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PARAMETRIZED HIGHER THEORY AND HIGHER ALGEBRA: EXPOSÉ I – ELEMENTS OF PARAMETRIZED HIGHER

CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH

Abstract. We introduce the basic elements of the theory of parametrized ∞-categories and between them. These notions are defined as suitable of ∞-categories and functors between them. We give as many examples as we are able at this stage. Simple operations, such as the formation of opposites and the formation of ∞-categories, become slightly more involved in the parametrized setting, but we explain precisely how to perform these constructions. All of these constructions can be performed explicitly, without resorting to such acts of desperation as straightening. The key results of this Exposé are: (1) a universal characterization of the 푇-∞-category of 푇-objects in any ∞-category, (2) the existence of an internal Hom for 푇-∞-categories, and (3) a parametrized .

Contents 1. Parametrized ∞-categories 1 2. Examples of parametrized ∞-categories 3 3. Parametrized opposites 7 4. Parametrized 7 5. Constructing 푇-∞-categories via pairings 8 6. A technical result: the strong pushforward 9 7. 푇-objects in ∞-categories 11 8. Parametrized fibrations 14 9. Parametrized functor categories 15 10. The parametrized Yoneda embedding 19 Appendix A. Notational glossary 20 References 21

1. Parametrized ∞-categories Suppose 퐺 a finite . At a minimum, a 퐺-∞-category should consist of an ∞- category 퐶 along with a weak action 휌 of 퐺. In particular, for every element 푔 ∈ 퐺, one should have an equivalence 휌(푔)∶ 퐶 ∼ 퐶, and for every 푔, ℎ ∈ 퐺, one should have a natu- ral equivalence 휌(푔ℎ) ≃ 휌(푔) ∘ 휌(ℎ), and these natural equivalences should then in turn be constrained by an infinite family of that express the higher associativity of 휌. Following [6], we will want to encode this data very efficiently as a cocartesian 퐶 퐵퐺. However, as we emphasize in the introduction [2], a 퐺-∞-category should contain yet more information. For example, one wishes also to retain the information of the “honest” (not ) fixed-point ∞-category 퐶퐻 of 퐶 for any subgroup 퐻 ≤ 퐺 along with the 퐻 ∼ 푔퐻푔−1 conjugation action equivalence 푐푔 ∶ 퐶 퐶 . All of these data should fit together via 1 2 , EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH the obvious restriction functors. These should be data in addition to a cocartesian fibration to 퐵퐺. It is a classical result of Tony Elmendorf and Jim McClure that to give a 퐺-equivariant space, one needs only to give the data of the honest fixed-point spaces, the residual actions, and their compatibilities with restriction. That is, again falling in line with the approach of [6], the of 퐺-spaces can be identified with the homotopy theory of left fibrations with target the opposite of the orbit category of 퐺. In the context of more general homotopy theories, Marc Stephan [13] and Julie Bergner [7] have proved versions of this theorem that exhibit equivalences of homotopy theories op between categories enriched in 퐺-spaces and cocartesian fibrations with target O퐺 . So we may simply take the latter as the definition of a 퐺-equivariant homotopy theory, and that is exactly what we will do.

