aa r X i v : . [ m a t h . C T ] J a n DUAL TANGENT STRUCTURES FOR ∞ -TOPOSES MICHAEL CHING
Abstract.
We describe dual notions of tangent bundle for an ∞ -topos,each underlying a tangent ∞ -category in the sense of Bauer, Burke andthe author. One of those notions is Lurie’s tangent bundle functor forpresentable ∞ -categories, and the other is its adjoint. We calculate thatadjoint for injective ∞ -toposes, where it is given by applying Lurie’s tangentbundle on ∞ -categories of points. In [BBC21], Bauer, Burke, and the author introduce a notion of tangent ∞ -category which generalizes the Rosick´y [Ros84] and Cockett-Cruttwell [CC14]axiomatization of the tangent bundle functor on the category of smooth man-ifolds and smooth maps. We also constructed there a significant example:the Goodwillie tangent structure on the ∞ -category C at diff ∞ of (differentiable) ∞ -categories, which is built on Lurie’s tangent bundle functor. That tan-gent structure encodes the ideas of Goodwillie’s calculus of functors [Goo03]and highlights the analogy between that theory and the ordinary differentialcalculus of smooth manifolds.The goal of this note is to introduce two further examples of tangent ∞ -categories: one on the ∞ -category of ∞ -toposes and geometric morphisms,which we denote T opos ∞ , and one on the opposite ∞ -category T opos op ∞ which,following Anel and Joyal [AJ19], we denote L ogos ∞ .These two tangent structures each encode a notion of tangent bundle for ∞ -toposes, but from dual perspectives. As described in [AJ19, Sec. 4], one canview ∞ -toposes either from a ‘geometric’ or ‘algebraic’ point of view. In theformer, an ∞ -topos is thought of as a generalized topological space ; for exam-ple, each actual topological space gives rise to an ∞ -topos S h ( X ) of sheaveson X (with values in the ∞ -category S of ∞ -groupoids), and each continuousmap X → Y determines a geometric morphism S h ( X ) → S h ( Y ). From thealgebraic perspective, an ∞ -topos is more like a category of (higher) groupoids with the ∞ -topos S being a prime example. The ‘algebraic’ morphisms be-tween two ∞ -toposes are those that preserve colimits and finite limits; i.e. theleft adjoints of the geometric morphisms. Our tangent structure on the ‘algebraic’ ∞ -category L ogos ∞ is simply therestriction of the Goodwillie tangent structure. For an ∞ -topos X , the tangentbundle T ( X ) is described by Lurie in [Lur17, 7.3.1]; the fibre of that bundle overeach object C ∈ X , i.e. the tangent space T C X , is the stabilization S p ( X /C ) ofthe slice ∞ -topos over C .The tangent structure on the ‘geometric’ category T opos ∞ is dual to that on L ogos ∞ ≃ T opos op ∞ in a sense described by Cockett and Cruttwell [CC14, 5.17].Lurie’s tangent bundle functor T : T opos op ∞ → T opos op ∞ has a left adjoint U op whose opposite U : T opos ∞ → T opos ∞ is the underlying functor for a tangentstructure. We call that structure the geometric tangent structure on T opos ∞ .The geometric tangent structure is representable in the sense of [CC14, Sec.5.2]. That is to say that the tangent bundle functor U : T opos ∞ → T opos ∞ isgiven by the exponential objects U ( X ) = X T for some object T in the ∞ -category T opos ∞ , with the tangent structure on U arising from so-called infinitesimal structure on T . This picture followsthe same pattern as the tangent category associated to a model of SyntheticDifferential Geometry (SDG) [CC14, 5.10], and we wonder which other featuresof SDG have a counterpart in the tangent ∞ -category T opos ∞ . That questionis not explored here.The infinitesimal object T is the ∞ -topos of ‘parameterized spectra’, alsoknown as the Goodwillie tangent bundle T ( S ) on the ∞ -category of spaces S ,with infinitesimal structure determined by the Goodwillie tangent structure.The exponential objects U ( X ) = X T ( S ) , however, do not seem to have a simpledescription, and we do not have a good understanding of the geometric tangentstructure in its entirety.We do offer a perspective on the geometric tangent structure by looking atits restriction to the subcategory of T opos ∞ consisting of those ∞ -toposesthat are injective in a sense that generalizes Johnstone’s notion of injective1-topos [Joh81]. We prove that the geometric tangent structure on injective ∞ -toposes is equivalent, via the ‘ ∞ -category of points’ functor, to the Good-willie tangent structure (restricted to those ∞ -categories that are presentableand compactly-assembled [Lur18, 21.1.2]). We therefore view the geometrictangent structure as an extension of the Goodwillie structure, to non-injective ∞ -toposes, in addition to being its dual.Here is an outline of the paper. In Section 1 we introduce the notion of ‘infin-itesimal object’ in an ∞ -category X with finite products (or, more generally, UAL TANGENT STRUCTURES FOR ∞ -TOPOSES 3 in any monoidal ∞ -category), and we describe what it means for a tangentstructure to be represented, or corepresented, by an infinitesimal object. Thesedefinitions extend to ∞ -categories the corresponding notions for tangent cat-egories due to Cockett and Cruttwell [CC14, Sec. 5].In Section 2 we turn to ∞ -toposes, and introduce the two perspectives givenby the ∞ -categories T opos ∞ and L ogos ∞ = T opos op ∞ . We then construct aninfinitesimal object T in T opos ∞ whose underlying ∞ -topos is T ( S ), and weshow that T represents and corepresents tangent structures on T opos ∞ and L ogos ∞ respectively. As mentioned, we prove that the latter of these is therestriction of the Goodwillie tangent structure of [BBC21] to the ∞ -toposes.Finally in Section 3 we give our description (Theorem 3.6) of the geometrictangent structure on the subcategory of T opos ∞ consisting of the injective ∞ -toposes, and make some explicit calculations that arise from that result. Acknowledgements.
