A bivariant Yoneda lemma and (∞,2)-categories of correspondences* Andrew W. Macpherson August 12, 2021 Abstract A well-known folklore states that if you have a bivariant homology theory satisfying a base change formula, you get a representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual adjunction, these data are actually equivalent. In other words, a 2-category of correspondences is the universal way to attach to a given 1-category a set of right adjoints that satisfy a base change formula . Through a bivariant version of the Yoneda paradigm, I give a definition of correspondences in higher category theory and prove an extension theorem for bivariant functors. Moreover, conditioned on the existence of a 2-dimensional Grothendieck construction, I provide a proof of the aforementioned universal property. The methods, morally speaking, employ the ‘internal logic’ of higher category theory: they make no explicit use of any particular model. Contents 1 Introduction 2 2 Higher category theory 7 2.1 Homotopy invariance in higher category theory . .............. 7 2.2 Models of n-categorytheory .................................. 9 2.3 Mappingobjectsinhighercategories . ............. 11 2.4 Enrichmentfromtensoring. .......... 12 2.5 Bootstrapping n-naturalisomorphisms . 14 2.6 Fibrations ...................................... ....... 15 2.7 Grothendieckconstruction . ........... 17 2.8 Evaluationmap................................... ....... 18 2.9 Adjoint functors and adjunctions in 2-categories . ................. 23 arXiv:2005.10496v2 [math.CT] 10 Aug 2021 3 Bivariance 25 3.1 Basechange...................................... ...... 26 3.2 Bivariantfunctors ............................... ......... 27 3.3 Universalbivariantextensions . ............ 30 3.4 Bi-Cartesianfibrations . .......... 32 3.5 Freebivariantfibrations . .......... 35 *AMS Math subject classification tags: 18G99 “None of the above, but in this section (derived categories)”; 18D20 “Enriched categories”; 18A40 “Adjoint functors”; 55U40 “Topological categories”; 55P65 “Homotopy functors”. The bulk of this work was supported by World Premier International Research Center Initiative (WPI), MEXT, Japan, and by JSPS KAKENHI Grant Number JP19K14522. Later revisions were supported by JSPS KAKENHI Grant Number JP16H06337. 1 3.6 TwistedCartesianfibrations . ........... 37 3.7 BivariantYonedalemma .. .. .. .. .. .. .. .. .. .. .. .. .. .. ........ 38 3.8 Localrepresentationofspans . ........... 40 3.9 Spansonabivariantfunctor . .......... 43 3.10 Omake: composition of correspondences . .............. 46 4 Universal property (†) 48 4.1 2-categoricalGrothendieckintegration . ................ 48 4.2 Spansonafamilyofbivariantfunctors . ............ 49 4.3 Correspondences and Cartesian fibrations . .............. 52 4.4 Monoidalstructure............................... ......... 55 5 Examples 57 5.1 Cartesian and co-Cartesian symmetric monoidal structures............... 57 5.2 Categoriesofcoefficients . ........... 59 1 Introduction Let D be a category with fibre products. We can define a category CorrD of correspondences in D as follows: Definition. The category CorrD of correspondences in D comprises the following data: (0) Objects of CorrD are objects of D. (1) A map X 9 Y in CorrD is a span K ❅ p ⑦⑦ ❅❅ q ⑦⑦ ❅❅ ⑦⑦ ❅❅ ⑦~ ⑦ ❅ X Y (◦) The composite of spans [X ← K → Y ] and [Y ← K ′ → Z] is calculated by the fibre product ′ K ×Y K ✈ ❍❍ ✈✈ ❍❍ ✈✈ ❍❍ ✈✈ ❍❍ ✈{ ✈ ❍$ X Z. of K with K ′ over Y . The associative law for composition can be deduced from that of fibre products. The reader may object that fibre products are defined only up to unique isomorphism and not on the nose: CorrD is more properly a (2,1)-category whose mapping objects are groupoids. This issue is, in a sense, the crux of the whole paper, and we’ll return to it shortly. Each map f : X → D in D defines two morphisms in CorrD: f • a ‘right-way’ map [X←˜ X → Y ] from X to Y , written f!; f • a ‘wrong-way’ map [Y ← X→˜ X] from Y to X, written f !. 2 These right and wrong way maps satisfy a compatibility condition called the base change formula, which says that if g¯ X ×Y Z / X f¯ f g Y / Z ! ¯! is a fibre product square in D, then g! f = f g¯!, since either composite is given by the same correspondence [X ← X ×Y Z → Y ]. This is in a precise sense the only relation: one can show, by explicitly checking relations, that to define a functor out of CorrD it is enough to define a covariant and a contravariant functor out of D such that this base change formula is satisfied. Correspondences and cohomology Students of the homology of manifolds, and its big city cousin, Grothendieck’s theory of motives, will be familiar with settings in which data of the above form is available. That is, we are