1.1. Definition. Suppose 퐺 a profinite group. Write O퐺 for the ordinary category of transi- tive, continuous 퐺-sets. In particular, its objects are, up to equivalence, the orbits 퐺/퐻 in which the stabilizer 퐻 ≤ 퐺 is an open subgroup. A 퐺-equivariant ∞-category or, more briefly, a 퐺-∞-category 퐶 is a pair (퐶, 푝) consisting of an ∞-category 퐶 and a cocartesian fibration op 푝∶ 퐶 O퐺 . The cocartesian fibration 푝 will be called the structure map for 퐶. From time to time, we will refer simply to 퐶 alone as a 퐺-∞-category, leaving the structure map implicit. If (퐶, 푝) and (퐷, 푞) are two 퐺-∞-categories, then a 퐺-functor (퐶, 푝) (퐷, 푞) is a functor op 퐶 퐷 over O퐺 that carries 푝-cocartesian edges to 푞-cocartesian edges. A 퐺-space is a 퐺-∞-category (퐶, 푝) in which every edge of 퐶 is 푝-cocartesian. In par- ticular, the structure map 푝 of a 퐺-∞-category 퐶 is a left fibration just in case (퐶, 푝) is a 퐺-space. A 퐺-functor between 퐺-spaces will simply be called a 퐺-map. 1.2. If a 퐺-∞-category 퐶 is the of an ordinary category, then this ordinary category is a Grothendieck opfibration that corresponds to a categorical coefficient system in the sense of Blumberg–Hill [8]. As we have described in the Introduction, we have cause to contemplate more general bases. This leads us to the following definition. 1.3. Definition. Suppose 푇 an ∞-category. Then a 푇-parametrized ∞-category – or more briefly, a 푇-∞-category – is a pair (퐶, 푝) consisting of an ∞-category 퐶 and a cocartesian fibration 푝∶ 퐶 푇op. Suppose (퐶, 푝) and (퐷, 푞) two 푇-∞-categories. A 푇-parametrized functor – or more briefly, a 푇-functor – (퐶, 푝) (퐷, 푞) is a functor 퐶 퐷 over 푇op that carries 푝-cocartesian edges to 푞-cocartesian edges. If (퐶, 푝) is a 푇-∞-category whose fibers are all 푛-categories in the sense of [9, §2.3.4], then we will say that (퐶, 푝) is a 푇-푛-category. A 푇-space is a 푇-∞-category (퐶, 푝) in which every edge of 퐶 is 푝-cocartesian. A 푇-functor between 푇-spaces will be called a 푇-map.

1.4. Notation. Suppose 푇 an ∞-category. If (퐶, 푝) is a 푇-∞-category and 푉 ∈ 푇0 is an object, then we write −1 퐶푉 ≔ 푝 {푉} for the fibre of 푝 over 푉. EXPOSÉ I – ELEMENTS OF PARAMETRIZED 3

The collection of 푇-∞-categories and 푇-functors defines an ∞-category Cat∞,푇: pre- + cisely, we define Cat∞,푇 as the simplicial nerve of the full simplicial of 푠Set/푇op spanned by the fibrant objects for the cocartesian model structure.

We also define Top푇 ⊂ Cat∞,푇 as the full subcategory spanned by the 푇-spaces. Accordingly, we may employ some notation for marked simplicial sets: for any 푇-∞-cate- gory (퐶, 푝), we write ♮퐶 for the corresponding marked simplicial (퐶, 휄푇op 퐶), where 휄푇op 퐶 ⊂ 퐶1 is the collection of cocartesian edges. If (퐶, 푝) and (퐷, 푞) are two 푇-∞-categories, then we write ♭ Fun푇op (퐶, 퐷) ≔ Map푇op (♮퐶, ♮퐷), and we write ♯ Map푇op (퐶, 퐷) ≔ Map푇op (♮퐶, ♮퐷).

Clearly Fun푇op (퐶, 퐷) is an ∞-category, and Map푇op (퐶, 퐷) ⊂ Fun푇op (퐶, 퐷) is the largest Kan complex contained therein.

1.5. Example. When 푇 = O퐺, we write Cat∞,퐺, Top퐺, Fun퐺(퐶, 퐷), and Map퐺(퐶, 퐷) for Cat∞,푇, Top푇, Fun푇op (퐶, 퐷), and Map푇op (퐶, 퐷), respectively. 1.6. Definition. Suppose 푇 an ∞-category. We say that a 푇-functor 퐶 퐷 is fully faith- ful, essentially surjective, or an equivalence just in case, for any object 푉 ∈ 푇0, the functor 퐶푉 퐷푉 is so. 1.7. Note that a 푇-∞-category (퐶, 푝) is classified by an essentially unique functor op C∶ 푇 Cat∞ whose value on an object 푉 ∈ 푇 is equivalent to the fiber 퐶푉. If (퐶, 푝) is a 푇-space, then C lands in the full subcategory Top ⊂ Cat∞. Furthermore, a 푇-functor (퐶, 푝) (퐷, 푞) corresponds to an essentially unique C D. In fact, the straightening/unstraightening equivalence yields equivalences of ∞-categories op op Cat∞,푇 ≃ Fun(푇 , Cat∞) and Top푇 ≃ Fun(푇 , Top). 1.8. We have chosen to define 푇-∞-categories as cocartesian fibrations 푝∶ 퐷 푇op in order maintain certain conventions down the road – in particular the theory of ∞-operads, where cocartesian edges rule the roost. However, this has two disadvantages: ▶ Notationally, it can be a bit burdensome to lug around a 푇op in subscripts. To address this, we will at times make a global declaration that 푆 ≔ 푇op, and use 푆 instead. ▶ Additionally, we will at times be presented with a cartesian fibration 푞∶ 퐷 푇, which encodes essentially the same information as a cocartesian fibration 퐶 푇op. In this case, one may apply the dualization construction of [5] to 푞 to obtain a cocartesian fibration 푞∨ ∶ 퐷∨ 푇op that classifies the same functor as 푞.