This paper is an extension of [BBC21], and hence owesmuch to conversations with Kristine Bauer and Matthew Burke. The questionof how the Goodwillie tangent structure is related to ∞ -toposes was suggestedby communications with Alexander Oldenziel and with Eric Finster. This workwas supported by the National Science Foundation under grant DMS-1709032.1. Representable tangent ∞ -categories The goal of this section is to extend Cockett and Cruttwell’s notion of rep-resentable tangent category [CC14, Sec. 5.2] to the tangent ∞ -categories of[BBC21]. We start by developing the notion of infinitesimal object in thesense of [CC14, 5.6]. As with extending other tangent category notions to ∞ -categories, we do this by re-expressing the Cockett-Cruttwell definition interms of the category of Weil-algebras that Leung used in [Leu17] to charac-terize tangent structures.Throughout this paper we rely on notation and definitions from [BBC21]. Inparticular, let W eil be the monoidal category of Weil-algebras of [BBC21, 1.1].The objects of W eil are the augmented commutative semi-rings of the form N [ x , . . . , x n ] / ( x i x j | i ∼ j )where the relations are quadratic monomials determined by equivalence rela-tions on the sets { , . . . , n } , for n ≥
0. Morphisms in W eil are semi-ring ho-momorphisms that commute with the augmentations, and monoidal structureis given by the tensor product which is also the coproduct. Certain pullback MICHAEL CHING squares in the category W eil play a crucial role in the definition of tangentstructure; see [BBC21, 1.10, 1.12]. We refer to those diagrams as the tangentpullbacks .We now give our notion of infinitesimal object in a monoidal ∞ -category. Definition 1.1.
Let X ⊠ be a monoidal ∞ -category, and let X op, ⊠ denote thecorresponding monoidal structure on the opposite ∞ -category X op . An infin-itesimal object in X ⊠ is a monoidal functor D • : W eil ⊗ → X op, ⊠ . for which the underlying functor W eil → X op preserves the tangent pullbacks(i.e. maps tangent pullbacks in W eil to pushouts in X ). These monoidalfunctors (and their monoidal natural transformations) form an ∞ -category(see [Lur17, 2.1.3.7]), whose opposite we refer to as the ∞ -category of infini-tesimal objects in X ⊠ , denoted I nf( X ⊠ ). Example 1.2. If X is any ∞ -category with finite products, then we refer toan infinitesimal object in the cartesian monoidal ∞ -category X × simply asan infinitesimal object in X . We write I nf( X ) for the ∞ -category of theseinfinitesimal objects. Example 1.3.
Let X be an ordinary category with finite products. Our notionof infinitesimal object in X agrees with that given by Cockett-Cruttwell [CC14,5.6] except that we do not require that the objects D A are exponentiable (theaxiom there labelled [ Infsml.6 ]). That condition is added in Proposition 1.8below to explain when D • represents a tangent structure on X . Remark 1.4.
In the language of [BBC21, 5.7], an infinitesimal object in amonoidal ∞ -category X ⊠ is precisely a tangent object in the ( ∞ , X op, ⊠ that has a single object, mapping ∞ -category X op , and composition givenby the monoidal structure ⊠ .In the case that the monoidal structure on X is given by the cartesian product,as in Example 1.2, there is a particularly simple way to identify infinitesimalobjects. Proposition 1.5.
