +Proof. Starting with the left-hand side of the equation, +(id × T.σ) ◦ (σ × c ) ◦ (id × T.ˆλ) ◦ (u,v) += (id × T.σ) ◦ (σ × c ) ◦ (u,T.ξ ◦ T.π◦ v,T.λv) += (id × T.σ) ◦ (σ ◦ (u,T.ξ ◦ ̺ ◦ u),T.λv) += (id × T.σ) ◦ (ξ ◦ π◦ u,0◦ ̺ ◦ u,T.λv) += (id × T.σ) ◦ (ξ ◦ π◦ u,0◦ T.π◦ v,T.λv) += (ξ ◦ π◦ u,T.σ ◦ ˆλ◦ v). +For the right-hand side: +(ˆλ,ℓ) ◦ σ ◦ (u,v) += (ξ ◦ π◦ π0 ◦ σ ◦ (u,v),(λ × ℓ) ◦ σ ◦ (u,v)) += (ξ ◦ π◦ p ◦ v,(λ × ℓ) ◦ σ ◦ (u,v)) += (ξ ◦ p ◦ T.π◦ v,(λ × ℓ) ◦ σ ◦ (u,v)) += (ξ ◦ p ◦ ̺ ◦ u,(λ × ℓ) ◦ σ ◦ (u,v)) += (ξ ◦ π◦ u,(λ × ℓ) ◦ σ ◦ (u,v)) += (ξ ◦ π◦ u,T.σ ◦ (0× c ◦ T.λ) ◦ (u,v)). +So the naturality equation 3.4 is equivalent to +T.σ ◦ (0× c ◦ T.λ) = (λ × ℓ) ◦ σ. +The category of involution algebroids is “tangent monadic” over the category of anchored bundles, +in the same sense that anchored bundles are tangent monadic over the category of differential bundles, +or internal categories over reflexive graphs in a category. The “tangent monadicity” leads to a similar +observation about T -limits of involution algebroids as in Observation 3.2.8. +66 + +Observation 3.3.5. The forgetful functor from involution algebroids to anchored bundles creates limits; +that is to say, the T -limits of the underlying anchored bundles give the limits of involution algebroids. +Recall that by Observation 3.2.13, the limit � (limAi ) = lim� (Ai ) in the category of anchored bundles, +and this induces a map between objects in � +limσi : lim� (Ai ) → lim� (Ai ). +The axioms for an involution algebroid are induced by universality. +Note that a point-wise tangent structure may be defined following the anchored bundles example +from Lemma 3.2.9: +Proposition 3.3.6. For a tangent category �, the category of involution algebroids has a “point-wise” +tangent structure that maps objects as follows: +(π : A → M ,ξ,λ,̺,σ) �→ (T.π,T.ξ,c ◦ T.λ,T.̺,σT ) +where σT is defined as +σT := (1×c c ) ◦ T.σ ◦ (1×c c ) : � (AT ) → � (AT ). +(3.5) +Proof. The tangent structure on involution algebroids is inherited from the functor Inv(�) → Anc(�). +Thus, it suffices to give the involution map for the tangent involution algebroid. +Note that given (π : A → M ,ξ,λ,̺,σ), the space of prolongations on (T.π,T.ξ, +c ◦T.λ,c ◦T.̺) is T Ac ◦T.̺×T 2πT 2A. We can construct an isomorphism between the objects � (T A) and +T (� (A)) in � using the cospan isomorphism +T A +T 2M +T 2A +T A +T 2M +T 2A +c ◦T.̺ +T 2.π +T.̺ +T 2.π +id +c +c +thus inducing a map +T Ac ◦T.̺×T 2πT 2A +id×c +−−→ T (A̺×T πT A) +T.σ +−→ T (A̺×T πT A) +id×c +−−→ T Ac ◦T.̺×T 2πT 2A +which we call σT . The linearity and involution axioms follow by construction. +Now check the rest of the axioms. For the unit: +(1×c c ) ◦ T.σ ◦ (1×c c ) ◦ (T.(ξ ◦ π),c ◦ T.λ) += (1×c c ) ◦ T.σ ◦ (T (ξπ),T λ) = (1×c c ) ◦ (T.(ξ ◦ π),T.λ) = (T.(ξ ◦ π),c ◦ T.λ). +For the anchor: +T.̺T ◦ π1 ◦ σT = T (c ◦ T ̺) ◦ π1 ◦ (1×c c ) ◦ T.σ ◦ (1×c c ) += T.c ◦ c ◦ T 2.̺ ◦ T.π1 ◦ T.σ ◦ (1×c c ) = T.c ◦ c ◦ T.c ◦ T 2.̺ ◦ T.π1 ◦ (1×c c ) += c ◦ T.c ◦ c ◦ T 2̺ ◦ c ◦ π1 = c ◦ T c ◦ T 2.̺ ◦ π1 = c ◦ T.̺T ◦ π1. +The Yang–Baxter equation is straightforward to check. +The tangent bundle is a canonical involution algebroid on every object in a tangent category, and +the anchor induces a morphism from an involution algebroid to the tangent involution algebroid on +its base space. The anchor acts as a reflector from involution algebroids in � to � itself. +67 + +Proposition 3.3.7. Any tangent category � is a reflective subcategory of the category of involution alge- +broids in �. +Proof. First, observe that the inclusion of � into Inv(�) (Definition 3.3.1) is fully faithful because the +anchor on the tangent involution algebroid is id : T M → T M . This means that the only involution +algebroid morphisms T A → T B are pairs (T f , f ), f : A → B. Now consider the functor Inv(�) → � that +sends (π : A → M ,ξ,λ,̺,σ) to M : this gives an endofunctor S : Inv(�) → Inv(�) along with a natural +transformation ̺ : id ⇒ S, so that S.̺ = ̺.S = id , given by the anchor map. Thus, the category � is +the category of algebras for an idempotent monad on Inv(�). +Corollary 3.3.8. Let +�A = (π : A → M ,ξ,λ,̺,σ) +be an involution algebroid in a tangent category �. +(i) The morphism +(T.π,π) : T A → T M +is an involution algebroid morphism from the tangent involution algebroid on A to the tangent +involution algebroid on M . +(ii) The morphism +(̺,id ) : �A → T M +is an involution algebroid morphism. +If pullback powers of p ◦ π1 : � (A) → A exist, then +(p ◦ π1 : � (A) → A,(ξ ◦ π,0),(λ × ℓ)) +is a differential bundle; note that π0 acts as an anchor. If the prolongations exist, then +(p ◦ π1 : � (A) → A,(ξ ◦ π,0),(λ × ℓ),π0,σ′) +is an involution algebroid; this follows from computing the pullback in the category of involution alge- +broids (σ′ is induced as in Observation 3.2.8). +The above corollary puts an involution algebroid structure on the differential bundle (� (A),λ × ℓ). +Note that the map σ gives an isomorphism of differential bundles +(� (A),λ × ℓ) → (� (A),0× c ◦ T.λ). +Martinez observed that the canonical involution σ puts a unique Lie algebroid structure on (� (A),0× +c ◦T.λ), and σ isa uniquelydetermined isomorphism ofinvolution algebroids(see de León et al. (2005)): +Corollary 3.3.9. For an involution algebroid (π : A → M ,ξ,λ,̺,σ), the isomorphism of differential +bundles +σ : (� (A),0× c ◦ T.λ) → (� (A),λ × ℓ) +induces a second involution algebroid structure on � (A). +68 + +Recall that Proposition 3.2.14 sketched out a proof that the category of anchored bundles in � has +an endofunctor � ′ and natural transformations p ′,0′,+′,ℓ′ satisfying the axioms of a tangent structure; +this endofunctor and the natural transformations all lift to Inv(�). The involution map σ is the missing +piece that gives a tangent structure on Inv(�). The construction may be spelled out here at a big-picture +level, but the actual proof brings up tricky coherence issues that make up the bulk of Chapter 4. +Proposition 3.3.10. The category of involution algebroids in a tangent category � has a second tangent +structure, where the tangent functor is given by +� ′ : Inv(�) → Inv(�) +and the tangent natural transformations are given as in Proposition 3.2.14, with the canonical flip +σ′ : � ′.� ′ → � ′.� ′ := � 2(A) +1×T.σ +−−−→ � 2(A). +Starting with an involution algebroid, � ′(A) and � ′.� ′(A) are given by +� ′(A) + + + + + + + + + + + + + + + +π′ : +� (A) +p◦π1 +−−→ A +ξ′ : +A +(ξ◦π,0) +−−−→ � (A) +λ′ : +� (A) +λ×ℓ +−−→ T.� (A) +̺′ : +� (A) +π1 +−→ T A +σ′ : +� 2(A) +σ×c +−−→ � 2(A) +� ′.� ′(A) + + + + + + + + + + + + + + + + + +π′′ +� 2(A) +(p◦π1,p◦π2): +−−−−−−−→ � (A) +ξ′′ : +� (A) +(ξ◦π◦π0,0◦π1,0◦π2) +−−−−−−−−−−−→ � 2(A) +λ′′ : +� 2(A) +(λ×ℓ×ℓ) +−−−−→ T.� 2(A) +̺′′ : +� 2(A) +(π1,π2) +−−−→ T.� (A) +σ′′ : +� 3(A) +(σ×c ×c ) +−−−−−→ � 3(A) +(where � 3(A) = A̺×T πT AT.̺×T 2.πT.(A̺×T πT A)). The tangent natural transformations are given by +• p : � ′ ⇒ id ; � (A) +π0 +−→ A +• 0 : id ⇒ � ′; A +(id,T.ξ◦T.π) +−−−−−−−→ � ′(A) +• + : � ′ +2 ⇒ � ′; A̺×T.π◦πi T2A +id×+ +−−−→ � (A) +• ℓ : � ′(A) ⇒ � ′.� ′; � (A) +1×T.(ξ◦π,λ) +−−−−−−→ � 2(A) +• ̺′ : � ′ ⇒ T +• c : � ′.� ′(A) ⇒ � ′.� ′; � 2(A) +1×T.σ +−−−→ � 2(A). +Proof. Deferred to Section 4.5. +3.4 +Connections on an involution algebroid +In this section we take an involution algebroid with a chosen linear connection on its underlying an- +chored bundle (Definition 3.3.1, 2.6.1), and rederive Martinez’s structure equations for a Lie algebroid +(Proposition 3.1.5). +In a tangent category � with negatives, there is a natural transformation +n : T ⇒ T +69 + +making each fibred commutative monoid (p : T M ⇒ M ,0,+,n) a fibred abelian group. Because the +additive bundle structure on differential bundles is induced via universality (Proposition 2.4.4), in a +tangent category with negatives the additive bundle structure for differential bundles will likewise have +negatives. We adopt the following notation for the “fibred linear algebra” used in this section, as there +are a significant number of computations done on bundles with multiple choices of additive bundle +structure (e.g. the second tangent bundle of a differential bundle has three additive bundles structures). +Notation 3.4.1. Let E be an object with multiple differential bundle structures (πi : E → M i ,ξi,λi) +in a tangent category with negatives. Addition over a specific differential bundle is written using infix +notation, where a subscript is added to the symbol denoting the projection of the differential bundle. +Letting x, y : X → E denote a pair of generalized elements for which the πi -addition operation is well- +defined; we set +x +π[i] y = X +(x,y ) +−−→ E π[i]×π[i]E ++i +−→ E . +Similar notation is used for subtraction: +x −π[i] y = X +(x,y ) +−−→ E π[i]×π[i]E +id×n[i] +−−−−→ E π[i]×π[i]E ++i +−→ E . +Throughout this section, we work in a tangent category � with negatives and a chosen anchored +bundle (π : A → M ,ξ,λ,̺), equipped with a connection (κ,∇), whose base object has a torsion-free +connection (κ′,∇′) and a morphism +σ : � (A) → � (A) +that exchanges the two projection maps p ◦ π1,π0, meaning that the following diagram commutes: +� (A) +A +A +� (A) +π0 +p◦π1 +σ +p◦π1 +π0 +The following notation will be useful when working with local coordinates. +Notation 3.4.2. First, recall the ∇-notation for morphisms of differential bundle where each morphism +has a choice of connection from Equation 3.3: +f : A → B +(κA,∇A,A,λA) +(κB,∇B,B,λB) +∇[f ] := κB ◦ T.f ◦ ∇A : A2 → B +Let +(π : A → M ,ξ,λ,̺),(q : B → N ,ζ,l ,ρ) +be a pair of anchored bundles equipped with connections. “Hatting” a map f : � (A) → � (B) refers to +the map: +�f : A3 +ˆν(π0,π2)+ ˆ∇(π0,π1) +−−−−−−−−−−→ � (A) +f−→ � (B) +(π0,p◦π1,κ◦π1) +−−−−−−−−→ B3. +Similarly, for f : T A → T B, +�f : T M p ×πAπ×πA +ν(π0,π2)+∇(π0,π1) +−−−−−−−−−−→ T A +f−→ T B +(π0,p◦π1,κ◦π1) +−−−−−−−−→ T M p ×q B π×q B +70 + +Clearly, � +f ◦ g = �f ◦ �g . Similarly, a map A3 → B3 may be “barred” to form a map � (A) → � (B): +g : � (A) +(π0,p◦π1,κ◦π1) +−−−−−−−−→ A3 +g−→ B3 +ˆν(π0,π2)+ ˆ∇(π0,π1) +−−−−−−−−−−→ � (B). +It is straightforward to see that �f = f , �g = g . +Lemma 3.4.3. σ : � (A) → � (A) induces a bracket on Γ(π): +λ ◦ [X ,Y ] = (σ ◦ (id ,T X ◦ ̺) ◦ Y −T π (id ,T Y ◦ ̺) ◦ X ) − 0◦ Y. +Proof. Let X ,Y ∈ Γ(π) and compute: +p ◦ π1 ◦ (σ ◦ (id ,T X ◦ ̺) ◦ Y −π0 (id ,T Y ◦ ̺) ◦ X ) += p ◦ (π1 ◦ σ ◦ (id ,T X ◦ ̺) ◦ Y −T π T Y ◦ ̺ ◦ X ) += p ◦ π1 ◦ σ ◦ (id ,T X ◦ ̺) ◦ Y −π p ◦ T Y ◦ ̺ ◦ X += π0 ◦ (id ,T X ◦ ̺) ◦ Y − Y ◦ p ◦ ̺ ◦ X += Y − Y = ξ, +π0 ◦ (σ ◦ (id ,T X ◦ ̺) ◦ Y −π0 (id ,T Y ◦ ̺) ◦ X ) += π0 ◦ σ ◦ (id ,T X ◦ ̺) ◦ Y −π π0 ◦ (id ,T Y ◦ ̺) ◦ X += p ◦ π1 ◦ (id ,T X ◦ ̺) ◦ Y −π X += p ◦ T X ◦ ̺ ◦ Y −π X = X −π X = ξ. +The universality of the lift induces a new section [X ,Y ] so that +λ ◦ [X ,Y ] = (σ ◦ (id ,T X ◦ ̺) ◦ Y −T π (id ,T Y ◦ ̺) ◦ X ) − 0Y . +Definition 3.4.4. The map σ : � (A) → � (A) is linear whenever the two bundle morphisms +σ : (� (A),λ × ℓ) → (� (A),0× c ◦ T λ) σ : (� (A),0× c ◦ T λ) → (� (A),λ × ℓ) +are linear, and cosymmetric if +σ ◦ �λ = �λ = (ξπ,λ). +Note that whenever σ is linear and cosymmetric, σ ◦ ˆµ(u,v) = ˆν(u,v). +Linearity and unit axioms, along with the connection on the differential bundle, force the existence +of a bilinear bracket 〈−,−〉 as in the definition of a Lie algebroid in Proposition 3.1.13. +Proposition 3.4.5. For an anchored bundle (π : A → M ,ξ,λ,̺) with connection (κ,∇), a cosymmetric +and bilinear σ is equivalent to a bilinear 〈−,−〉 : A2 → A, with the correspondence given by +〈−,−〉 : κ ◦ σ ◦ ∇−π κ ◦ ∇ +σ : � (A) −−−−−−−−−−→ +(π1,π0,π2+〈π0,π1〉) +� (A) +71 + +Proof. Derive +�σ ◦ (x, y,z) = (y, x,κ ◦ σ ◦ (∇(̺x, y ) +p◦π1 µ(y,z))) += (y, x,κ ◦ σ ◦ ∇(̺x, y ) +π κ ◦ σ ◦ µ(y,z)) += (y, x,κ ◦ σ ◦ ∇(̺x, y ) +π κ ◦ ν ◦ (y,z)) += (y, x,κ ◦ σ ◦ ∇(̺x, y ) +π z) +=: (y, x,〈x, y 〉 +π z) +where 〈−,−〉 is certainly bilinear. The converse is immediate: take +�σ(u,v,w ) := (v,u,w + 〈u,v〉). +It is easy to see that σ is linear and cosymmetric. +The linear bracket must be involutive for the bilinear bracket to be alternating. +Proposition 3.4.6. If σ is cosymmetric and linear, then the bilinear bracket 〈−,−〉 is alternating if and +only if σ ◦ σ = id . +Proof. First, note that σ ◦ σ = id if and only if �σ ◦ �σ = id . Then check that +�σ ◦ �σ(u,v,w ) = �σ(v,u,w + 〈u,v〉) = (u,v,w + 〈u,v〉 + 〈v,u〉). +By the cancellativity of addition on A, +w = w + 〈u,v〉 + 〈v,u〉 ⇐⇒ 0 = 〈u,v〉 + 〈v,u〉. +Observation 3.4.7. A bilinear 〈−,−〉 on an anchored bundle with a connection induces the maps +{v, x}(κ,∇) := κ′ ◦ T.