2. Examples of parametrized ∞-categories We list a few basic examples of 퐺-∞-categories and 푇-∞-categories. We will use these in the sequel. op 2.1. Example. Of course O퐺 , which lies over itself via the identity, is a 퐺-space. Of course it’s the terminal object in the ∞-category of 퐺-spaces, so we shall write ∗퐺 for this object. Similarly, 푇op lying over itself via the identity is the terminal 푇-space, and we shall write ∗푇 for it. 4 CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH

2.2. Example. More generally, suppose 퐶 an ∞-category. Then one has the constant 푇-∞-cate- gory 퐶 × 푇op, which lies over 푇op by the projection functor. The functor that classifies it is op the constant functor 푇 Cat∞ at 퐶. Among the constant 푇-∞-categories is the empty 푇-∞-category ∅푇. 2.3. Example. If (퐶, 푝) and (퐷, 푞) are two 푇-∞-categories, then the fiber product

퐶 × 퐷 ≔ (퐶 ×푇op 퐷,(푝, 푞)) is the product 푇-∞-category.

2.4. Example. For any object 푉 ∈ 푇0, the forgetful functor op op 푉 ≔ (푇 )푉/ 푇 op exhibits (푇 )푉/ as a 푇-space, which we shall simply denote. The fiber of 푉 over an object 푊 ∈ 푇 is of course Map푇(푊, 푉), and indeed the functor that classifies 푉 is the functor represented by 푉. By the Yoneda lemma, for any 푇-∞-category 퐶, one has a natural equivalence

Fun푇op (푉, 퐶) ≃ 퐶푉. 2.5. Example. If 푇 is an ∞- (i.e., a Kan complex), then any categorical fibration 퐶 푇op is automatically a 푇-∞-category, and any 푇-space 푋 푇op is a . 2.6. Example. Suppose 푖∶ 푈 푇 a functor, and suppose 푞∶ 퐷 푈op a 푈-∞-category. Then there exist a 푇-∞-category 푝∶ 퐶 푇op and a 푈-functor

op 퐶 ×푇op 푈 퐷 such that for any object 푡 ∈ 푇0, the natural functor

퐶푡 lim 퐷푠 푠∈푈푡/ is an equivalence of ∞-categories. The functor that classifies 푝 is right Kan extended from the functor that classifies 푞 along 푖. Let’s give an explicit construction of the 푇-∞-category 퐶 right Kan extended along 푖. We’ll need a spot of notation. 2.7. Notation. If 푓∶ 푀 푆 and 푔∶ 푁 푆 are two maps of simplicial sets, then let us write 1 푀 ↓푆 푁 ≔ 푀 × Fun(훥 , 푆) × 푁. Fun(훥{0},푆) Fun(훥{1},푆)

A vertex of 푀 ↓푆 푁 is thus a vertex 푥 ∈ 푀0, a vertex 푦 ∈ 푁0, and an edge 푓(푥) 푔(푦) in 푆1. When 푀 and 푁 are ∞-categories, the 푀 ↓푆 푁 is a model for the lax pullback of 푓 along 푔.

2.8. Example. Suppose 푆 an ∞-category, and suppose 푥 ∈ 푆0 a vertex. The simplicial set 푥/ {푥} ↓푆 푆 is isomorphic to the “alternative” undercategory 푆 [9, §4.2.1], and, dually, the /푥 simplicial set 푆 ↓푆 {푥} is isomorphic to the 푆 . Consequently, for any two maps 푓∶ 푀 푆 and 푔∶ 푁 푆, one has

푥/ /푥 {푥} ↓푆 푁 ≅ 푆 ×푆 푁 and 푀 ↓푆 {푥} ≅ 푀 ×푆 푆 . In light of [6, Prp. 6.5], we have the following EXPOSÉ I – ELEMENTS OF PARAMETRIZED HIGHER CATEGORY THEORY 5