Let X be an ∞ -category with finite products. Taking under-lying functors determines an equivalence between I nf( X ) and the full subcate-gory of Fun( W eil op , X ) whose objects are the functors D • : W eil op → X suchthat We take opposites here to ensure that morphisms in I nf( X ⊗ ) are built from morphismsin X rather than X op . UAL TANGENT STRUCTURES FOR ∞ -TOPOSES 5 (1) D N is a terminal object in X ; (2) for Weil-algebras A, A ′ there is an equivalence in X D A ⊗ A ′ ˜ −→ D A × D A ′ , induced by the canonical maps from A and A ′ to their coproduct A ⊗ A ′ ; (3) D • maps the tangent pullbacks in W eil to pushouts in X .Proof. Since the monoidal structures on both W eil and X op are given by thecoproduct, we can apply [Lur17, 2.4.3.8] to identify the ∞ -category of lax monoidal functors W eil ⊗ → X op, × with the ∞ -category of all functors D • : W eil → X op . Conditions (1) and (2) on D • correspond to the case where thatlax monoidal functor is monoidal, and (3) is the condition that this monoidalfunctor is an infinitesimal structure. (cid:3) We now consider two ways in which an infinitesimal object can determine atangent structure, corresponding to those described in [CC14, Prop. 5.7 andCor. 5.18].
Proposition 1.6.
Let X ⊠ be a monoidal ∞ -category for which the monoidalstructure ⊠ commutes with pushouts in its first variable, and let D • be aninfinitesimal object in X ⊠ . Then there is a tangent structure on the ∞ -category X op given by the W eil -action map T : W eil × X op → X op ; T A ( C ) = D A ⊠ C for a Weil-algebra A and C ∈ X .Proof. The monoidal structure on X determines a monoidal functor ⊠ : X op, ⊠ → End( X op ) ◦ ; D D ⊠ − which, by hypothesis, preserves pullbacks. Here End( X op ) ◦ denotes the ∞ -category of endofunctors on X op with monoidal structure given by composition.Composing the infinitesimal object D • : W eil ⊗ → X op, ⊠ with the monoidalfunctor ⊠ , we get a monoidal functor D • ⊠ − : W eil ⊗ → End( X op ) ◦ which preserves the tangent pullbacks. Thus T is a tangent structure on X op ;see [BBC21, 2.1]. (cid:3) Definition 1.7.
We say the tangent structure T in Proposition 1.6 is corepre-sented by the infinitesimal object D • . A tangent structure is corepresentable if it is equivalent to a tangent structure corepresented by some infinitesimalobject. MICHAEL CHING
Proposition 1.8.
Let X ⊠ be a monoidal ∞ -category such that the monoidalproduct ⊠ preserves pushouts in its first variable. Let D • be an infinitesimalobject in X ⊠ such that for each Weil-algebra A , the functor D A ⊠ − : X → X admits a right adjoint Map ⊠ X ( D A , − ) . Then there is a tangent structure on the ∞ -category X given by U : W eil × X → X ; U A ( C ) = Map ⊠ X ( D A , C ) . Proof.
Note that we do not assume that the monoidal structure ⊠ as a wholeis closed , only that certain specific functors admit a right adjoint. Our firsttask is to show that those right adjoints can be chosen functorially.Let End L ( X ) ◦ ⊆ End( X ) ◦ be the full (monoidal) subcategory whose objectsare the left adjoint functors X → X , and similarly for End R ( X ). Then there isan equivalence of monoidal ∞ -categoriesadj : End L ( X ) ◦ ˜ −→ End R ( X ) ◦ that sends a functor to some choice of its right adjoint. Such an equivalencecan be constructed in a similar manner to that in [Lur09, 5.5.3.4].By hypothesis, the composite W eil ⊗ / / D • X op, ⊠ / / ⊠ End( X op ) ◦ / / ∼ op End( X ) ◦ takes values in End L ( X ) ◦ . Therefore we can form the composite monoidalfunctor W eil ⊗ → End L ( X ) ◦ / / ∼ adj End R ( X ) ◦ → End( X ) ◦ . It remains to show that the underlying functor A Map ⊠ X ( D A , − ) preserveseach of the tangent pullbacks in W eil. Suppose A A A A is one of those tangent pullbacks. Then it is sufficient to show that for all C , E ∈ X , the following diagram is a pullback of mapping spaces:Hom X ( E , Map ⊠ X ( D A , C )) Hom X ( E , Map ⊠ X ( D A , C ))Hom X ( E , Map ⊠ X ( D A , C )) Hom X ( E , Map ⊠ X ( D A , C )) UAL TANGENT STRUCTURES FOR ∞ -TOPOSES 7 We can write this diagram equivalently asHom X ( D A ⊠ E , C ) Hom X ( D A ⊠ E , C )Hom X ( D A ⊠ E , C ) Hom X ( D A ⊠ E , C )which is a pullback since − ⊠ E preserves pushouts by hypothesis. (cid:3) Definition 1.9.
We say that the tangent structure U in Proposition 1.8 is represented by the infinitesimal object D • . A tangent structure is representable if it is equivalent to one represented by some infinitesimal object. Definition 1.10.