̺ ◦ ∇(v, x), +{v, x, y }(κ,∇) := κ ◦ T (〈−,−〉(κ,∇)) ◦ ∇A2 ◦ (v, x, y ) +from Equation (3.2) in Proposition 3.1.13. +Proposition 3.4.8. Let σ be cosymmetric and bilinear, with the associated bracket 〈−,−〉. Then +T ̺ ◦ π1 ◦ σ = c ◦ T ̺ ◦ π1 +if and only if the Leibniz equation is satisfied: +̺ ◦ 〈u,v〉 + {̺v,u} = {̺u,v.} +(3.6) +Proof. Following the given notation and using the hypothesis that the connection is torsion-free on +M , +� +T ̺ ◦ π1(u,v,w ) = (̺u,̺v,w + {u,v}) +�c (x, y,z) = (y, x,z). +Computing each side, +� +T ̺ ◦ � +π1 ◦ �σ ◦ (u,v,w ) = � +T ̺ ◦ � +π1 ◦ (v,u,w + 〈u,v〉) += � +T ̺ ◦ (̺v,u,w + 〈u,v〉) += (̺v,̺u,̺w + ̺〈u,v〉 + D[̺](̺v,u)) += (̺v,̺u,̺w + ̺〈u,v〉 + {v,u}), +�c ◦ � +T ̺ ◦ � +π1(u,v,w ) = �c (̺u,̺v,̺w + {u,v}) += (̺v,̺u,̺w + {u,v}), +so it follows that the two terms are equal if and only if the desired equality holds. +72 + +Lemma 3.4.9. Let σ be linear and cosymmetric. Then +(i) T (〈−,−〉) : T A2 → T A satisfies +� +T (〈−,−〉)(ax,uy ,uz,ux y ,ux z) += (ax ,〈uy ,uz〉,{ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉); +(ii) T.σ satisfies +� +(id × T (�σ))(ux ,uy ,ux y ,uz,ux z,uy z,ux y z) += (ux ,uz,ux z,uy ,ux y ,uy z + 〈uy ,uz〉, +ux y z + {ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉). +Proof. +(i) By the universality of the vertical lift and bilinearity of 〈−,−〉, the outer squares below are pull- +backs: +A4 +A2 +T (A) +T (A) +T M +T M +M +M +µA2 +(〈π0,π1〉,〈π0,π3〉+〈π1,π2〉) +µA +T 〈,〉 +T π◦T πi +T π +0 +0 +so that +T (〈,〉)(0,uy ,uz,ux y ,ux z) = µA(〈uy ,uz〉,〈uy ,ux z〉 + 〈ux y ,uz〉). +Now we compute +T (〈,〉)(µA2((uy , yz),(ux y ,ux z)) +p ∇A2(ax,(uy ,uz))) += T (〈,〉)µA2((uy ,uz),(ux y ,ux z)) +p T (〈,〉) ◦ ∇(ax,(uy ,uz)) += µA(〈uy ,uz〉,〈uy ,ux z〉 + 〈ux y ,uz〉) +p T (〈,〉) ◦ ∇(ax,(uy ,uz)) +and then postcompose this with (T π,p,κ) to obtain +(0,〈uy ,uz〉, 〈uy ,ux z〉 + 〈ux y ,uz〉) +p (ax,〈uy ,uz〉,{ax,uy , yz}) += (ax,〈uy ,uz〉,{ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉). +(ii) Consider the following diagram: +T M p ×πA3π×πA3 +T M p ×πA3π×πA3 +T (A3) +T (A3) +T ◦ � (A) +T ◦ � (A) +� +T �σ +∇A3+p µA3 +T �σ +T (π1,π0,π2+〈π0,π1〉) +T (∇′+π0µ′)) +(T (π3),p,κA3) +T σ +T (π0,p◦π1,κ◦π1) +73 + +We want to find � +T �σ = +� +T (π1,π0,π2 + 〈π0,π1〉). Note that +T (π1,π0,π2 +π 〈π0,π1〉) = T (π1,π0,π2) +T π T (ξππi ,π0,〈π0,π1〉). +The left term is straightforward: +� +T (π0,π1,π2)(ax ,(uy ,ux y ),(uz,ux z),(uy z,ux y z)) += (ax,(uz,ux z),(uy ,ux y ),(uy z,ux y z)) +and for the right term, use part (i) of this lemma: +� +T 〈−,−〉◦ � +T (π0,π1))(ax,(uy ,ux y ),(uz,ux z),(uy z,ux y z)) += � +T 〈−,−〉(ax,uy ,ux y ,uz,ux y z) += (ax ,〈uy ,uz〉,{ax,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉) +Then compute +� +T �σ(ax ,uy ,ux y ,uz,ux z,uy z,ux y z) += (ax ,uz,ux z,uy ,ux y ,uy z + 〈uy ,uz〉,ux y z + q) +where q = {ax ,uy , yz} + 〈uy ,ux z〉 + 〈ux y ,uz〉, giving the desired equation. +Proposition 3.4.10. Let σ be cosymmetric, doubly linear, and involutive, and satisfy the target axiom. +Then σ satisfies the Yang–Baxter equation if and only if 〈−,−〉 and {−,−,−} satisfy the Bianchi identity: +0 = +� +γ∈Cy(3) +〈xγ0,〈xγ1, xγ2〉〉 + +� +γ∈Cy(3) +{̺xγ0, xγ1, xγ2.} +(3.7) +Proof. We expand � +id × T σ and � +σ × c . Start with T (id × σ), which was derived in Lemma 3.4.9: +σ1(u) = +� +(id × T (�σ))(ux,uy ,ux y ,uz,ux z,uy z,ux y z) += (ux ,uz,ux z,uy ,ux y ,uy z + 〈uy ,uz〉, +ux y z + 〈uy ,ux z〉 + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz}). +Using the fact that κ′ is torsion free, so ˆc (ux z,uy z,ux z y ) = (uy z,ux z,ux y z), it follows that +σ2(ux,uy ,ux y ,uz,ux z,uy z,ux y z) = (uy ,ux,ux y + 〈ux,uy 〉,uz,uy z,ux z,ux y z). +Then compute +σ2σ1σ2(u) += +� +uz,uy ,uy z + 〈uy ,uz〉,ux,ux z + 〈ux,uz〉,ux y + 〈ux ,uy 〉,z1 +� +, +σ1σ2σ1(u) += +� +uz,uy ,uy z + 〈uy ,uz〉,ux,ux z + 〈ux,uz〉,ux y + 〈ux ,uy 〉,z2 +� +. +74 + +Note that the first five terms are equal, so it suffices to check z1 = z2 for z1 = π6σ1σ2σ1(u),z2 = +π6σ2σ1σ2(u). +z1 = ux y z + 〈uy ,ux z〉 + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } ++ 〈ux,uy z + 〈uy ,uz〉〉 + 〈ux z + 〈ux ,uz〉,uy 〉 += ux y z + 〈uy ,ux z〉 + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } ++ 〈ux,uy z〉 + 〈ux,〈uy ,uz〉〉 + 〈ux z,uy 〉 + 〈〈ux ,uz〉,uy 〉 += ux y z + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } ++ 〈ux,uy z〉 + 〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉, +z2 = ux y z + 〈ux,uy z〉 + {̺ ◦ uy ,ux,uz} + 〈ux y + 〈ux ,uy 〉,uz〉 += ux y z + 〈ux,uy z〉 + {̺ ◦ uy ,ux,uz} + 〈ux y ,uz〉 + 〈〈ux,uy 〉,uz〉. +So z1 = z2 is equivalent to requiring +ux y z + 〈ux y ,uz〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } ++ 〈ux,uy z〉 + 〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉 += ux y z + 〈ux,uy z〉 + {̺ ◦ uy ,ux,uz} + 〈ux y ,uz〉 + 〈〈ux,uy 〉,uz〉; +cancelling alike terms, this is equivalent to +〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉 + {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } += {̺ ◦ uy ,ux,uz} + 〈〈ux,uy 〉,uz〉. +Using the fact that 〈−,−〉 and {−,−,−} are alternating in the last two arguments, this is equivalent to +0 = 〈ux,〈uy ,uz〉〉 + 〈〈ux,uz〉,uy 〉 + 〈uz,〈ux,uy 〉〉 ++ {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } + {̺ ◦ uy ,uz,ux} += 〈ux,〈uy ,uz〉〉 + 〈uy ,〈uz,ux〉〉 + 〈uz,〈ux,uy 〉〉 ++ {̺ ◦ ux,uy ,uz} + {̺ ◦ uz,ux,uy } + {̺ ◦ uy ,uz,ux} +giving the desired identity. +Corollary 3.4.11. Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category with negatives, +with anchored connection (∇,κ) on A and torsion-free affine connection (∇′,κ′) on M . An involution +algebroid structure on A is equivalent to a bilinear map +〈−,−〉 : A2 → A +with derived maps +{−,−} : Aπ×p T M → T M := κ′ ◦ T ̺ ◦ (π0,π1), +{−,−,−} : T M p ×π◦πi A2 → A;{a,u1,u2} := κ ◦ T (〈−,−〉) ◦ (∇(a,u1),∇(a,u2)) +satisfying +(i) 〈−,−〉 is linear and cosymmetric, +(ii) 〈−,−〉 is alternating, +75 + +(iii) 〈−,−〉 and {−,−} satisfy the Leibniz equation, Equation 3.6. +(iv) 〈−,−〉,{−,−}, and {−,−,−} satisfy the Bianchi identity, Equation (3.7). +Morphisms of involution algebroids may also be characterized by preservation of the tensor. +Proposition 3.4.12. Let A,B be a pair of involution algebroids with chosen connections in a tangent +category with negatives. Then an anchored bundle morphism f : A → B is an involution algebroid +morphism if and only if (recalling the notation from Equation 3.3) +∇[f ](x, y ) + 〈f ◦ x, f ◦ y 〉 = ∇[f ](y, x) + f ◦ 〈x, y 〉. +Proof. Note that +(σ ◦ � (f ) = � (f ) ◦ σ) +⇐⇒ (π1,π0,π2 + 〈π0,π1〉) ◦ (f ◦ x, f ◦ y, f ◦ z + ∇[f ](x, y )) += (f , f , f ◦ π2 + ∇[f ](π0,π1)) ◦ (y, x,z + 〈x, y 〉) +while the second condition reduces to +∇[f ](x, y ) + 〈f ◦ x, f ◦ y 〉 = ∇[f ](y, x) + f ◦ 〈x, y 〉. +3.5 +The isomorphism of Lie and involution algebroid categories +Sections 3.1 and 3.4 have made the relationship between involution algebroids and Lie algebroids clear. +It is important to note that while the proofs used connections as a tool to identify the local coherences +satisfied by involution and Lie algebroids, the construction of a Lie algebroid from an involution alge- +broid (or vice versa) is independent of the choice of connection. +Theorem 3.5.1. There is an isomorphism of categories between Lie algebroids and involution algebroids +in smooth manifolds. +Proof. For the equivalence of categories, note that by Propositions 3.1.13 and 3.1.14, Corollary 3.4.11, +and Proposition 3.4.12 there is an isomorphism of categories between involution algebroids with a +choice of connection and Lie algebroids with a choice of connection (morphisms are not restricted to +connection preserving morphisms). This allows us to chain together isomorphisms +Inv(SMan) ∼= Inv(SMan)ChosenConn ∼= LieAlgdChosenConn ∼= LieAlgd. +To complete the proof, we must show that the assignment that sends an involution algebroid to a +Lie algebroid whose bracket is given by +λ ◦ [X ,Y ]∗ = +� +(π1 ◦ σ ◦ (id ,T.X ◦ ̺) ◦ Y −p T.Y ◦ ̺◦) −T.π 0◦ Y +� +, +(3.8) +is a bijection on objects, which brings up some subtleties. First, while an involution map +σ : A̺×T πT A → A̺×T πT A +76 + +is defined with respect to a particular choice of pullback A̺ ×T πT A, the category of involution alge- +broids does not distinguish between different choices of this pullback (and therefore different repre- +sentations of the map σ), and it is not part of the data of an involution algebroid. It is immediate by +universality that the left-hand-side of Equation 3.8 is independent of the choice of pullback A̺×T πT A. +Now, recall thatthe canonical involution ofa Lie algebroid isuniquelybyTheorem 4.7ofde León et al. +(2005) (this was also mentioned in Corollary 3.3.9). Once we make a choice of prolongation A̺×T πT A, +we have made a choice of pullback � (A) in LieAlgd, which uniquely determines the canonical involu- +tion +σ : A̺×T πT A → A̺×T πT A. +While the exact map σ depends on the choice of pullback A̺ ×T π T A, they all determine the same +involution algebroid, thus proving the bijection correspondence of objects. +77 + +Chapter 4 +The Weil nerve of an algebroid +The first three chapters of this thesis demonstrated that tangent categories allow for an essentially alge- +braic description of Lie algebroids by axiomatizing the behaviour of the tangent bundle, and showing +that a Lie algebroid over a manifold M is a “generalized tangent bundle”, namely an involution al- +gebroid. This chapter will make precise the sense in which an involution algebroid is a generalized +tangent bundle, by showing that the category of involution algebroids in a tangent category � is equiv- +alent to a certain tangent-functor category from the free tangent category over a single object to �, or +more generally that there is a fully faithful functor +Inv(�) �→ TangLax(FreeTangCat(∗),�). +This functor, the Weil nerve of an involution algebroid, builds a functor from the free tangent category +over a single object to � using a span construction. This chapter primarily builds on two pieces of work: +Leung’s construction of the free tangent category Weil1 (Leung (2017)) , and Grothendieck’s original +nerve construction (first published in Segal (1974)). +To understand Leung’s construction of the free tangent category, and more generally his actegory- +theoretic presentation of tangent categories (Section 4.2), we first look at Weil’s original insight relating +the kinematic and operational descriptions of the tangent bundle in SMan. The definition of a tangent +vector on a manifold M as an equivalence class of curves (Definition 1.2.8) puts a bijective correspon- +dence between tangent vectors and �-algebra homomorphisms from the ring of smooth functions +C ∞(M ) to the ring of dual numbers: +C ∞(M ) → �[x]/x 2. +The hom-set �Alg(C ∞(M ),�[x]/x 2) is precisely the set of derivations on C ∞(M ), which defines the +operational tangent bundle discussed in Definition 1.2.16: there is a natural smooth manifold structure +on this set. The Weil functor formalism, most notably developed in Kolár et al. (1993), extends this ob- +servation to a general class of endofunctors on SMan. For example, the fibre product T2M corresponds +to �-algebra morphisms, +C ∞(M ) → R[x, y ]/(x 2, y 2, x y ), +while the second tangent bundle corresponds to �-algebra morphisms into the tensor product, +C ∞(M ) → R[x]/(x 2) ⊗ R[y ]/(y 2) = R[x, y ]/(x 2, y 2). +By applying Milnor’s exercise (Problem 1-C Milnor and Stasheff (1974)), which states that the C ∞ func- +tor +SMan → �Algop; M �→ C ∞(M ) = SMan(M ,�) +78 + +is fully faithful, the structure maps occur as natural transformations. For example, the tangent projec- +tion is induced by the morphism +p : �[x]/x 2 a+b x�→a +−−−−−→ �, +so that +T M +p−→ M = [C ∞(M ),�[x]/x 2] +(p)∗ +−→ [C ∞(M ),�]. +The zero map and addition are similarly induced by +0 : � +a�→a+0x +−−−−−→ R[x]/x 2 and + : R[x, y ]/(x 2, y 2, x y ) +a+b x+c y +−−−−−−−→ +�→a+(b+c )x R[x]/x 2, +respectively, while the lift and flip are induced by the morphisms +ℓ : �[x]/x 2 +a+b x +−−−−−→ +�→a+b x y �[x, y ]/(x 2, y 2) +and +c : �[x, y ]/(x 2, y 2) +a+b x+c y +d x y +−−−−−−−−−−−→ +�→a+c