given a contravariant functor H : D → Eop which has, at least for a subclass SD of morphisms in D, also a covariant behaviour in the form of transfer or Gysin maps. Perhaps after imposing further conditions, these satisfy a base change formula and hence define a functor op Spex(H) : Corr(D; SD) → E where Corr(D; SD) ⊆ Corr(D) is the subcategory where morphisms are those spans whose wrong- way component belongs to SD. Example (Cohomology). Let SmC denote the category of smooth quasi-projective varieties over C and let pr|sm be the class of projective submersions. De Rham cohomology with complex coeffi- cients defines a functor • C Z op HdR(−; ): SmC → (VectC) into the category of Z-graded C-vector spaces. For morphisms f : X → Y in pr|sm there is also • •−dimR(X/Y ) a transfer map f! : H (X;C) → H (Y ;C) defined using Poincaré duality, and this map satisfies the base change compatibility with pullbacks. (The transfer map is actually defined for any projective morphism in SmC, but we cannot always pull back projective morphisms and keep the domain smooth.) We therefore obtain a functor • op Spex(⊕•HdR) : Corr(SmC,pr|sm) → VectC where we summed over the grading to avoid having to deal with the dimension shift. This con- struction parallels that of HdR as a functor on Grothendieck’s category of Chow motives. Example (Cochains). The homotopy coherence issue one encounters when attempting to construct a chain level version of HdR is familiar. Although this can be solved using a sufficiently robust model for de Rham cochains, the modern approach is to find a native ∞-categorical construction. Introducing Gysin maps and base change, the coherence issue is further compounded: the ∞-functor lift of Spex(HdR) must somehow encode coherences between base change homotopies in arbitrary composite grids of base change squares, in a 2-dimensional version of that classic problem of higher category theory. (The 2-cells in the (2,1)-category CorrD would also have to come into play to ‘see’ these base change homotopies.) Example (Derived source). If we want to generalise the domain to include objects of a homotopical nature, such as derived manifolds or orbifolds — and we do actually want to do this in applications to Gromov-Witten theory [MR17] — then the source category is also an ∞-category; now we have to explain how to construct an ∞-category of correspondences from an input ∞-category, and how to construct ∞-functors out of it.1 1Another, completely separate, difficulty is entailed in this case, which is that Poincaré duality fails even at the level of homology. We don’t address this here, but see [STV11]. 3 In the first instance, by-hand methods are quite sufficient for constructing a functor from the correspondences category. Subsequent examples, however, highlight two questions that we must answer before correspondences can be made available in a homotopical context: i) When constructing functors from CorrD into a higher category, how can we express the compatibility needed between the covariant and contravariant parts? ii) When D is an ∞-category, the given Definition cannot be regarded as defining an ∞- category; it only specifies the 0- and 1-simplices, and the inner face map from two to one. How can we express the full associative structure of CorrD? There is an obvious way to attack problem ii) by extending the Definition to define a complete Segal space SpanD,• whose associated ∞-category deserves to be called CorrD. This is the ap- proach taken, for example, in [DK19, Chap. 10]. However, when it comes to problem i), we are still left with trying to manipulate an infinite hierarchy of compatibilities. While it is surely possible with enough effort to chase down these compatibilities directly, a more satisfying strategy is to try to define a ‘machine’ that churns out the necessary relations automatically. For the present example, we need a machine that manufactures relations between the covariant and contravariant aspects of de Rham cohomology. To construct this machine, we will pass into the world of (∞,2)-category theory. Categorification We can extend the preceding Definition and enhance CorrD to a 2-category of correspondences in D by defining 2-cells as follows: (2) A morphism from [X ← K → Y ]to[X ← K ′ → Y ] is a commutative diagram K ▼ qqq ▼▼▼ qx qq ▼▼& Xf 8 Y ▼▼▼ rr ▼▼▼ rrr K ′ r in D. (There are other interesting possibilities for 2-cells in CorrD [GR17; Hau14], but we will not pursue them in this paper.) With this class of 2-cells, each ‘right-way’ map is left adjoint, in the sense internal to the 2-category CorrD, to the corresponding ‘wrong-way’ map. The unit, respectively counit of the ! adjunction f! ⊣ f is depicted X ×Y X X ✈ O ❍❍ ⑦ ❅❅ ✈✈ ❍❍ ⑦⑦ ❅❅ ✈✈ e ❍❍ ⑦⑦ ǫ ❅❅ ✈✈ ❍❍ ⑦ ❅❅ ✈{ ✈ ❍# ⑦ ⑦