Tangent structures on X and X op are dual if they are, re-spectively, represented and corepresented by the same infinitesimal object. Itfollows from comparing the hypotheses in Propositions 1.6 and 1.8 that anyrepresentable tangent structure on X has a dual tangent structure on X op .2. ∞ -toposes The main purpose of this paper is to construct dual tangent structures on acertain ∞ -category T opos ∞ and its opposite, whose objects are ∞ -toposes.In this section we introduce that ∞ -category and construct the infinitesimalobject T • that (co)represents those tangent structures. Our main reference forthe ∞ -category T opos ∞ is [Lur09, Sec. 6.3] where it is denoted RT op ∞ .To define the ∞ -category T opos ∞ we have to pay some attention to size issues.We assume three nested Grothendieck universes and refer to simplicial sets ofthese sizes as small , large and very large , respectively. Let C at ∞ denote the(very large) ∞ -category of large ∞ -categories. One of the objects in C at ∞ isthe (large) ∞ -category S of small spaces, i.e. Kan complexes. Definition 2.1. An ∞ -topos is a (large) ∞ -category X that is an accessibleleft exact localization of the ∞ -category P ( C ) = Fun( C op , S ) of presheaves onsome small ∞ -category C . In other words, there is some small ∞ -category C and an adjunction(2.2) P ( C ) X fg ⊣ such that g is fully faithful and accessible (preserves κ -filtered colimits forsome small regular cardinal κ ), and f preserves finite limits. MICHAEL CHING
Example 2.3.
The presheaf ∞ -category P ( C ), for a small ∞ -category C , isan ∞ -topos. In particular S is an ∞ -topos which, by [Lur09, 6.3.4.1], is aterminal object in the ∞ -category T opos ∞ which we now introduce. Definition 2.4.
Let T opos ∞ denote the subcategory of C at ∞ whose objectsare the ∞ -toposes and whose morphism are the geometric morphisms , i.e.those functors F : X → Y which admit a left adjoint Y → X that preservesfinite limits.The opposite ∞ -category T opos op ∞ is also equivalent to a subcategory of C at ∞ which, following Anel and Joyal [AJ19], we denote by L ogos ∞ . Definition 2.5.
Let L ogos ∞ be the subcategory of C at ∞ whose objects are the ∞ -toposes and morphisms are the functors that preserve small colimits andfinite limits. There is an equivalence L ogos ∞ ≃ T opos op ∞ that is the identity onobjects and maps a functor Y → X in L ogos ∞ to the right adjoint guaranteedby the Adjoint Functor Theorem [Lur09, 5.5.2.9], a geometric morphism X → Y ; see [Lur09, 6.3.1.8].The remainder of this section is devoted to the construction of an infinitesimalobject in T opos ∞ ≃ L ogos op ∞ , i.e. a monoidal functor T • : W eil ⊗ → L ogos ⊠ ∞ where ⊠ denotes the coproduct in the ∞ -category L ogos ∞ , or equivalently theproduct in T opos ∞ .The infinitesimal object T • is derived from the Goodwillie tangent structureconstructed in [BBC21] which we now recall. See [BBC21, 2.1] for the notionof tangent structure on an ∞ -category, and see [BBC21, Sec. 7] for the basicfacets of the Goodwillie tangent structure. Definition 2.6.
Let C at diff ∞ be the ∞ -category of (large) differentiable ∞ -categories and sequential-colimit-preserving functors. Let S fin , ∗ denote the(small) ∞ -category of pointed finite spaces.The Goodwillie tangent structure on C at diff ∞ is a map T : W eil × C at diff ∞ → C at diff ∞ given on a Weil-algebra A with n generators, and differentiable ∞ -category C ,by the subcategory(2.7) T A ( C ) = Exc A ( S n fin , ∗ , C ) ⊆ Fun( S n fin , ∗ , C ) An ∞ -category is differentiable if it has finite limits and sequential colimits which com-mute. Any ∞ -topos is differentiable by [Lur09, 7.3.4.7]. UAL TANGENT STRUCTURES FOR ∞ -TOPOSES 9 of functors S n fin , ∗ → C that are A -excisive in the sense described in [BBC21,7.1]. By [BBC21, 7.5], the inclusion (2.7) admits a left adjoint P A whichpreserves finite limits.The action of T on morphisms (in W eil and C at diff ∞ ) is described in detailin [BBC21, 7.7 and 7.14]. For a Weil-algebra morphism φ : A → A ′ and(sequential-colimit-preserving) functor G : C → D , we have T φ ( F ) : T A ( C ) → T A ′ ( D ); L P A ′ ( GL ˜ φ )where ˜ φ : S n ′ fin , ∗ → S n fin , ∗ is a functor built to the same pattern as the algebrahomomorphism φ ; see [BBC21, 7.12]. Proposition 2.8.