x+b y +d x y �[x, y ]/(x 2, y 2). +More generally, there is a monoidal category of Weil algebras (Definition 4.1.1) which has a monoidal +action on the category of smooth manifolds. The Weil functor formalism, then, studies differential ge- +ometric structures from the perspective of the endofunctors and natural transformations induced by +this action. Leung’s insight is that there is an analogous category of commutative rigs1 built by replac- +ing �[x]/x 2 with �[x]/x 2, called Weil1 (Definition 4.1.3); a tangent structure is precisely a monoidal +action by Weil1 satisfying some universal properties. In particular, this category Weil1 is precisely the +free tangent category over a point, FreeTang(∗), so that every object A in a tangent category � deter- +mines a strict tangent functor +T (−)A : Weil1 → �;V �→ T V A +and morphisms f : A → B are in bijective correspondence with tangent-natural transformationsT (−)A ⇒ +T (−)(B). +The axioms of an involution algebroid in a tangent category � correspond bijectively with those +of the tangent bundle - this suggests that an involution algebroid should determine a tangent functor +from Weil1 to �. A first guess would lead one to think that p : �[x]/x 2 → � is sent to π : A → M , 0 to +ξ, and + to +A. As the space of prolongations � (A) = A̺ ×T πT A plays the role of the second tangent +bundle, we can see that +ℓ : N [x]/x 2 → N [x, y ]/(x 2, y 2) �→ (ξ ◦ π,λ) : A → � (A), +and +c : N [x, y ]/(x 2, y 2) → N [x, y ]/(x 2, y 2) �→ σ� (A) → � (A). +This pattern may be neatly summed up using span composition - we will construct a functor that sends +�[x]/x 2 to the span +A +M +T M +̺ +π +1A rig is a ring without negatives, i.e. a commutative monoid equipped with a bilinear multiplication. +79 + +and the tensor product �[x]/x 2 ⊗ �[x]/x 2 to the span composition (e.g. the pullback) +� (A) +A +T A +M +T M +T 2M +̺ +π +T.π +T.̺ +⌟ +which is the space of prolongations. This leads to the first major result of this chapter, the Weil Nerve +(Theorem 4.3.9), which states there is a fully faithful embedding +NWeil : Inv(�) �→ [Weil1,�]. +This bears a strong similarity to Grothendieck’s original nerve theorem, which takes an internal cate- +gory s,t : C → M and constructs a functor ∆op → � (where ∆op is the monoidal theory of an internal +monoid) by sending tensor to span composition, and the composition and unit maps given span mor- +phisms +C2 +M +M +M +M +M +C +C +s◦π0 +t ◦π1 +s +t +m +s +t +e +while the unit and associativity axioms for a category are exactly the unit and associativity laws for +a monoid in this setting. The Segal conditions identify exactly the functors C : ∆op → � that lie in +the image of the nerve functor N as those whose [n]t h object is sent to the wide pullback C ([n]) = +C [2]t ×s C [2]... t ×s C [2]. The corresponding result for involution algebroids is found in Theorem 4.4.8, +which states that a tangent functor (A,α) : Weil1 → � is the nerve of an involution algebroid if and +only if A preserves tangent limits and α is a T -cartesian natural transformation (this forces A.(W ⊗n) = +A̺×T.πT A ...T n−1̺×T nπT nA). The similarity between the Weil nerve and Grothendieck/Segal’s nerve +runs deep, and in Chapter 5 we demonstrate that the enriched perspective on tangent categories puts +these both into the same formal framework. +Sections4.1 and 4.2give a detailed introduction tothe Weil functorformalism (Kolár et al.(1993),Bertram and Souvay +(2014)) and Leung’s unification of Weil functors with tangent categories Leung (2017). The rest of the +chapter contains contains new results developed by the author. Section 4.3 proves the embedding +part of the Weil nerve theorem, that the category of involution algebroids embeds into the category +of tangent functors and tangent natural transformations [Weil1,�]. Section 4.4 identifies exactly those +tangent functors (A,α) : Weil1 → � that are the nerve of an involution algebroid, completing the proof +of the Weil nerve theorem. Section 4.5 uses the Weil nerve to develop a novel tangent structure on the +category of involution algebroids in a tangent category (in particular, the category of Lie algebroids will +have this novel tangent structure). +4.1 +Weil algebras and tangent structure +This section gives a more thorough introduction to the Weil functor formalism of Kolár et al. (1993), +and in particular how the structure maps of a tangent category may be teased out of it. We begin by +80 + +introducing Weil algebras, and the prolongation of a smooth manifold by a Weil algebra. (The relation- +ship with prolongations of involution algebroids from Definition 3.2.1 will be made clear in Section +4.3.) +Definition 4.1.1. An �-Weil algebra2 is a finite-dimensional �-algebra V so that V = � ⊕ ˙V as �- +modules and ˙V is a nilpotent ideal. The category �Weil is the full subcategory of �Alg spanned by the +�-Weil algebras. The prolongation of a manifold by a Weil algebra V is given by the manifold +T V M := �Alg(C ∞(M ,�),V ). +Eck (1986) showed that every product-preserving endofunctor on the category of smooth mani- +folds is constructed as the Weil prolongation by some Weil algebra. Consequently, �-algebra homo- +morphisms induce natural transformations between these product-preserving endofunctors on the +category of smooth manifolds. +Example 4.1.2. Consider the following �-Weil algebras and their associated prolongation functors. +(i) Prolongation by � induces the identity functor, and the tangent bundle is given by �[x]/x 2. The +tangent projection, then, is equivalent to the �-algebra morphism +p : �[x]/x 2 → �; p(a + b x) = a +while the 0-map induces the zero vector field: +0 : � → �[x]/x 2; 0(a) = a + 0x. +(ii) The algebra �[xi ]1≤i≤n/(xi x j )1≤i≤j ≤n = (�[x]/x 2)n is the wide pullback TnM = T M p×pT M ...p×p +T M . In particular, prolongation by �[x, y ]/(x 2, y 2, x y ) gives the bundle T2M = T M p×pT M . The +�-algebra morphism ++ : �[x, y ]/(x 2, y 2, x y ) → R[x]/x 2; +(a0 + a1x + a2y ) = a0 + (a1 + a2)x +corresponds to the addition of tangent vectors. +(iii) The algebra �[x, y ]/(x 2, y 2) = (�[x]/x 2) ⊗ (�[x]/x 2) is the second tangent bundle T 2M = T T M . +The vertical lift T → T 2 is induced by the morphism +ℓ : �[x]/x 2 → �[x, y ]/(x 2, y 2); ℓ(a + b x) = a + b x y. +(iv) The monoidal symmetry map induces c : T 2 ⇒ T 2, as follows: +c : (�[x]/x 2) ⊗ (�[y ]/y 2) → (�[y ]/y 2) ⊗ (�[x]/x 2); +c (a + b1x + b2y + b3x y ) = a + b2x + b1y + b3x y. +(v) For n ≥ 2, the algebra �[x]/x n gives the n-jet bundle. Note that this is the equalizer of ⊗n�[x]/x 2 +by the symmetry actions of Sn. +2Not to be confused with the normal usage of “Weil algebra” in Lie theory, e.g. Meinrenken and Pike (2021). +81 + +Further examples may be found in the monograph Kolár et al. (1993). Tangent categories bridge +the gap between the Weil functor approach to studying the differential geometry of smooth manifolds +and the synthetic differential geometry approach of axiomatizing a tangent bundle using nilpotent in- +finitesimals. The main structure axiomatized here is that of monoidal action of a symmetric monoidal +category on a category M × � → �, or equivalently, a lift from a category to the category of complexes +� → [M ,�], which involves translating a bit of classical category theory to the 2-categorical setting. +Example 4.1.2 leads to the classical theorem that the category of smooth manifolds has an action by +the category of �-Weil algebras that preserves all connected limits that exist. These “natural” universal +properties (in the sense of Kolár et al. (1993)) is foundational to synthetic differential geometry; see, +for example, Chapter Two of Lavendhomme (1996). Unfortunately, Weil algebras are not an ideal syn- +tactic presentation: they are not a finitely presentable category, and it is not immediately clear when +a diagram is a connected limit.3 Moving from �-algebras to commutative rigs and restricting to an +appropriate subcategory solves this problem. +Definition 4.1.3 (Definition 3.1 Leung (2017)). The category Weil1 is defined to be the full subcategory +of commutative rigs, CRig, generated by the rig of dual numbers W := �[x]/x 2, constructed as follows: +1. Start with finite product powers of W in CRig, and make a strict choice of presentation: +W0 = �, Wn := �[xi ]/(xi x j )i≤j ,0 ≤ i < n. +2. Then take the closure of Wn under coproduct of commutative rigs, written ⊗. Again, make a strict +choice of presentation: +Wn(0) ⊗ ...⊗ Wn(m−1) := �[xi,j ]/(xi j xik)j ≤k,0 < i < m,0 < j < n(i). +Note that we will often suppress the tensor product ⊗ and simply write +U V :=U ⊗ V. +Proposition 4.1.4 (Definition 3.3 Leung (2017) ). +(i) The category Weil1 is a symmetric strict monoidal category with unit � and coproduct ⊗. +(ii) � is a terminal object in Weil1. +Note that there is a forgetful functor +Weil1 → (CMon/�) → CMon +that reflects connected limits. This gives the following class of limits, identified in Leung (2017). +Definition 4.1.5. We say the following pullback diagrams in Weil1 are transverse: +W +W +W 2 +W +W 2 +W W +W +W +W +� +� +W +id +id +id +id +⌟ +π0 +π1 +p +p +⌟ +p◦πi +µ +0 +pW +where µ(a +a1x +a2y ) = a +a1x +a2x y . The ⊗-closure of these three pullbacks is the set of transverse +squares, and they are also pullback squares by Leung (2017). +3It should be noted that Nishimura and Osoekawa (2007) made progress applying techniques from computer algebra to +latter problem. +82 + +To see that each transverse square in the ⊗-closure is a pullback diagram, take the two non-identity +squares and rewrite them in CMon: +� × (� × �) +� × � +� × (� × �) +� × (� × � × �) +� × � +� +� +� × � +�×π0 +�×π1 +π0 +π0 +⌟ +π0 +�×(π0,0◦!,π1) +(id,0◦!) +�×π1 +⌟ +The coproduct of Weil algebras is the tensor product of the underlying commutative monoids, which +are finite-dimensional and free, so these limits are closed under ⊗. +Proposition 4.1.6 (Leung (2017) Proposition 4.1). The category Weil1 is a tangent category, where the +tangent functor is +T := W ⊗ (_) : Weil1 → Weil1 +and the natural transformations are given by +p : W ⊗ (_) +p⊗(_) +−−→ (_), 0 : (_) +0⊗(_) +−−→ W ⊗ (_), + : W2 ⊗ (_) ++⊗(_) +−−→ W ⊗ (_), +ℓ : W ⊗ (_) +ℓ⊗(_) +−−→ W ⊗ W ⊗ (_), +c : W ⊗ W ⊗ (_) +p⊗(_) +−−→ W ⊗ W ⊗ (_)). +The category Weil1 is, in some sense, a finitely presented theory. It is precisely the free tangent +category on a single object: +Proposition4.1.7(Proposition 9.5, Leung(2017)). The categoryWeil1 isgenerated by the maps{p,0,+,ℓ,c } +from Example 4.1.2, closed under composition, tensor, and maps induced by transverse limits. +Corollary 4.1.8. The category Weil1 is the free tangent category over a single object: every object C in a +tangent category � determines a strict tangent functor T−.C : Weil1 → �, mapping +V = W n1 ⊗ ...⊗ W n(k) �→ Tn(1).