The Goodwillie tangent structure on C at diff ∞ restricts to atangent structure on the subcategory L ogos ∞ ⊆ C at diff ∞ .Proof. Suppose first that X is an ∞ -topos and A is a Weil-algebra with n generators. By [BBC21, 7.5], the ∞ -category T A ( X ) is an accessible left exactlocalization of the ∞ -topos Fun( S n fin , ∗ , X ), hence T A ( X ) is an ∞ -topos.Now let G : X → Y be a morphism in L ogos ∞ and φ : A → A ′ a Weil-algebramorphism. We have to show that the functor T φ ( G ) : T A ( X ) → T A ′ ( Y ); L P A ′ ( GL ˜ φ )is also in L ogos ∞ . Finite limits in T A ( X ) and T A ′ ( Y ) are calculated objectwise,and both G and P A ′ preserve those finite limits, so T φ ( G ) preserves finite limits.Let ( L α ) be a diagram in T A ( X ) with colimit L . Then we have an equivalence L ≃ P A (colim L α ) where colim denotes the (objectwise) colimit calculated inFun( S n fin , ∗ , C ). Then there is a sequence of equivalences: P A ′ ( GL ˜ φ ) ≃ P A ′ ( GP A (colim L α ) ˜ φ ) ≃ P A ′ ( G (colim L α ) ˜ φ ) ≃ P A ′ (colim GL α ˜ φ )where we have used equivalences of the form [BBC21, 7.11 and 7.25] to identifythe first and second lines, and the fact that G preserves colimits to identifythe second and third.Therefore T φ ( G ) preserves colimits, which completes the proof that T φ ( G ) isa morphism in L ogos ∞ , and hence that the W eil-action on C at diff ∞ restricts to A functor is excisive if it maps pushout squares to pullbacks. The notion of A -excisiveis a multivariable generalization of excisive that reflects the structure of the Weil-algebra A . a functor T : W eil × L ogos ∞ → L ogos ∞ . It remains to show that T preserves the tangent pullbacks in W eil. Sincelimits in L ogos ∞ are calculated in C at ∞ by [Lur09, 6.3.2.3], that claim followsfrom [BBC21, 7.36 and 7.38]. (cid:3) We now construct the desired infinitesimal object.
Definition 2.9.
Define a functor T • : W eil → L ogos ∞ by T A := T A ( S ) , i.e. by evaluating the Goodwillie tangent structure at the ∞ -topos S of spaces. Proposition 2.10.
The functor T • is the underlying functor of an infinitesi-mal object in T opos ∞ ≃ L ogos op ∞ .Proof. We apply Proposition 1.5:(1) By [Lur09, 6.3.4.1], T N = S is the terminal object in T opos ∞ .(2) The canonical map T A ⊗ A ′ ( S ) → T A ( S ) ⊠ T A ′ ( S )is an equivalence of ∞ -toposes; this claim follows from Lemma 2.15below by taking X = T A ′ ( S ) and noting that T A ⊗ A ′ ( S ) = T A ( T A ′ ( S )).(3) For each tangent pullback in W eil, the corresponding diagram T A ( S ) T A ( S ) T A ( S ) T A ( S )is a pullback in L ogos ∞ (and hence a pushout in T opos ∞ ). This claimis part of the condition that T is a tangent structure on L ogos ∞ , asverified in the last part of the proof of Proposition 2.8. (cid:3) To show that the infinitesimal object T • represents, and corepresents, tangentstructures on the ∞ -categories T opos ∞ and L ogos ∞ respectively, we verify theconditions of Propositions 1.6 and 1.8. UAL TANGENT STRUCTURES FOR ∞ -TOPOSES 11 Proposition 2.11.
The product ⊠ on T opos ∞ preserves pushouts in eachvariable individually.Proof. We use the fact, e.g. see [AL18, 2.15], that the coproduct ⊠ in L ogos ∞ is given by the tensor product of cocomplete ∞ -categories; see [Lur17, 4.8.1].Then [AL18, 4.24] tells us that for ∞ -toposes Y , X , we have Y ⊠ X ≃ Fun lim ( X op , Y )where the right-hand side is the ∞ -category functors Y op → X that preservesmall limits.We therefore have to show that Fun lim ( X op , − ) preserves pushouts in T opos ∞ which, by [Lur09, 6.3.2.3], are pullbacks in C at ∞ . That claim is a consequenceof [RV20, 6.4.12] which implies that pullbacks in the ∞ -cosmos of ∞ -categorieswith small limits are given by pullbacks in the ∞ -cosmos of all ∞ -categories. (cid:3) Proposition 2.12.
For each Weil-algebra A , the functor T A ⊠ − : T opos ∞ → T opos ∞ admits a right adjoint.Proof. Anel and Lejay [AL18, 4.37] show that any compactly-generated ∞ -topos is exponentiable, so it is sufficient to show that T A = Exc A ( S n fin , ∗ , S ) iscompactly-generated. It follows from [Lur09, 5.3.5.12] that the presheaf ∞ -category Fun( S n fin , ∗ , S ) is compactly-generated, so by [Lur09, 5.5.7.3] it is suffi-cient to note that Exc A ( S fin , ∗ , S ) is closed under filtered colimits in Fun( S n fin , ∗ , S ),which follows from the fact that filtered colimits in S commute with pull-backs. (cid:3) Theorem 2.13.