(...).Tn(k).C = T V C +so that there isan isomorphism of categoriesbetween � and the category of strict tangent functorsWeil1 → +� with tangent natural transformations as morphisms. +Notation 4.1.9. Throughout this section, the notation T V C will refer to the action of the Weil algebra +V on an object C in a tangent category. In particular, we will make use of the isomorphism T U .T V C = +T U V C . +4.2 +Tangent structures as monoidal actions +The presentation of Weil1 as the free tangent category situates the formal theory of tangent categories +as an instance of more general categorical machinery, namely monoidal actions. Recall that in a sym- +metric monoidal category (�,⊗,I ), an internal monoid (C ,•,i) determines a monad: +(C ⊗ _ : � → �,µ : � ⊗ (� ⊗ _) +•⊗_ +−→ � ⊗ _,η : _ +ρ−→ I ⊗ _ +e ⊗_ +−−→ C ⊗ _). +83 + +The category of algebras for this monad is exactly the category of C -modules, objects with an associa- +tive and unital action by C . A morphism will be a map on the base object that preserves the action: +C +M ⊗ C +M ⊗ M ⊗ C +M ⊗ C +C +M ⊗ C +C +M ⊗ C +M ⊗ D +C +D +M ⊗∝ +•⊗M +∝ +∝ +(u,C ) +∝ +f +∝C +∝D +M ⊗f +Strict actegories are the 2-categorical generalization of modules over a monoid. The coherences for a 2- +monad follow from the coherences from a strict monoidal category in the 2-category of categories. The +following proposition relies on a few facts from enriched category theory (treating the cartesian closed +category Cat as a Cat-enriched category, per Kelly (2005)) but a more general treatment of non-strict +actegories may be found in Janelidze and Kelly (2001): +• A 2-functor and 2-natural transformations are exactly a functor and natural transformations +that satisfy extra coherences. These coherences follow for free by constructing the monad and +comonad � × _,[� ,_]. +• An algebra of the underlying 1-monad is exactly an algebra of the 2-monad (the same result holds +for comonads). +When working with algebras of a 2-monad, four different notions of morphisms can come into play +(Lack and Power (2009)). These arise through using the 2-categorical data to weaken the notion of a +morphism: +(i) Strict: this is exactly a morphism of the underlying algebras. Write the 2-category of strict � - +actegories. +(ii) Strong: the morphisms preserve the action to an isomorphism: +� × � +� × � +� +� +� ×F +∝C +∝D +F +α +(iii) Lax: the 2-cell is no longer an isomorphism: +� × � +� × � +� +� +� ×F +∝C +∝D +F +α +(iv) Oplax: the 2-cell travels in the opposite direction (these will not figure into this account). +2-cells between actegory morphisms must satisfy a coherence between the natural transformation +parts of the actegory morphisms. +84 + +Definition 4.2.1. In the case of strict, strong, and lax tangent functors, the same notion of a 2-cell applies: +a natural transformation γ : F ⇒ G satisfying the following coherences with the natural transformations +α and β: +� × � +� × � +� × � +� × � +� +� +� +� +� ×F +∝� +∝� +F +α +G +� ×G +� ×F +∝� +∝� +G +β +γ +� ×γ += +We call these actegory natural transformations. +Note that for strict actegory morphisms, this condition holds for any natural transformation γ : F ⇒ +G . Now consider the following three 2-categories. +Definition 4.2.2. Let (� ,⊗,I ) be a strict monoidal category. Define the following three 2-categories. +1. �Actstrict: the 2-category of strict � -actegories, strict actegory morphisms, and natural transfor- +mations. +2. �Actstrong: the 2-category of strict � -actegories, strong actegory morphisms, and actegory nat- +ural transformations. +3. �Actlax: the 2-category of strict � -actegories, lax actegory morphisms, and actegory natural +transformations. +Note that the inclusions of these 2-categories are locally fully faithful, so only the 1-cells differ. +The case where the action preserves certain limits in the monoidal category is of particular interest. +A small category equipped with a class of chosen limits is known as a sketch. The previous correspon- +dence restricts to the class of limit-preserving actions in this case. +Definition 4.2.3. A sketch is a small category with a class of chosen limits, and a sketch morphism is a +functor sending chosen limits to the chosen limits in the domain (up to isomorphism). The category of +models of a sketch � in a category �, Mod(� ,�), is the full subcategory [� ,�] whose functors preserve +the chosen limits. A monoidal sketch, then, is a sketch (� ,� ) equipped with a symmetric monoidal +category structure on � so that _ ⊗ _ preserves limits in each argument. +Now use the fact that the category Weil1 is a monoidal sketch, since it is a small, strict monoidal +category equipped with a class of limits stable under the tensor product. +Theorem 4.2.4 (Theorem 14.1, Leung (2017)). Let � be a category. The following are equivalent: +(i) A tangent structure on �, +(ii) A sketch action ∝: Weil1 × � → �. +Observation 4.2.5. There is a coalgebraic perspective on tangent categories, coming from the equiva- +lence between algebrasof the 2-monad (Weil1×(_),⊗,I : 1 → Weil1) and the 2-comonad ([Weil1,_],[⊗,_],[I ,_]). +For any category �, there is a free tangent category given by +Weil1 × � +85 + +and this agrees with the free Weil1-actegory. However, for the cofree tangent category, take +Mod(Weil1,�), +the category of transverse-limit-preserving functors Weil1 → �. +We can use Leung’s theorem to induce a monoidal functor Weil1 → � when a tangent structure is +induced by a single object. +Corollary 4.2.6. Let (�,⊗,I ) be a strict monoidal category. If an additive bundle (p : A → I ,+ : A2 → +A,0 : I → A) equipped with morphisms +A ⊗ A +c−→ A ⊗ A +A +ℓ−→ A ⊗ A +determines a tangent structure on � using the endofunctor A⊗(−), then A determines a strict, transverse- +limit-preserving, monoidal functor +A(−) : Weil1 → �;Wn[1] ⊗ ...⊗ Wn[k] �→ An[1] ⊗ ...⊗ An[k] = Tn[1]...Tn[k].I +Note that this allows for a more conceptual description of representable tangent structure. +Proposition 4.2.7. In a symmetric monoidal closed category, an infinitesimal object is exactly a strict +symmetric monoidal functor D : Weil1 → �. +This presentation of an infinitesimal object makes it tautological that �op has a tangent structure. +Corollary 4.2.8. Given a strict symmetric monoidal functor +D : Weil1 → �op +there is a strict action of Weil1 on �op given by +Weil1 × �op D op ⊗� +−−−−→ �op × �op +⊗−→ �op . +There is a clear correspondence between the notions of a (strict, strong, lax) tangent functor and +a (strict, strong, lax) actegory morphism. This proposition extends to the following equivalence of 2- +categories. +Corollary 4.2.9 (Theorem 14.1 Leung (2017)). The following pairs of 2-categories are equivalent. +(i) The 2-category of tangent categoriesand strict tangent functorsisthe full sub-2-category ofWeil1ActStrict +spanned by sketch actions. +(ii) The 2-category of tangent categoriesand strong tangent functorsisthe full sub-2-category ofWeil1Actstrong +spanned by sketch actions. +(iii) The 2-category of tangent categoriesand lax tangent functorsisthe full sub-2-category ofWeil1ActLax +spanned by sketch actions. +86 + +4.3 +The Weil nerve of an involution algebroid +The construction in this section is analagous to the nerve of an internal category—hence the “Weil +nerve” construction—and deals with similar technical issues. In particular, the construction in this +section will mimic the nerve construction for internal categories by replacing the tensor product of +Weil1 with span composition in the domain category. Recall that every anchored bundle or internal +category has a canonical span associated with it: +A +C +M +T M +M +M +π +̺ +s +t +In any category �, there is a category of spans in � as well as span composition. +Definition 4.3.1. A span from A to B in a category � is a diagram of the form +X +A +B +l +r +There is a notion of span composition, so given a span X : A → B and Y : B → C , then the composition +of X and Y is the pullback (if it exists): +Z +X +Y +A +B +C +l +r +l ′ +r ′ +⌟ +l ′′ +r ′′ +A morphism of spans is a commuting diagram of the form +A +X +B +C +Y +D +l +r +l ′ +fl +fr +r ′ +fc +Note that if f and g are span morphisms with fr = gl , then the horizontal composition may be formed +if each respective span composition exists: +• +X +W +A +B +E +C +D +F +Y +Z +• +l +r +l ′ +fl +fr =gl +r ′ +fc +gr +gc +⌟ +⌟ +87 + +When discussing span composition in a tangent category, it is assumed that the pullback is a T -pullback. +Observation 4.3.2. Note that the category of spans in � is a functor category, so that limits are computed +pointwise in �. This also means that the horizontal composition operation, when it exists, preserves +limits in either argument. +These span constructions can be helpful in constructing functors from a monoidal category into +a non-monoidal category �, by forming a monoidal category from � using spans. In the case of an +internal category over M , one takes the slice category �/(M × M ) where the tensor product is span +composition. An internal category s,t : C → M is a monoid in this category of spans over M , so that it +determines a monoidal functor +C : ∆op → �/(M × M ), +remembering that ∆op is the monoidal theory for monoid (every monoid in a monoidal category � +determines a monoidal functor ∆ → �). The construction of the corresponding monoidal category for +spans is more nuanced, as the category Weil1 is not �-indexed. Observe that the prolongation of an +anchored bundle is constructed as a span composition: +� (A) +A +T A +M +T M +T 2M +π +̺ +T π +T ̺ +π0 +π1 +⌟ +The third prolongation is given by span composition as well: +� 2(A) +A +T.� (A) +M +T M +T 2M +π +̺ +T.π◦π0 +T.̺◦π1 +π0 +π1 +⌟ +This horizontal composition will play the same role as the tensor product in �/(M × M ). +Definition 4.3.3. In a tangent category �, consider a pair of spans +X : M → T U M , Y : M → T V M . +Define X ⊠ Y to be the horizontal composition (when it exists): +Z +X +T U Y +M +T U M +T U V M +lX +rY +T U .lY +T U .r +⌟ +88 + +(recall that we will often suppress the ⊗ in Weil1 to save space). So the span composition is +M +X−→ T U M +T U .Y +−−−→ T U .T V M +M +X +T U M +M +Y +T U M +M +A +T V +M +B +T V ′ +lX +lY +θ.M +rX +rY +f +rA +rB +ψ.M +g +lB +lA +(4.1) +The horizontal composition f ⊠ g is defined as f ×θ.M θ.g : +Z +X +Y +M +T U M +T U .T V M +M +T U ′M +T U ′.T V ′M +A +T U ′B +C +lX +lA +θ.M +rX +rA +f +T U .rY +T U ′.rB +θ.ψ.M +θ.g +T U .lY +T U ′.lB +⌟ +⌟ +In any tangent category with a tangent display system (Definition 1.5.3), the category of spans on +M whose maps are of the form given by Equation 4.1 with l ∈ � is a monoidal category. Any anchored +bundle in a tangent category gives rise to a monoidal category after a strict choice of T -pullbacks (as- +suming those T -pullbacks exist). +Definition 4.3.4. Let (π : A → M ,ξ,λ,̺) be an anchored bundle in a tangent category �. Write the span +�A.Wn := (M +π◦πi +←−− An +̺n +−→ TnM ), +�A.� := (M = M = M ). +A choice of prolongations for (π,ξ,λ,̺) is a strict choice of horizontal composition for each V ∈ Weil1: +�A.V = �A.(Wn[1]...Wn[k]) := �A.Wn[1] ⊠ ··· ⊠ �A.Wn[k]. +We will write the span as follows: +A.V +M +T V .M +πV +̺V +(notice that the apex is not hatted). Given a choice of prolongations for an anchored bundle (π,ξ,λ,̺), +the category Span(π,ξ,λ,̺) is defined as follows: +89 + +• Objects are �A.V for V ∈ Weil1. +• Morphisms are given by pairs +(f ,φ) : �A.V → �A.U +where f : A.V → A.U and φ : V →U determine a span morphism of the form +M +A.V +T V .M +M +A.U +T U .M +πV +̺V +φ.M +πV +̺U +f +as discussed in Definition 4.3.1. +• Tensor structure: The tensor product is defined using the horizontal composition ⊠ as defined in +Definition 4.3.1. +The idea is to show that an involution algebroid structure on an anchored bundle induces a tangent +structure on the monoidal categoryofprolongations, and then toapplyLeung’stheorem. The following +two lemmas will simplify this proof. +Lemma 4.3.5. Let (π : A → M ,ξ,λ,̺) be an anchored bundle with chosen prolongations in a tangent +category �, and identify the monoidal category Span(π,ξ,λ,̺). +(i) There is a functor +U ̺ : Span(π,ξ,λ,̺) → � +constructed by sending a span morphism to the morphism between the objects at its apex: +M +A.V +T V .M +A.V +M +A.U +T U .M +A.U +πV +̺V +φ.M +πV +̺U +f +f +(ii) Suppose we have a square +�A.U +�A.Y +�A.X +�A.Z +(f ,φ) +(g ,ψ) +(l ,α) +(r,β) +whose image under U ̺ is a T -pullback in �, and so that the square in Weil1 is a transverse T - +pullback: +A.U +A.Y +U +Y +A.X +A.Z +X +Z +f +g +l +r +φ +ψ +β +α +⌟ +⌟ +Then U ̺ reflects the limit; that is, the original square in Span(π,ξ,λ,̺) is a T -pullback. +(iii) T -pullbacks of the form described in (ii) are closed under ⊠. +90 + +Proof. The functor in in (i) is straightforward to construct, as it simply forgets the left and right legs +of the spans. For (ii), note that because the Weil1 part of the diagram is a transverse T -pullback, then +given a pair of maps +�A.V +�A.U +�A.Y +�A.X +�A.Z +(f ,φ) +(g ,ψ) +(l ,α) +(r,β) +(x,ω) +(y,γ) +a unique span morphism �A.V → �A.U may be induced using the apex map from �, and the unique +map induced in Weil1 by the universality of transverse squares (this square is also universal in �), so +the span morphism diagram will commute by universality. +For (iii), T -pullback squares of the form in (ii) are closed under ⊠ as transverse squares in Weil1 +are closed under ⊗, so the result follows by the commutativity of limits and by applying part (ii) of this +lemma. +Observation 4.3.6. It will be useful to have a “flat” presentation of the prolongation A.Wn(1) ...Wn(k). +The higher prolongations of an anchored bundle may be concretely described as the T -pullback of the +zig-zag below: +ˆA.Wn(1)...Wn(k) +ˆA.Wn(1) +Tn[1]. ˆA.Wn(2) +Tn(1)...n(k−1) ˆA.Wn(k) +Tn(1)M +... +Tn(1).̺n(2) +Tn(1)...n(k).(π◦π0) +̺n(1) +Tn[1].(π◦π0) +so that the prolongation A.W n[1]...W n[k] may be written concretely as +(u1,...,uk) : An[1]̺′×T.π′Tn[1].An[2]T.̺×T 2.π...̺′×T.π′Tn[1]...n[k−1]An[k]. +Furthermore, the choice of prolongation identifies the following limits: +ˆA.U V +T U . ˆA.V +ˆA.U +T U M +̺U +T U .πV +⌟ +ˆA.U V +T U ˆA.V +T U M +T U M +ˆA.U +T U M +⌟ +̺U +T U .πV +so that +A.U V = A.U ⊠ A.V = A.U ⊠ idM ⊠ A.V +where idM is the span M = M = M . +Note that a ̺ sends the involution algebroid structure map to its corresponding tangent structure +map. Each of the structure maps, then, gives a span morphism where ⊠ is well defined: +91 + +Definition 4.3.7. Let (π : A → M ,ξ,+q,λ,̺,σ) be an involution algebroid in � with chosen prolonga- +tions. Then define the following maps in Span(π,ξ,λ,̺): +• The projection p : ˆA.W → ˆA.�, +M +A +T M +M +M +M +̺ +π +p +π +• The zero map 0 : ˆA.� → ˆA.W , +M +M +M +M +A +T M +0 +π +̺ +ξ +• The addition map + : ˆA.W2 → ˆA.W , +M +A2 +T2M +M +A +T M +̺2 +π◦πi ++.M +π +̺ ++ +• The lift map ℓ : ˆA.W → ˆA.W W , +M +A +T M +M +A̺×T πT A +T 2M +π +̺ +ℓ.M +π◦π0 +T.̺◦π1 +(ξπ,λ) +• The flip map c : ˆA.W W → ˆA.W W , +M +A̺×T πT A +T 2M +M +A̺×T πT A +T 2M +c .M +π◦π0 +̺◦π1 +π◦π0 +̺◦π1 +σ +The idea is to show that the monoidal category of chosen prolongations Span(π,ξ,λ,̺) for an in- +volution algebroid has a tangent structure generated by the structure maps in Definition 4.3.7 and the +endofunctor �A.W ⊠(−). Using the flat presentation, we can then show thatU ̺ will determine a tangent +functor in �. The following lemma about ̺ will be useful in constructing the natural transformation +part of a tangent functor. +Definition 4.3.8. Let (π : A → M ,ξ,λ,̺) be an anchored bundle with chosen prolongations. Recall that +by Definition 4.3.4, the right leg of A.U is written ̺, so it induces a span map: +M +A.U +T U M +M +T U M +T U M +̺U +̺U +πU +pU +92 + +This map has a flat presentation as +A.U V +A.U ̺U ×T U .πV T U .A.V +T U .A.V +T U M id ×T U .πV T U .A.V +̺U ⊠A.V +πU ×T U .A.V +We write the map +̺U .V := ̺U ⊠ ( �A.V ) +which corresponds to the following span morphism: +A.U V +A.U +T U .A.V +M +T U M +T U V M +T U M +T U .A.V +T U .A.V +̺U +T U .̺V +T U .πV +̺U +πU +⌟ +pU .M +T U .πV +Theorem 4.3.9 (The Weil Nerve). There is a fully faithful functor +NWeil : Inv(�) → [Weil1,�] +that sends an involution algebroid to the transverse-limit-preserving tangent functor: +( �A,α) : Weil1 → � +Proof. For the first step of this proof, we show that an involution algebroid structure on an anchored +bundle (π : A → M ,ξ,λ,̺) determines a tangent category structure on the monoidal category Span(π : +A → M ,ξ,λ,̺). +We check that the endofunctor ˆA ⊠ (−) determines a tangent structure, with the structure maps +given by Definition 4.3.7: +[TC.1] Additive bundle axioms: +(i) Use Lemma 4.3.5 to see that +ˆA.W2 = ˆA.W p ×p ˆA.W Aπ×πA; +this is preserved by ˆA.V ⊠ (−). +(ii) The triple ( ˆA.+, ˆA.p, ˆA.0) = (+q ,π,ξ) is an additive bundle induced by Proposition 2.4.4, +and ⊠ preserves pullbacks (and therefore additive bundles), so the additive bundle axioms +hold. +93 + +[TC.2] Symmetry axioms: +(i) ˆA.c ◦ ˆA.c = id follows from the involution axiom σ ◦ σ = id . +(ii) For Yang–Baxter, note that +( ˆA ⊠ c ) ◦ (c ⊠ ˆA) ◦ ( ˆA ⊠ c ) = (c ⊠ ˆA) ◦ ( ˆA ⊠ c ) ◦ (c ⊠ ˆA) +follows from the Yang–Baxter equation on an involution algebroid +(σ × c ) ◦ (id × T.σ) ◦ (σ × c ) = (id × T.σ) ◦ (σ × c ) ◦ (id × T.σ), +since +σ × c .A = ( ˆA.c ) ⊠ ( ˆA.W ) and id × T.σ = ( ˆA.W ) ⊠ c . +(iii) For the naturality conditions: +(a) The interchanges of +,0,p all follow from the fact that +σ : (A.W W,λ × ℓ) → (A.W W,id × c ◦ T.λ) +is linear, and so is an additive bundle morphism. +(b) The axiom +ℓ.T ◦ c = T.c ◦ c .T ◦ T.ℓ +is equivalent to the equation +(σ × c ) ◦ (1× T.σ) ◦ (ˆλ× ℓ) = (1× T ˆλ) ◦ σ +which is equivalent to the double linearity axiom on σ by Proposition 3.3.4. +[TC.3] The lift axioms: +(i) The additive bundle equations are a consequence of λ being a lift and + being the addition +induced by the non-singularity of λ. +(ii) The coassociativity axiom +ℓ.T ◦ ℓ = T.ℓ◦ ℓ +is equivalent to +(ˆλ× ℓ) ◦ ˆλ = (id × T.ˆλ) ◦ ˆλ +proved in (i) of Proposition 3.2.6. +(iii) The symmetry of comultiplication, c ◦ℓ = ℓ, is given by the unique equation for an involu- +tion algebroid, so that σ ◦ (ξ ◦ π,λ) = (ξ ◦ π,λ). +(iv) The universality of the lift follows from part (ii) of Proposition 3.2.6; Lemma 4.3.5 ensures +that for any V ∈ Weil1, �A.V ⊠ µ and µ ⊠ �A.V are universal. +This lemma puts a tangent structure on Span(π,ξ,λ,̺). Now consider the functor sending spans to +the apex map, +U ̺ : Span(π,ξ,λ,̺) → �. +The family of maps +{̺U .V : ̺U ⊠ �A.V |U ,V ∈ Weil1} +94 + +gives a family of natural transformations +̺U : A.T U ⇒ T U .A, +so that the following pair constitute a tangent functor +(U ̺,̺) : Span(π,ξ,λ,̺) → �. +Because the universality conditions on Span(π,ξ,λ,̺) followed by reflecting limits in � using Lemma +4.3.5, it follows that (U ̺,̺) will preserve the tangent-natural limits in Span(π,ξ,λ,̺) corresponding to +transverse limits in Weil1. +By Leung’s Theorem 4.1.7 (by way of Corollary 4.2.6), the tangent structure on Span(π,ξ,λ,̺) in- +duces a strict, monoidal, transverse-limit-preserving functor +¯A : Weil1 → Span(π,ξ,λ,̺) +that sends the tensor product ⊗ to the span composition ⊠. By composing the strict tangent functor +( ¯A,id ) and (U ̺,̺), we have a lax, transverse-limit-preserving, tangent functor: +(A,̺) : Weil1 → �;V �→ A.V +Now, check the bijection on morphisms. Starting with an involution algebroid morphism (f ,m) : +A → B, note that this gives a span morphism ˆf : +M +A +T M +N +B +T N +T.m +m +f +This gives a natural definition of ˆf .V using the horizontal composition of span morphism, so that +ˆf .(U V ) = ˆf .U ⊠ ˆf .V and ˆf .� = m, +(4.2) +giving a family of maps { ˆfU : U ∈ objects(Weil1)}. Because f will commute with the structure maps +{π,ξ,+,(ξ ◦ π,λ),σ}, it follow immediately that ˆf is a natural transformation, because the following +calculation holds for each θ : X → Y ∈ {p,0,+,ℓ,c }: +ˆf .U Y V ◦ ( ˆA.U ⊠ θ ⊠ ˆA.V ) += ( ˆf .U ⊠ ˆf .Y ⊠ ˆf .V ) ◦ ( ˆA.U ⊠ θ ⊠ ˆA.V ) += ˆf .U ⊠ ( ˆf .Y ◦ θ) ⊠ ˆf .V += ˆf .U ⊠ (θ ◦ ˆf .X ) ⊠ ˆf .V += ( ˆA.U ⊠ θ ⊠ ˆA.V ) ◦ ˆf .U X V +Tangent naturality will follow by the preservation of the anchor map by f . The equality, for any Weil +algebra U , of the diagrams +M +A.U +T U M +M +A.U +T U M +N +B.U +T U N +M +T U M +T U M +N +T U N +T U N +N +T V N +T V N +T U .m +m +πU +̺U +̺U +ˆf .U +̺U +pU +πU +pU +̺U +pU +T U .m +T U .m +m +̺U += +95 + +is precisely the tangent-naturality condition from Definitions 1.3.12, 4.2.1. +For the inverse of this mapping, consider a tangent natural transformation (Definition 1.3.12) +γ : (A,α) → (B,β), +A ◦ T (A) +B ◦ T (A) +T ◦ A(A) +T ◦ B(A) +γT A +αA +βT A +T γA +where (A,α) and (B,β) are tangent functors Weil1 → � built out of involution algebroids with chosen +prolongations. For any U ,V , the map γ.U V decomposes as γ.U ⊠ γ.V : +�A.T.T +T. �A.T +�B.T.T +T. �B.T +�A.T +T. �A +�B.T +T. �B +�A.p.T =π0 +α.T =π1 +�A.p +T. �A.p +�B.p +T. �B.p +�B.p.T =π0 +β.T =π1 +γ.T +T.γ +T.γ.T +γ.T T +Applying this relationship inductively, it is clear that the base maps γ.W and γ.� determine the entire +morphism γ.V : +M +A.U +T U M +N +B.U +T U N +M +An[1] +Tn[1].M +... +Tn[1]...Tn[k].M +N +Bn[1] +Tn[1].M +... +Tn[1]...Tn[k].N +Tn[1]...Tn[k].γ.� +γ.� +̺n[1] +π◦πi +π◦πi +̺n[1] +γ.Wn[1] +Tn[1].γ.� +Tn[1].(π◦πi) +Tn[1].(π◦πi) +... +T U .γ.� +πU +πU +̺U +̺U +γ.� +γ.U += +Thus, every tangent-natural transformation is constructed out of a pair +(γ.� : M → N ,γ.W : A → B) +using the ⊠ construction from Equation 4.2. All that remains to show is that this pair is an involution +algebroid morphism. +Tangent naturality gives the following two coherences: +̺B ◦ γ.W = T.γ.� ◦ ̺A and σB ◦ γ.W W = γ.W W ◦ σA +since ̺B = β.W,̺A = α.W,σB = B.c , and σA = A.c by construction. The following diagram proves +that γ.W preserves the lifts, so that (γ.W,γ.�) is an involution algebroid morphism: +T.A.T +T.A.T +T.A.T +T.B.T +T.B.T +T.B.T +A.T.T +A.T.T +B.T.T +B.T.T +A.T +A.T +A.T +B.T +B.T +B.T +A.ℓ +α.T +(ξπ,λA) +π1 +T.γ +γ.T +m +B.ℓ +β.T +(ξπ,λB ) +β.T +λA +λB +96 + +Thus, a tangent natural transformation ( �A,α) → ( �B,β) is exactly a morphism of involution algebroids +A → B, proving the theorem. +Now, the projection for a Lie algebroid is a submersion, as we may make a choice of prolongations +for each U ∈ Weil1. These prolongations lead to a new observation about Lie algebroids: they embed +into a category of functors into smooth manifolds. +Corollary 4.3.10. Using the Weil nerve construction, the category of Lie algebroids embeds into the +tangent-functor category: +LieAlgd �→ [Weil1,SMan]. +4.4 +Identifying involution algebroids +This section identifies those tangent functors +(A,α) : Weil1 → � +that are involution algebroids as precisely those where A preserves transverse limits and α is a T - +cartesian natural transformation (Definition 4.4.1). These conditions will force each A.V to be the +V -prolongation of the underlying anchored pre-differential bundle: +(A.p : A.T → A, A.0 : A → A.T, A.T +A.ℓ +−→ A.T.T +α.T +−→ T.A.T, α : A.T → T.A) +(these conditions also ensure that this tuple is an anchored differential bundle). +Initially, it is only clear that α is T -cartesian for the projection p. Indeed, recall that the prolonga- +tion A.U V is defined to be the T -pullback of the cospan: +�A.U +αU +−→ T U .A.� +T U .A.p V +←−−−−− T U . �A.V +Then consider the following diagram: +�A.U W V +T U . �A.T.T V +�A.U .V +T U . �A.V +�A.T U +T U . �A.� +T U . ˆA.p.V +�A.T U .p.T V +�A.T U .p V +T U . �A.p V +⌟ +⌟ +This means that every naturality square of α for p is a T -pullback; natural transformations satisfying +this property for every map in the domain category are called T -cartesian. +Definition 4.4.1. A natural transformation γ : F ⇒ G is cartesian whenever each naturality square +F C +G C +F D +G D +F.f +γ +γ +G .f +⌟ +is a pullback. A natural transformation between functors into a tangent category is T -cartesian when- +ever each component square is a T -pullback (we will generally suppress the T when the context is clear). +97 + +Now, recall that the Weil complex determined by an involution algebroid has A.U .V determined +by the following T -pullback squares: +�A.U .V +T U . �A.V +�A.T U +T U . �A.� +�A.T U .p V +T U . �A.p V +⌟ +Then it is not difficult to show that the T -cartesian condition on a Weil complex forces it to be an +involution algebroid. We first need: +Definition 4.4.2. A T -cartesian Weil complex in � is a tangent functor +(A,α) : Weil1 → � +for which A sends transverse limits to T -limits and α is a T -cartesian natural transformation. +The first condition to check is that a T -cartesian Weil complex gives a natural anchored bundle ˆA +whose Weil prolongations coincide with the functor assignments on objects. +Proposition 4.4.3. Let (A,α) be a T -cartesian Weil complex. Then we have an anchored bundle +(M := A.�, +ˆA := A.W, π := A.π, ξ := A.ξ, λ := α.T ◦ A.ℓ). +Furthermore, +� ( ˆA) = A.W W, +� 2( ˆA) = A.W W W. +Proof. Suppose we have a tangent functor (F,α) : � → � and a differential bundle (π,ξ,λ) in �. If F pre- +serves T -pullbacks of π, it preserves the additive bundle structure on (π,ξ,+), so to show (F.π,F.ξ,α ◦ +F.λ) is universal it suffices to show that the following diagram is a T -pullback in �: +F.A2 +T.F.A +F.M +T.F.M +F.π2 +µα◦F.λ +F.0 +F.T.π +Expand this to +F.A2 +F.T.A +T.F.A +F.M +F.T.M +T.F.M +F.π2 +F.µλ +F.0 +T.F.π +F.T.π +α.A +α.M +In this case, it restricts to the diagram +A.T2 +A.T 2 +T.A.T +A +A.T +T.A +A.