The infinitesimal object T • represents a tangent structure U on the ∞ -category T opos ∞ and corepresents a tangent structure T on the ∞ -category L ogos ∞ .Proof. We apply Propositions 1.8 and 1.6, respectively, using the results ofPropositions 2.11 and 2.12. (cid:3)
We refer to the tangent structure U on T opos ∞ as the geometric tangentstructure , and we begin the study of that structure in the next section. Thecorepresented tangent structure T turns out to be much more familiar. Proposition 2.14.
The corepresentable tangent structure of Theorem 2.13 isequivalent to the restriction of the Goodwillie tangent structure of [BBC21] tothe subcategory L ogos ∞ ⊆ C at diff ∞ , as described in Proposition 2.8.Proof. We have to show that the following diagram of monoidal functors com-mutes up to monoidal equivalence. W eil ⊗ L ogos ⊠ ∞ End( L ogos ∞ ) ◦ T • ⊠ T This claim is a consequence of the following lemma. (cid:3)
Lemma 2.15.
Let X be an ∞ -topos, and A a Weil-algebra. Then there is acanonical equivalence in L ogos ∞ T A ( S ) ⊠ X ˜ −→ T A ( X ) built from the maps T A (!) : T A ( S ) → T A ( X ) and T η ( X ) : X → T A ( X ) , where ! : S → X and η : N → A are the maps from the initial objects in L ogos ∞ and W eil respectively.Proof. Using the approach from the proof of Proposition 2.11, it is sufficientto show that the canonical map T A ( X ) → Fun lim ( X op , T A ( S )); L Hom X ( − , L )is an equivalence. Noting that limits in T A ( S ) = Exc A ( S n fin , ∗ , S ) are calculatedobjectwise, we can rewrite the target of that map as the ∞ -category of A -excisive functors S n fin , ∗ → Fun lim ( X op , S ), since pullbacks in Fun lim ( X op , S ) arealso calculated objectwise. Our claim then follows by noting that the map X → Fun lim ( X op , S ); Hom X ( − , L )is an equivalence by [AL18, 4.24] again. (cid:3) The geometric tangent structure
The aim of this section is to begin a study of the geometric tangent structure on T opos ∞ given by Thorem 2.13. By definition, the tangent bundle constructionfor this tangent structure, U ( X ) = X T ( S ) , UAL TANGENT STRUCTURES FOR ∞ -TOPOSES 13 is the exponential object for the ∞ -topos T ( S ). These exponential objectsdo not appear to be easy to calculate, though a general construction can begleaned from the proofs of [AL18, 4.33] or [Lur18, 21.1.6.12].Those approaches proceed by first calculating the exponential object for acollection of ∞ -toposes that are injective in the sense of Definition 3.1 below.Since any ∞ -topos X can be written as the pullback of a diagram of injective ∞ -toposes [Lur18, 21.1.6.16], and since U preserves pullbacks, one might beable to recover an explicit description of U ( X ) from those calculations, thoughwe do not attempt that here.It turns out that the geometric tangent structure for injective ∞ -toposes hasa compelling description. We prove in Theorem 3.6 that the ‘ ∞ -category ofpoints’ construction determines an equivalence between that geometric tangentstructure and the Goodwillie tangent structure restricted to the presentablecompactly-assembled ∞ -categories of [Lur18, 21.1.2]. Thus, on injective ∞ -toposes at least, one can view the geometric tangent structure as simply adifferent incarnation of the Goodwillie structure. Definition 3.1. An ∞ -topos X is injective if X is a retract, in T opos ∞ , ofa presheaf ∞ -category P ( D ) where D is a small ∞ -category that has finitelimits. Let I nj T opos ∞ be the full subcategory of T opos ∞ consisting of theinjective ∞ -toposes. Remark 3.2. An ∞ -topos is injective if and only if it satisfies the equivalentconditions of [Lur18, 21.1.5.4]; our definition is an intermediate step in proving(4) implies (1) in that result. Our definition is also equivalent to that of Aneland Lejay in [AL18, 4.6]; combine (4) of [Lur18, 21.1.5.4] with [AL18, 2.6].The attraction of injective ∞ -toposes is that they can be recovered from their ∞ -categories of ‘points’. Definition 3.3.
The ∞ -category of points of an ∞ -topos X is the ∞ -category p ( X ) := Fun ∗ ( X , S )of functors X → S that preserve small colimits and finite limits, i.e. thegeometric morphisms S → X . Since S is the terminal object in T opos ∞ ,the objects of p ( X ) are indeed the ‘generalized points’ of the ∞ -topos X .By [Lur18, 21.1.1.6], the construction of p ( X ) extends to a functor p : T opos ∞ → C at acc ,ω ∞ whose target is the subcategory of C at ∞ consisting of the ∞ -categories thatare accessible and admit filtered colimits, with morphisms the filtered-colimit-preserving functors. Proposition 3.4.