µ +α.T +A.T.p +α +T.A.p +A.0 +⌟ +98 + +Each square is a T -pullback by hypothesis, so the universality of the lift follows by the T -pullback +lemma. Because the complex is T -cartesian, the assignment A.V gives a coherent choice of prolonga- +tions by the T -pullback +A.U .V +T U .A.V +A.U +T U .A +A.U .p V +αU +T U .A.p V +αU +There is, of course, a natural candidate for the involution map. +Corollary 4.4.4. Let (π : A → M ,ξ,λ,̺) be the anchored bundle induced by a T -cartesian Weil complex +in a tangent category �. Then we have an involution map +σ : � (A) +A.c +−→ � (A). +The equations for an involution algebroid should follow immediately by functoriality; one need +only ensure that the maps take the correct form. +Lemma 4.4.5. Let A be a T -cartesian Weil complex in a tangent category �, with (π : A → M ,ξ,λ,σ) its +underlying anchored bundle. Then we have: +(i) A.c .T = σ × c , +(ii) A.T.c = 1× T.σ, +(iii) A.ℓ.T = ˆλ × ℓ.A, +(iv) A.T.ℓ = id × T.ˆλ. +Proof. +(i) Consider the diagram +A.T.T.T +A.T.T.T +T.T.A.T +T.T.A.T +T.T.A +T.T.A +A.T.T +A.T.T +A.T.T.p +αT T +αT T +T.T.A.p +A.T.T.p +T.T.A.p +c .A.T +c .A +A.c +A.c .T +⌟ +⌟ +Observe that this forces A.c .T = A.c × c .A.T = σ × c . +99 + +(ii) Likewise, the diagram +A.T.T.T +A.T.T.T +T.T.A.T +T.T.A.T +T.A +T.T.A +A.T +A.T +A.T.p.p +α.T T +αT +T.T.A.p +A.T.p.p +αT +T.T.A.p +α.T T +T.A.c +A.T.c +⌟ +⌟ +forces A.T.c = id × T.A.c = id × T.σ. +(iii) The diagram +A.T.T +A.T.T.T +T.A.T +T.T.A.T +T.A +T.T.A +A.T +A.T.T +A.ℓ.T +α.T +αT T .T +ℓ.A.T +A.T.p +T.A.p +T.T.A.p +ℓ.A +α +A.ℓ +A.T.T.p +αT T +⌟ +⌟ +⌟ +⌟ +forces A.ℓ.T = A.ℓ× ℓ.A = ˆλ × ℓ. +(iv) As α is T -cartesian, the following diagram is a T -pullback: +A.T.T +A.T.T.T +T.A.T +T.A.T.T +α.T +A.T.ℓ +α.T.T +T.A.ℓ +⌟ +Using previous results, this means that A.ℓ.T is the unique map making the following diagram +commute: +A̺×T.πT A +A̺×T.πT AT.̺×T 2.πT 2A +T A +T AT.̺×T 2.πT 2A +(π1,π2) +T.ˆλ +π1 +which we can see is id × ˆλ. +Pulling together this lemma and the previous proposition, the following is now clear: +Proposition 4.4.6. A T -cartesian Weil complex determines an involution algebroid. +However, we have not yet exhibited an isomorphism of categories between the image of the Weil +nerve functor and T -cartesian Weil complexes. At first glance, the Weil nerve construction only gives +a Weil complex that is T -cartesian for the tangent projection p ∈ Weil1. Being T -cartesian for p is, +100 + +however, sufficient: a Weil complex that is T -cartesian for tangent projections will be T -cartesian for +every map in Weil1 (a similar result appears in the context of differentiable programming languages; +see Cruttwell et al. (2019)). +Proposition 4.4.7. A lax tranverse-limit-preserving tangent functor (F,α) : Weil1 → � for which F pre- +serves pullback powers of each T U .p is T -cartesian if and only if each +F.T.T +T.F.T +F.T +T.F +F.T.p +αA +T.F.p +αB +is a T -pullback. +Proof. We only check the converse since the forward implication is trivial. We make use of the T - +pullback lemma. +(i) c is an isomorphism, so its naturality square is a T -pullback. +(ii) For projections T2 → T , the retract of a T -pullback diagram is a T -pullback, so the following +diagram is universal: +F.T.T2 +F.T.T2 +F.T.T2 +F.T.T +T.F.T2 +T.F.T +F.T.T2 +F.T.T2 +α +α +F.T.πi +F.T.πi +α +α +T.F.πi +F.T.πi +(iii) For 0, observe that the following two diagrams are equal: +F.T +F.T.T +F.T +F.T +F.T +T.F +T.F.T +T.F +T.F +T.F +α +F.T.p +T.F.p +F.T.0 +T.F.0 +α +α +α +α +⌟ +⌟ +The right diagram is a T -pullback, and the right square of the left diagram is a T -pullback by +hypothesis. By the T -pullback lemma, the left square of the left diagram is a T -pullback. +(iv) For ℓ, observe that +F.T.T +F.T.T.T +F.T.T +F.T.T +F.T +F.T.T +T.F.T +T.F.T.T +T.F.T +T.F.T +T.F +T.F.T +F.T.ℓ +α.T +T.F.ℓ +α.T.T +F.T.p.T +T.F.p.T +α.T +⌟ +F.T.p +T.F.p +α.T +α +⌟ +F.T.0 +T.F.0 +α.T +⌟ +The outer perimeter of the right diagram is a T -pullback (left square by hypothesis, right square +by (ii)), as is the right square of the left diagram (by hypothesis). By the T -pullback lemma, the +left square of the left diagram is a T -pullback. +101 + +(v) For +, observe that +F.T.T2 +F.T.T +F.T.T +F.T.T2 +F.T.T +F.T +T.F.T2 +T.F.T +T.F.T +T.F.T2 +T.F.T +T.F +F.T.+ +α.T +T.F.+ +α.T +F.T.p +T.F.p +α.T +⌟ +F.T.πi +T.F.πi +α.T2 +α.T +⌟ +F.T.p +T.F.p +α +⌟ +The outer diagram on the right is a T -pullback by composition, and the right square on the left +diagram is a T -pullback by hypothesis, so the result follows. +To check that the naturality square is a T -pullback for every map in Weil1, we once again use Leung’s +characterization of maps in Weil1 from Proposition 4.1.7. Inductively, the set of maps generated by +{p,0,+,ℓ,c } closed under ⊗ and ◦ follows as T -pullback squares are closed to composition. For maps +induced by a tranverse limit in Weil1, F preserves transverse limits so this follows by the commutativity +of limits. +Theorem 4.4.8. For any tangent category �, the replete image of the Weil nerve functor +Inv(�) �→ [Weil1,�] +is precisely the category of T -cartesian Weil complexes. +Corollary 4.4.9. That α : A.T ⇒ T.A is T -cartesian is equivalent to requiring that the tangent functor +(A,α) : Weil1 → � +restricts to an anchored bundle +(π : A.T +A.p +−→ A.�, ξ : A +A.0 +−→ A.T, λ : A.T +A.ℓ +−→ A.T T +α.T +−→ T.A.T, ̺ : A.T +α−→ T.A) +and each A.T V is the V -prolongation of this anchor bundle. +Remark 4.4.10. The condition in Corollary 4.4.9 is analogous to the Segal conditions identifying those +simplicial complexes +∆ → � +that are internal categories. Note that every simplicial object has an underlying reflexive graph +tr1(X ) := (s,t : X ([1]) → X ([0]),i : X ([0]) → X ([1])) +where X ([n]) is isomorphic to the object of n-composable arrows for the underlying reflexive graph. +Remark 4.4.11. Notably, being T -cartesian for p is enough to force that a natural transformation is +T -cartesian for the other tangent-structural natural transformations. This has consequences when one +uses partial maps to combine topological notions with tangent categories. In this context, a partial map +N → X with domain M �→ N is a span +M +N +X +f +m +whose right leg is monic. The intuition is that the map f is defined on the subobject M of N , which +introduces a new problem: what is the proper notion of a subobject in a tangent category? Such a notion +102 + +should give rise to a stable class of monics: one that is closed under horizontal span composition. One +answer is the notion of etale monics: a morphism is etale whenever the naturality square for p is a T - +pullbacks: +T M +T N +M +T N +T.f +T.f +p +p +⌟ +Geometrically, this means that the morphism is a local diffeomorphism; for example, an etale subobject +of �n in the Dubuc topos is precisely an open subset in the usual sense. An endofunctor lifts to the partial +map category whenever it preserves the class of monics. A natural transformation lifts to endofunctors +on the partial map category whenever it is T -cartesian for the class of monics, and the same proof will +show that this property holds for etale monics Cruttwell et al. (2019). +4.5 +The prolongation tangent structure +One of the most important consequences of the Weil Nerve Theorem 4.3.9 is that the category of in- +volution algebroids (with chosen prolongations) may be equipped with two tangent structures. The +first tangent structure is the pointwise tangent structure described in Proposition 3.3.6. The tangent +functor sends +(A,α) �→ (T.A : Weil1 → �,c .A ◦ T.α : T.A.T ⇒ T.T.A) +(recall the composition of tangent functors given in Example 1.3.11 (ii)). The structure morphisms will +be given by whiskering, so in this case θ.A,θ ∈ {p,0,+,ℓ,c }. The restriction to tangent functors that +preserve transverse limits along with the fact that the natural part α is T -cartesian, however, ensures +that precomposition with the tangent functor +(A,α) �→ (A.T : Weil1 → �,T.α ◦ A.c : A.T.T ⇒ T.A.T ) +returns an involution algebroid. The structure maps are once again given by whiskering, with the pre- +composition tangent structure A.θ,θ ∈ {p,0,+,ℓ,c }. Preservation of transverse limits guarantees that +this tangent structure will satisfy the necessary universality conditions. +Proposition 4.5.1 (Proposition 3.3.10). The category of involution algebroids with chosen prolongations +in a tangent category � has a second tangent structure, where the action by Weil1 is given by +(A,α) �→ (A.T : Weil1 → �,α.T ◦ ˆA.c : ˆA.T.T ⇒ T. ˆA.T ). +Proof. The proposition statement means that the structure morphisms for this new involution alge- +broid are given by +(A.T,α.T ◦ A.c ) ∼= + + + + + + + + + + + + + + + +α.T ◦ A.c = ̺′ : +� (A) +π1 +−→ T A +A.T.p = π′ : +� (A) +p◦π1 +−−→ A +A.T.0 = ξ′ : +A +(ξ◦π,0) +−−−→ � (A) +̺′ ◦ A.T.ℓ = λ′ : +� (A) +λ×ℓ +−−→ T.� (A) +A.T.c = σ′ : +� 2(A) +σ×c +−−→ � 2(A) +103 + +Similarly, we can see that +(A.T.T,α.T.T ◦ A.c .T ◦ A.T.c ) += + + + + + + + + + + + + + + + + + +α.T.T ◦ A.c .T ◦ A.T.c = ̺′′ : +� 2(A) +(π1,π2) +−−−→ T.� (A) +A.T.p = π′′ : +� 2(A) +(p◦π1,p◦π2): +−−−−−−−→ � (A) +A.T.0 = ξ′′ : +� (A) +(ξ◦π◦π0,0◦π1,0◦π2) +−−−−−−−−−−−→ � 2(A) +̺′′ ◦ A.T.ℓ = λ′′ : +� 2(A) +(λ×ℓ×ℓ) +−−−−→ T.� 2(A) +A.T.T.c = σ′′ : +� 3(A) +(σ×c ×c ) +−−−−−→ � 3(A) +These coincide with the involution algebroids � ′(A),� ′.� ′(A) in Proposition 3.3.10: the second tan- +gent structure follows from the fact that the natural transformations for the tangent structure there are +given by +A.φ : A.U ⇒ A.V,φ :U → V ∈ {p,0,+,ℓ,c }. +The result follows as a corollary of Theorem 4.3.9. +The Jacobi identity for involution algebroids +Classically, the theory of Lie algebroids uses the algebra of sections Γ(π). One key observation is that +when using the Lie tangent structure (Inv(�),� ), sections of π are in bijective correspondence with +χ� (A). This observation allows for different statements about Lie algebroids to be translated into for- +mal statements about the tangent bundle in (Inv(�),� ). +Proposition 4.5.2. Let A be an involution algebroid in �. There is a bijection between the sections of π +in � and the vector fields on A in Inv(�): +X ∈ Γ(π) �→ ((id ,T X ◦ ̺),X ) : A → TL(A); +ˆX ∈ χ� (A) �→ ˆXR : A.R → A.T. +Proof. Recall the coherence for tangent natural transformations γ : (H ,φ) ⇒ (G ,ψ): +H .T +K .T +T.H +T.K +γ.T +φ +ψ +T.γ +We specify this to a morphism ˆX : (A,α) ⇒ TL(A,α) = (A.T,α.T ) at �,W : +A +A̺×T πT A +T M +T A +X .T +̺ +π1 +T.X +and so infer that, if we set X := XR , we have π1 ◦ X .T = T (X ) ◦ ̺. Furthermore, the condition that +pL ◦ X = id forces id = π0 ◦ X .T ; thus, we can see that every section X of pL is given by a morphism of +the form ((id ,T X ◦ ̺),X )) on the underlying involution algebroids, where π◦ X = id . +We now show that every X ∈ Γ(π) gives rise to a section of πL. Observe that the following is a mor- +phism of involution algebroids: +A +A̺×T πT A +M +A +π +(id,T X ◦̺) +p◦π1 +X +104 + +Note that it is well typed, as T.π◦ T.X ◦ ̺ = ̺ ◦ id . Check that it is a bundle morphism: +p ◦ π1 ◦ (id ,T.X ◦ ̺) = p ◦ T.X ◦ ̺ = X ◦ p ◦ ̺ = X ◦ π +and that it is linear: +(λ ◦ π0,ℓ◦ π1) ◦ (id ,T.X ◦ ̺) = (λ,ℓ◦ T.X ◦ ̺) += (λ,T 2.X ◦ ℓ◦ ̺) = (λ,T 2.X ◦ λ) = T (id ,T X ) ◦ λ. +Then check that it preserves the anchor: +π1 ◦ (id ,T X ◦ ̺) = T X ◦ ̺ +and the involution: +(π0,π1,T 2X ◦ T ̺π1) ◦ σ = (σ,T 2X ◦ T ̺ ◦ π1σ) += (σ,T 2X ◦ c T ̺ ◦ π1) = (σ(π0,π1),c π3) ◦ (id ,T 2X ◦ T ̺ ◦ π1). +Thus we have that (id ,T X ◦̺π1) is a morphism of involution algebroids, inducing a morphism of +T -cartesian Weil complexes. Lastly, we check that it is a section of p L +A , but this is clear, since +π0 ◦ (id ,T X ◦ ̺) = id ; +thus we have the desired bijection. +Recall that given an involution on an anchored bundle, there is a bracket on its set of sections (see +the explicit construction in Section 3.4). Given an X ,Y ∈ Γ(π), there is a bracket defined as follows: +ˆλ ◦ [X ,Y ]A +1 (ξπ,0) ◦ Y = ((σ ◦ (id ,T Y ◦ ̺) ◦ X −2 (id ,T X ◦ ̺) ◦ Y )). +A direct proof of the Jacobi identity is a detailed calculation (see the original preprint on involution al- +gebroids Burke and MacAdam (2019)) and still relies on Cockett and Cruttwell’s result for an arbitrary +tangent category with negatives. As a result of Proposition 4.5.2, we can instead use Cockett and Crut- +twell’s result directly: +Corollary 4.5.3. Let A be a complete involution algebroid in a tangent category � with negatives. There +is a Lie bracket defined on Γ(π), [−,−] induced by +ˆλ ◦ [X ,Y ]A +1 (ξπ,0) ◦ Y = ((σ ◦ (id ,T Y ◦ ̺) ◦ X −2 (id ,T X ◦ ̺) ◦ Y )). +Proof. The bracket is induced by Rosicky’s universality diagram, as +0 = p ◦ ((σ ◦ (id ,T Y ◦ ̺) ◦ X − (id ,T X ◦ ̺) ◦ Y )) − 0Y += T p ◦ ((σ ◦ (id ,T Y ◦ ̺) ◦ X − (id ,T X ◦ ̺) ◦ Y )) − 0Y . +We look at the Lie tangent structure for Inv∗(A); this is precisely the vector field induced by +e vR([ ˆX , ˆY ]). +We complete the proof by using the result that for any A in a tangent category with negatives, the +bracket on χ(A) that is defined by +ℓ◦ [X ,Y ] = (c ◦ T.X ◦ Y − T.Y ◦ X ) − 0X +satisfies the Jacobi identity. +105 + +Identifying categories of involution algebroids +Section 4.4 identified whenever a functor Weil1 → � is an involution algebroid, whereas this section +identifies tangent categories � that embed into the category of involution algebroids in some tangent +category �. We call this structure an abstract category of involution algebroids. This notion involves +some 2-category theory, using a modified notion of codescent (see Bourke (2010) for a development of +codescent). +Recall that for any tangent category �, the category of involution algebroids has � as a reflective +subcategory. Furthermore, because limits of involution algebroids are computed pointwise, this reflec- +tor is left-exact. This left-exact reflection is the main structure we axiomatize. +Definition 4.5.4. An abstract category of involution algebroids is a tangent category � with a left-exact +T -cartesian tangent localization (Z ,̺) : � → �, where L satisfies a codescent condition: +TangCatStrict(Weil1,�) �→ TangCatLax(Weil1,�) +L∗ +−→ TangCatLax(Weil1,�) +(where L∗ denotes post-composition by L) is fully faithful. +Example 4.5.5. The category of involution algebroids in any tangent category � is an abstract category +of involution algebroids using (Inv(C ),� ). The reflector is the functor sending an involution algebroid +to its base space; the T -cartesian natural transformation is the anchor map. Any tangent subcategory of +Inv(�) that contains � as a full subcategory will give rise to an abstract category of involution algebroids. +Proposition 4.5.6. Let � �→ � be an abstract category of involution algebroids. Then there is an embed- +ding � �→ Inv(�). +Proof. The proof follows by treating objects in � as strict tangent functors Weil1 → � and morphisms +as tangent natural transformations. +Weil1 +� +� +Z +� (−,A) +� (−,B) +� (−,f ) +The natural part of Z is T -cartesian, and the functor part preserves limits, so Z .� (−,A) =: Z [A] de- +termines an involution algebroid in �, and f a morphism of involution algebroids. The embedding is +guaranteed by the codescent condition so that the post-composition functor is fully faithful. +Corollary 4.5.7. An abstract category of involution algebroids � �→ � is exactly a full subcategory � �→ +� �→ Inv(�). +106 + +Chapter 5 +The infinitesimal nerve and its realization +The main thrust of Chapters 2, 3, and 4 has been that the tangent categories framework allows for Lie +algebroids to be regarded as tangent functors +Weil1 → SMan +which satisfy certain universality conditions. This chapter, which is more experimental than the pre- +vious four chapters and represents work still in progress, puts Lie algebroids into the framework of +enriched functorial semantics. This new perspective on algebroids uses Garner’s enriched perspective +on tangent categories (Garner (2018)) and the enriched theories paradigm from Bourke and Garner +(2019). The functorial-semantics presentation of the Lie functor will generalize the Cartan–Lie the- +orem (that the category of Lie algebras is a coreflective subcategory of Lie groups) into a statement +within the general theory of functorial semantics. +The goal is to show that the infinitesimal approximation of a groupoid, as discussed in Example +3.1.2, may be constructed as a nerve, just like Kan’s original simplical approximation of a topological +space. The nerve of a functor K : � → � approximates objects and morphisms in � by � -presheaves, +so it sends an object in � to the � -presheaf +NK : � → �;C �→ �(K −,C ). +Thus there will be an infinitesimal object, +∂ : Weilop +1 +→ Gpd(� ) +where Gpd(� ) denotes groupoids in the category � of Weil spaces (formally defined in Section 5.1). +The nerve of ∂ has a left adjoint, the Lie realization, given by the left Kan extension (just as Kan’s geo- +metric approximation of a simplicial set is, in Kan (1958)): +Weilop +1 +Gpd(� ) +[Weil1,� ] +� +∂ +Lan� ∂ +(5.1) +Theorem 5.5.13 (The Lie Realization). There is a tangent adjunction between the category of involution +algebroids and groupoids in � , where each functor preserves products and the base spaces. +Gpd(� ) +Inv(� ) +N∂ +|−|∂ +⊣ +107 + +Note, however, that the left Kan extension in Equation 5.1 does not immediately give the desired +adjunction of Theorem 5.5.13, as there is no guarantee that the nerve functor N∂ lands in the category +of algebroids. To prove this we must revisit the work in Section 4.3 presenting involution algebroids as +those functors A : Weil1 → � for which each A(V ) is the prolongation of its underlying anchored bundle; +this leads naturally to the formalism for enriched theories developed in Bourke and Garner (2019). +The presentation of algebroids as models of an enriched theory requires situating the categories +of differential bundles, anchored bundles, and involution algebroids in � as full subcategories of � - +presheaf categories on small � -categories. Section 5.1 reviews the work in Garner (2018) characteriz- +ing tangent categories as categories enriched in � = Mod(Weil1,Set) (the cofree tangent category on +Set, by Observation 4.2.5). +Section 5.2 reconfigures the content of Chapter 2 using the enriched perspective, so that lifts are � - +functors from the � -monoid D + 1, while differential bundles are a reflective subcategory of functors +from its idempotent splitting. Similarly, the category of anchored bundles are a reflective subcategory +of the category [Weil1 +1,�], where Weil1 +1 is the category of 1-truncated Weil algebras (Definition 5.2.8). +Section 5.3 reviews the basic idea of an enriched nerve/approximation. A particularly important ex- +ample is the linear approximation of a reflexive graph, the functor introduced in Example 3.2.7, which +is the nerve of a functor +∂ : (Weil1 +1)op → Gph(� ). +Section 5.4 appliesthe enriched theoriesframework introduced in Bourke and Garner(2019), where +a dense subcategory of a locally presentable � �→ � forms the “arities”, and a bijective-on-objects func- +tor � → � is the theory. The category of models is the pullback of (enriched) categories: +�� +[� ,� ] +� +[� op,� ] +The first step is to freely complete the � -category Weil1 +1 of truncated Weil algebras (Definition 5.2.8) +so that its base anchored bundle has all prolongations; call this � -category � . Then, in every tangent +category�, the categoryofanchored bundleswith chosen prolongations(Definition 4.3.4) in � embeds +fully and faithfully into the functor category: +Anc� (�) �→ [� ,�] +(here, Anc� (�) denotes the category of anchored bundles with chosen prolongations). In particular, +the tangent bundle on � in Weil1 determines a bijective-on-objects functor +� → Weil1 +so that the category of involution algebroids in any tangent category � is the pullback in � Cat: +Inv∗(�) +[Weil1,�] +Anc∗(�) +[� op,�] +⌟ +This means that the category of involution algebroids in � is monadic over the category of anchored +bundles in � using the monad-theory correspondence from Bourke and Garner (2019). +108 + +The final section looks at the category of � -groupoids. Essentially, the free groupoid over the lin- +ear approximation of a graph will now give an infinitesimal object Weil1 → Gpd(� ). The nerve of +∂ : Weilop +1 +→ Gpd(� )—the infinitesimal approximation—has a right adjoint via the realization of the +nerve functor from Definition 5.3.5. This is used to prove the culminating Theorem 5.5.13. +The individual pieces of categorical machinery used in this chapter are not new (the enriched per- +spective on tangent categories, enriched nerve constructions, enriched theories). However, all of the +results dealing with the application of enriched nerve constructions and enriched theories to tangent +categories is original work of the author. +5.1 +Tangent categories via enrichment +This section gives a quick introduction to Garner’s enriched perspective on tangent categories. The +enriched approach to tangent categories first appeared in Garner (2018) and builds on the category +perspective on tangent categories introduced in Leung (2017). Garner was able to exhibit some of the +major results from synthetic differential geometry as pieces of enriched category theory; for example, +the Yoneda lemma implies the existence of a well-adapted model of synthetic differential geometry. +The category of Weil spaces is the site of enrichment for tangent categories and is closely related to +Dubuc’s Weil topos from his original work on models of synthetic differential geometry Dubuc (1981); +a deeper study of this topos may be found in Bertram (2014). Recall that the category Weil1 is the free +tangent category over a single object. The category of Weil spaces is the cofree tangent category over +Set, which is the category of transverse-limit-preserving functors Weil1 → Set by Observation 4.2.5. +Call this the category of Weil spaces, and write it � . Just as a simplicial set S : ∆ → Set is a gadget +recording homotopical data, a Weil space records infinitesimal data. +Definition 5.1.1. A Weil space is a functor Weil1 → Set that preserves transverse limits (Definition 4.1.5): +that is, the ⊗-closure of the set of limits + + + + + + + +Tn+m +Tm +Tn +� +⌟ +, +T2 +T 2 +� +T +0 +µ +⌟ +, +Tn +Tn +Tn +Tn + + + + + + + +A morphism of Weil spaces is a natural transformation. Write the category of Weil spaces as � . +Example 5.1.2. +(i) Every commutative monoid may be regarded as a Weil space canonically. Observe that for every V ∈ +Weil1 and commutative monoid M , one hasthe free V -module structure on M given by |V |⊗CMonM +(here |V | is the underlying commutative monoid of V ). The commutative monoid |V | is exactly +�dimV, so that +|V |⊗CMon M ∼= ⊕dimV M . +This agrees with the usual tangent structure on a category with biproducts. +(ii) Following (i), any tangent category � that is concrete—that is, admitting a faithful functorU : � → +Set—will have a natural functor into Weil spaces (copresheaves on Weil1). Every object A will have +an underlying Weil space V �→ USet(T V (A)), and whenever U preserves connected limits (such as +the forgetful functor from commutative monoids to sets), each of the underlying copresheaves will +be a Weil space. +109 + +(iii) Consider a symmetric monoidal category with an infinitesimal object, which by Proposition 4.2.7 is +a transverse-colimit-preserving symmetric monoidal functor D : Weil1 → �. Then for every object +X , the nerve (Definition 5.3.1) ND (X ) : �(D −,X ) : Weil1 → Set is a Weil space. +(iv) For any pair of objects A,B in a tangent category, �(A,T (−)B) : Weil1 → Set is a Weil space by the +continuity of �(B,−) : � → Set. +Unlike the category of simplicial sets, the category of Weil spaces is not a topos. The category of +Weil spaces does, however, inherit some nice properties from the topos of copresheaves on Weil1 by +applying results from Section 5.3, as it is locally presentable. The basics of locally presentable categories +can be found in the Appendix 6.2. Roughly speaking, a cocomplete category � has a subcategory of +finitely presentable objects �f p, those C so that +�(C ,−) : � → Set +preserves filtered colimits (i.e. those colimit diagrams that commute with all finite limits in Set). � is +locally finitely presentable whenever every object is given by the coend +C ∼= +� X ∈�f p +�(X ,C ) · X +where S · X is the (possibly infinite) product X |S| with |S| the cardinality of the set S. This means that +� = Lex(�op +f p,Set), where Lex means the category of finite-limit-preserving functors. +The third pointofCorollary5.1.4 below, that� islocallyfinitelypresentable asa cartesian monoidal +category, means that the category �f p is closed under products. This implies that � is locally pre- +sentable as a � -category (this ends up being an important technical condition whereby locally pre- +sentable � -categories make sense). Whenever we discuss an arbitrary � -category, we will assume +that � is locally presentable as a monoidal category. +Proposition 5.1.3 (Garner (2018)). The category of Weil spaces is a cartesian-monoidal reflective sub- +category of [Weil1,Set]. +Corollary 5.1.4. The category of Weil spaces is +(i) a cartesian closed category; +(ii) a representable tangent category, where the infinitesimal object is given by the restricted Yoneda +embedding � : Weilop +1 +→ � ; +(iii) Locally finitely presentable as a cartesian monoidal category. +The cofree tangent structure on � is given by precomposition, that is: +T U .M .