The functor p restricts to an equivalence of ∞ -categories p : I nj T opos ∞ ˜ −→ C at pr , ca ∞ ⊆ C at acc ,ω ∞ whose target consists of those ∞ -categories C that are both presentable andcompactly-assembled, in the sense of [Lur18, 21.1.2.1] . The inverse to p mapssuch an ∞ -category C to the ∞ -topos Fun ω ( C , S ) of filtered-colimit-preservingfunctors C → S .Proof. The inverse map Fun ω ( − , S ) is fully faithful by [Lur18, 21.1.5.3], essen-tially surjective by [Lur18, 21.1.5.4(1)], and has inverse p by [Lur18, 21.1.5.1]. (cid:3) Remark 3.5.
An alternative approach to the proof of Proposition 3.4 isin [AL18, 4.9] which identifies C at pr , ca ∞ with the full subcategory of C at acc ,ω ∞ consisting of retracts of the preasheaf ∞ -categories.We now prove the main result of this section, giving a calculation of the geo-metric tangent structure for injective ∞ -toposes. Theorem 3.6.
The equivalence p of Proposition 3.4 underlies an equivalenceof tangent structures p : ( I nj T opos ∞ , U ) ˜ −→ ( C at pr , ca ∞ , T ) between the geometric tangent structure on I nj T opos ∞ ⊆ T opos ∞ and theGoodwillie tangent structure on C at pr , ca ∞ . Proof.
We start by showing that the Goodwillie tangent structure restrictsto C at pr , ca ∞ . Suppose C is presentable and compactly-assembled. Then, byRemark 3.5, C is a retract, in C at acc ,ω ∞ , of a presheaf ∞ -category. HenceFun( S n fin , ∗ , C ) is a retract of a presheaf ∞ -category, so is also presentable andcompactly-assembled. Finally, the map P A of Definition 2.6 displays T A ( C )as a retract, in C at acc ,ω ∞ , of Fun( S n fin , ∗ , C ), so T A ( C ) is also presentable andcompactly-assembled. It is unclear to this author whether an arbitrary presentable compactly-assembled ∞ -category C is differentiable, so C at pr , ca ∞ is perhaps not a subcategory of C at diff ∞ . However,the construction of the Goodwillie tangent structure in [BBC21] can be carried out with C at diff ∞ replaced by the ∞ -category C at pr ∞ of presentable ∞ -categories and filtered-colimit-preserving functors, of which C at pr , ca ∞ is a full subcategory. Alternatively the reader mayrestrict attention to the compactly-generated ∞ -categories which correspond to those ∞ -toposes that are presheaves on a small ∞ -category that has finite limits. UAL TANGENT STRUCTURES FOR ∞ -TOPOSES 15 Now let q : C at pr , ca ∞ → I nj T opos ∞ be the inverse to p given by q ( C ) =Fun ω ( C , S ). We then define natural equivalences α : qT A ˜ −→ U A q with components α C : Fun ω ( T A ( C ) , S ) → U A (Fun ω ( C , S )) = Fun ω ( C , S ) T A ( S ) as follows.First note that the proof of Lemma 2.15 relies purely on the identification of ⊠ with the tensor product for presentable ∞ -categories, and so extends to givea canonical equivalence T A ( S ) ⊠ C ˜ −→ T A ( C )for all presentable ∞ -categories. By [Lur18, 21.1.4.3], and the argumentof [Lur18, 21.1.6.9], we also have equivalences of ∞ -toposes of the formFun ω ( X ⊠ C , S ) ˜ −→ Fun ω ( C , S ) X . Combining these two maps, with X = T A ( S ), yields the desired equivalence α C . Note that the existence of these equivalences also verifies that U restrictsto a tangent structure on the subcategory I nj T opos ∞ ⊆ T opos ∞ .The construction of α C is natural (in A and C ) and monoidal (with respectto the tensor product of Weil-algebras), so the maps α yield an equivalenceof tangent structures with underlying functor q , whose inverse is the requiredtangent equivalence p . (cid:3) Corollary 3.7.
For an injective ∞ -topos X : U X ≃ Fun ω ( T ( p X ) , S ) . Corollary 3.8.