(V ) = M .(U ⊗ V ) = M .T U .V. +This tangent coincides with the representable tangent structure induced by the Yoneda embedding. +The proof is an application of the Yoneda lemma. Observe that +[D,M ](V ) ∼= [Weil1,Set](D × � (V ),M ) ∼= [Weil1,Set](� (W V ),M ) ∼= M (W V ) +where D ×� (V ) = � (W V ) follows because the tensor product in Weil1 is cocartesian and the reflector +is cartesian monoidal. +110 + +Notation 5.1.5. The tangent category � is a representable tangent category, where D = � W (Definition +1.3.7). We will write the Yoneda functor � : Weilop +1 +→ � as D(−), so it is closer to the usual notation +used in representable tangent categories or synthetic differential geometry (a single D may be used as +shorthand for D(W ), D (n) for D(Wn), etc.). Note that +D(V ) = D (⊗kWn(i)) = +K +� +D(ni) +and that D(ℓ) = ⊗,D(c ) = (π1,π0), D(+) = δ, and so on. +At this point, we are ready to move to the enriched perspective on tangent categories. The basics +of enriched category theory may be found in the Appendix 6.2, but one definition in particular is im- +portant to include here. +Definition 5.1.6. Let � be a � -category for � a closed symmetric monoidal category. For J ∈ � , the +power by J of an object C ∈ � is an object C J so that the following is an isomorphism: +∀D : � (J ,�(D,C )) ∼= �(D,C J ) +whereas the copower is given by +∀D : � (J ,�(C ,D)) ∼= �(J • C ,D). +Let � ←� � be a full monoidal subcategory of � . A � -category � has coherently chosen powers by � if +there is a choice of � -powers (−)J so that +(C J )K = C J ⊗K . +Likewise, coherently chosen copowers are a choice of � -copowers so that +K • (J • C ) = (K ⊗ J ) • C . +The sub-2-categories of � -categories equipped with coherently chosen powers and copowers are � Cat� +and � Cat� , respectively. +Wood (1978) proved that the 2-category of actegories over a monoidal � is equivalent to the 2- +category of [�,Set]-enriched categories with powers by representable functors (Definition 5.1.6), us- +ing the monoidal structure on [�,Set] induced by Day convolution1. Moreover, Garner (2018) showed +that a monoidal reflective subcategory � �→ +ˆ +� exhibits the 2-category of � -categories as a reflective +sub-2-category of � -categories; this proves that tangent categories are equivalent to a particular class +of enriched category. +Proposition 5.1.7 (Garner (2018)). A tangent category is exactly a � -category with powers by repre- +sentables. +Proof. For every A,B ∈ �0, the Weil space is defined as +�(A,B) :=U �→ �(A,T U B). +1The Day convolution tensor product of presheaves X ,Y : �op → Set is given by X �⊗Y := Lan⊗(X ⊠Y ), where (X ×Y )(V ) = +X (V ) × Y (V ) (Day (1970)). +111 + +The functor �(A,−) is continuous and T −B is an infinitesimally linear functor, so this is a Weil space. +The following diagram gives the composition map : +�(B,C ) × �(A,B) +�(A,C ) +�(B,T U C ) × �(A,T V B) +�(A,T V U C ) +�(T V B,T V .T U C ) × �(A,T V B) +T V ×id +m +Note that it is natural in U and V . +By the Yoneda lemma (as the internal hom in � is the internal hom of copresheaves on Weil1), +�(A,T V B) = (U �→ �(A,B)(V ⊗U )) = [D (V ),�(A,B)] +so this category has coherently chosen powers by representable functors. +Now the original notions of (lax, strong, strict) tangent functors can be shown to correspond to +power-preservation properties of � -functors between � -categories with coherently chosen powers: +Theorem 5.1.8 (Garner (2018)). We have the following equivalences of 2-categories: +(i) the 2-category of � -categories with coherently chosen powers and TangCatLax; +(ii) the 2-category of � -categories with coherently chosen powers and power-preserving � functors +and TangCatStrong; +(iii) The 2-category of � -categories with coherently chosen powers and chosen-power-preserving � +functors and TangCatStrict. +Note that for a lax tangent functor (F,α) : � → �, the map +α.X : F.T.X → T.F.X +can be seen as the unique morphism induced by universality. Conversely, for a strong tangent functor, +the natural isomorphism α is the isomorphism from a power F.T.X to the coherently chosen power +T.F.X . In contrast, a strict tangent functor preserves the coherent choice of powers. +The ability to work with � -categories that do not have powers by representables allows for signif- +icant flexibility. It is useful to observe that there are � -categories which are not tangent categories. +Example 5.1.9. +(i) Every monoid (M ,m,e ) in � gives rise to a one-object � -category whose hom-object is M , com- +position is m, and unit is e . +(ii) Given a tangent category �, it is possible to take the full � -category over some set of objects �, +even though D may not be closed under iterated applications of the tangent functor. +(iii) The dual of a � -category is a � category, where +�op (A,B) = �(B,A) : Weil1 → Set. +Dually, in the case that � is a tangent category, �op will have coherent copowers by representables. +112 + +(iv) If a cartesian category � hasan infinitesimal object D (Definition 1.3.7), it hasa natural � -category +structure (with coherent copowers by representables) +�(A,B) : Weil1 → Set := �(A × D (−),B). +Using the dual tangent structure on �op from Proposition 1.3.8, the enrichment on � is exactly the +enrichment found by regarding � as the dual � -category of the tangent category �op . +As a final remark, note that the Yoneda lemma applies to � -categories, so there is an embedding +� �→ [�op ,Set]. +The powers and copowers by representables are computed pointwise in a presheaf category, so they +inherit the coherent choice. Thus the following holds: +Corollary 5.1.10 (Garner (2018)). Every tangent category embeds into a � -cocomplete representable +tangent category. +(In fact, showing that the Yoneda embedding applies to tangent categories is the main theorem of +Garner (2018).) +5.2 +Differential and anchored bundles as enriched structures +This section gives an enriched-categorical reinterpretation of the work in Chapter 2 regarding differen- +tial bundles and Chapters 3 and 4 regarding anchored bundles. +Differential bundles as enriched structures +As a first case study using the enriched perspective for tangent categories, consider differential bundles. +Most of the work in Chapter 2 uses the intuition that differential bundles are some sort of tangent- +categorical algebraic theory; this section will make that intuition concrete. Recall that a lift (Definition +2.2.3) is a map λ : E → T E . Using the enriched perspective and treating T E as a power (recall Defini- +tion 5.1.6), this gives the following correspondence: +λ : 1 → �(E ,E D ) +ˆλ : D → �(E ,E ) +The commutativity condition for ˆλ, then, is translated as follows: +E +T E +D × D +�(E ,E ) × �(E ,E ) +T E +T 2E +D +�(E ,E ) +⊙ +mE ,E ,E +ˆλ׈λ +ˆλ +λ +λ +T.λ +ℓ +That is, ˆλ is a semigroup morphism D → �(E ,E ). In any cartesian closed category with coproducts, +a semigroup may be freely lifted to a monoid using the "exception monad" (−) + 1 from functional +programming (see, for example, Seal (2013)). +Definition 5.2.1. Regard the following monoid as the one-object � -category Λ: +D × D + D + D + 1 +(ιL ◦m|ιL|ιL |ιR ) +−−−−−−−−→ D + 1 +m : (D + 1) × (D + 1) → D + 1 +113 + +Thus, a lift λ is exactly a functor ˆλ : Λ → �. Now check that morphisms are tangent natural transfor- +mations. Note that the semigroup D is commutative, so Λ = Λop; the choice of using Λop in the next +lemma is to be consistent with conventions used in Section 5.3. +Lemma 5.2.2. The category of lifts in a tangent category � is isomorphic to the category of � -functors +and � -natural transformations Λop → �. +Proof. Check that a � -natural transformation is exactly a morphism of lifts f : λ → λ′. Start with the +� -naturality square: +D + 1 +�(A,A) × �(A,B) +�(A,B) × �(B,B) +�(A,B) +(λ,f ◦!) +(f ◦!,l ) +mAB B +mAAB +Now, rewriting D + 1 → �(A,B) as a semigroup map D → �(A,B), we have: +D +(λ,f ◦!) +−−−→ �(A,A) × �(A,B) +mAAB +−−−→ �(A,B) +1 +(λ′,f ) +−−→ �(A,T A) × �(A,B) +1×T +−−→ �(A,T A) × �(T A,T B) +mA,T A,T B +−−−−−→ �(A,T B) +1 +T f ◦λ′ +−−−→ �(A,T B) +Similarly, the other path is exactly λ′ ◦ f . Thus, a � -natural transformation is exactly a morphism of +lifts. +It is a classical result in synthetic differential geometry that the object D has only one point. In the +case of � , this follows from the Yoneda lemma (regarding � as a Set-category): +� (1,D) = Weil1(�[x]/x 2,�) = {! : �[x]/x 2 → �} +Note that the natural idempotent e : id ⇒ id from Proposition 2.2.8, then, must be the point 0 : 1 → D. +Note that this idempotent is an absorbing element of the monoid D + 1, so for any f : X → D + 1, it +follows that m(f ,0◦!) = m(0◦!, f ) = 0◦!. A pre-differential bundle is exactly a lift with a chosen splitting +of the natural idempotent p ◦ λ. +Lemma 5.2.3. For every lift ¯λ : Λop → �, the natural idempotent e = p ◦ λ is exactly +1 +0−→ D +ιL−→ D + 1 → �(E ,E ). +Now that the natural idempotent is understood as a map in Λ, that idempotent splits to give the +theory of a pre-differential object. +Definition 5.2.4. The � -category Λ+ is given by the set of objects {0,1} with hom-Weil-spaces. Specifi- +cally, +• The hom-spaces are Λ+(1,1) = D + 1, otherwise Λ+(i, j) = 1. +• Composition (writing the original composition from Λ as m) is given by +m111 : (D + 1) × (D + 1) +m +−→ (D + 1) +m101 : 1× 1 +ιR−→ (D + 1) +otherwise: mi j k =! +114 + +Idempotent splittings are absolute (co)limits, and are preserved by all functors; as this is a limit +completion, we have the following. +Lemma5.2.5. The category of pre-differential bundles is exactly the category of �-functors Λ+ → � (that +is, �-valued presheaves). +It is straightforward to exhibit the category of differential bundles as a reflective subcategory of +pre-differential bundles in [Λ+,�] (so long as � has equalizers). +Proposition 5.2.6. The category of differential bundles in a tangent category � with T -equalizers and +T -pullbacks is a reflective subcategory of [Λ+,�]. +Proof. The category of pre-differential bundles in � is isomorphic to [Λ+,�]. By Corollary 2.4.8, the +category of differential bundles is the category of algebras for an idempotent monad on the category +of pre-differential bundles in �. The reflector sends a pre-differential bundle to the T -equalizer: +A +T E +T E +e .E +T.e +This equalizer will always exist if � has equalizers, and pullbacks of the projection will exist if � has +pullbacks, so the pullback is a differential bundle. This reflection gives a left-exact idempotent monad +on [Λ+,�] whose algebras are differential bundles. +Now, in the case that � is a locally presentable tangent category (such as � ), [Λ+,�] is locally pre- +sentable and so is the reflective subcategory of differential bundles; thus the following holds. +Corollary 5.2.7. If � is locally presentable, then DBun(�) is a locally presentable category. +Anchored bundles +There are two ways to think about anchored bundles: +(i) an anchored bundle is a differential bundle with an anchor A → T M , or +(ii) an anchored bundle is an involution algebroid without an involution. +These two perspectives can be unified by regarding A as a cylinder for the weighted limit T M , so that +the anchor is induced by the unique map A.T.0 → T.A.0: +Λ+ +� +A +T M +̺ +That is, the syntactic category for anchored bundles is constructed as a full � -category of Weil1 that +doesn’t include the map c . This may be found by taking the full subcategory of Weil1 whose objects +are constructed out of Wn,n ∈ �. +Definition 5.2.8. A Weil algebra has width k ∈ � if it can be written +V = ⊗0≤i