Let X be an injective ∞ -topos, and let x : S → X be a gener-alized point in X . Then the geometric tangent space U x X (in T opos ∞ ) existsand has ∞ -category of points p ( U x X ) ≃ T x ( p X ) . Note, however, that we have no reason to believe that U x X is injective.Proof. By definition the tangent space is the pullback in T opos ∞ of the form U x X U XS X ǫ X x This pullback exists, and is preserved by each U A , since T opos ∞ has all limits,and U A is a right adjoint. The functor p : T opos ∞ → C at acc ,ω ∞ is a right adjointby [Lur18, 21.1.1.6], so applying p we get a pullback diagram in C at acc ,ω ∞ , andhence in C at ∞ , of the form p ( U x X ) T ( p X ) ∗ p X ǫ p X x which identifies p ( U x X ) with the Goodwillie tangent space T x ( p X ), as claimed. (cid:3) Lurie’s proof that a compactly-generated ∞ -topos is exponentiable relies ona lemma [Lur18, 21.1.6.16] that says every ∞ -topos is a pullback of injective ∞ -toposes; see also [AL18, 2.8]. Since the geometric tangent bundle functor U is a right adjoint, we could in principle use such pullbacks, together withthe calculation in Corollary 3.7, to give an explicit description of U X for any ∞ -topos X .Theorem 3.6 also gives us a different perspective on the relationship betweenthe geometric and Goodwillie tangent structures. We saw in Theorem 2.13that those tangent structures are dual, but now we can also view the geomet-ric tangent structure on T opos ∞ as an extension of the Goodwillie tangentstructure. If we think of an ∞ -topos as an ∞ -category (of points) togetherwith additional information, then the geometric tangent bundle on an injective ∞ -topos simply is the Goodwillie tangent bundle. The full geometric tangentstructure on T opos ∞ extends the Goodwillie structure to ∞ -toposes for whichthat additional information is nontrivial.We conclude by giving some simple calculations of the geometric tangent struc-ture based on Corollary 3.7. Example 3.9.
For the terminal ∞ -topos S , we have U ( S ) ≃ S . Since U is a right adjoint and S is a terminal object, we did not need The-orem 3.6 to prove this fact. However, we now see that it corresponds to thecalculation T ( ∗ ) ≃ ∗ for the Goodwillie tangent bundle on the trivial ∞ -category. Example 3.10.
Let C be a small ∞ -category, and let A C := P (¯ C ) = Fun(¯ C op , S ) UAL TANGENT STRUCTURES FOR ∞ -TOPOSES 17 be the affine ∞ -topos of [AL18, 2.7], where ¯ C is obtained by freely addingfinite limits to C . The ∞ -category of points of A C is p ( A C ) ≃ P ( C op ) = Fun( C , S )and so the geometric tangent bundle is given by U ( A C ) ≃ Fun ω (Fun( C , T ( S )) , S ) . By Corollary 3.8, the geometric tangent space U x ( A C ), for a functor x : C → S ,has ∞ -category of points p ( U x A C ) ≃ Fun x ( C , T ( S ))the ∞ -category of functors which lift x to T ( S ) along the projection map T ( S ) → S . Example 3.11.
Taking C = ∗ in Example 3.10 we obtain an affine ∞ -topos A ∗ whose ∞ -category of points is S . Therefore U ( A ∗ ) ≃ Fun ω ( T ( S ) , S )and, for an ∞ -groupoid x ∈ S , p ( U x A ∗ ) ≃ T x S = S p ( S /x )the ∞ -category of spectra parameterized over x . References [AJ19] Mathieu Anel and Andr´e Joyal,
Topo-logie , available at http://mathieu.anel.free.fr/mat/doc/Anel-Joyal-Topo-logie.pdf (2019).[AL18] Mathieu Anel and Damien Lejay,
Exponentiable higher toposes , arxiv:1802.10425.[BBC21] Kristine Bauer, Matthew Burke, and Michael Ching,
Tangent ∞ -categories andgoodwillie calculus , arxiv:2101.07819.[CC14] J. R. B. Cockett and G. S. H. Cruttwell, Differential structure, tangent structure,and SDG , Appl. Categ. Structures (2014), no. 2, 331–417. MR 3192082[Goo03] Thomas G. Goodwillie, Calculus. III. Taylor series , Geom. Topol. (2003), 645–711 (electronic). MR 2 026 544[Joh81] Peter T. Johnstone, Injective toposes , Continuous Lattices (Berlin, Heidelberg)(Bernhard Banaschewski and Rudolf-Eberhard Hoffmann, eds.), Springer BerlinHeidelberg, 1981, pp. 284–297.[Leu17] Poon Leung,
Classifying tangent structures using Weil algebras , Theory Appl.Categ. (2017), Paper No. 9, 286–337. MR 3613087[Lur09] Jacob Lurie, Higher Topos Theory , Annals of Mathematics Studies, vol. 170,Princeton University Press, Princeton, NJ, 2009. MR 2522659 (2010j:18001)[Lur17] ,
Higher Algebra , available at , version datedSeptember 18, 2017.[Lur18] ,
Spectral Algebraic Geometry , available at ,version dated February 3, 2018. [Ros84] J. Rosick´y,
Abstract tangent functors , Diagrammes (1984), JR1–JR11.MR 800500[RV20] Emily Riehl and Dominic Verity, Elements of ∞ -Category Theory , available at , version dated September 29, 2020. Email address : [email protected]@amherst.edu