2.9. Proposition. Suppose 푖∶ 푈 푇 a functor, and suppose 퐷 a 푈-∞-category. Define a functor 퐶 푇op via the following : for any map 휂∶ 퐾 푇op, we demand a bijection op Mor/푇표푝 (퐾, 퐶) ≅ Mor/푈op (퐾 ↓푇op 푈 , 퐷), op natural in 휂. Then 퐶 is a 푇-∞-category, and the natural 푈-functor 퐶 ×푇op 푈 퐷 exhibits 퐶 as the right of 퐷 along 푖.

2.10. Notation. Write F for the 1-category of finite sets. For any ∞-category 푇, write F푇 for F the ∞-category denoted 풫∅ (푇) in [9, §5.3.6]. That is, F푇 is the full subcategory of the ∞- category Fun(푇op, Top) spanned by those objects that are equivalent to a finite of representables. In particular, it enjoys the following universal property: for any ∞-category 퐷 with all finite , the Yoneda embedding 푗∶ 푇 F푇 induces an equivalence ⊔ ∼ Fun (F푇, 퐷) Fun(푇, 퐷), where Fun⊔ denotes the ∞-category of functors that preserve finite coproducts. We call an object of F푇 a finite 푇-set. The ∞-category F푇 may be described as follows. The objects may be thought of as pairs (퐼, 푋퐼) consisting of a finite set 퐼 and a collection 푋퐼 = {푋푖}푖∈퐼 of objects of 푇. The mapping space Map ((퐽, 푌 ), (퐼, 푋 )) can be identified with the disjoint union F푇 퐽 퐼 Map ((퐽, 푌 ), (퐼, 푋 )) ≃ ∐ ∏ Map (푌 , 푋 ). F푇 퐽 퐼 푇 푗 휙(푗) 휙∈MapF(퐽,퐼) 푗∈퐽

We abuse notation slightly by treating the Yoneda embedding 푗∶ 푇 F푇 as if it were an inclusion of a full subcategory. Left Kan extension of the projection 푇 훥0 along j yields a functor Orbit∶ F푇 F that carries (퐼, 푋퐼), whence one obtains a decomposition of any finite 푇-set 푈 as 푈 ≅ ∐ 푉. 푉∈Orbit(푈) Observe that if 푇 = O퐺, then of course F푇 = F퐺, the category of finite 퐺-sets. ̃ 2.11. Example. Consider the twisted arrow ∞-category 풪(F푇) of the ∞-category of finite 푇-sets [3]. This comes equipped with a left fibration ̃ op 풪(F푇) F푇 × F푇. Now for any finite 푇-set 푈, one may pull back this left fibration along the inclusion op op op 푇 ≅ 푇 × {푈} F푇 × F푇 to obtain a 푇-space op 푝푈 ∶ 푈 푇 . op It is easy to see that the functor 푇 Top that classifies 푝푈 carries an object 푉 ∈ 푇 to the space 푈(푉) ≃ Map (푉, 푈). We therefore call this the discrete 푇-space attached to 푈. F푇 The following result is a simple consequence of the Yoneda lemma and the fact that

푈 ≃ ∐푉∈Orbit(푈) 푉. 2.12. Lemma. Suppose (퐶, 푝) a 푇-∞-category, and suppose 푈 a finite 푇-set. Then the forma- tion of the fibers over each orbit 푉 ∈ Orbit(푈) induces a trivial fibration ∼ Fun푇op (푈, 퐶) ∏ 퐶푉. 푉∈Orbit(푈) It will from time to time be helpful for us to select a section of this trivial fibration. Of particular importance is the case in which 푈 is terminal: 6 CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH

2.13. Definition. Suppose (퐶, 푝) a 푇-∞-category. Then a cocartesian section of 퐶 is a 푇- functor 푇op 퐶. In particular, if 푇 has a terminal object 1 (in which case the corresponding left fibration is an equivalence), then the lemma above yields a trivial fibration from the ∞-category of cocartesian sections of a 푇-∞-category 퐶 and the fiber ∞-category 퐶1. 2.14. Example. Pull back the target functor 1 {1} 풪(F푇) = Fun(훥 , F푇) Fun(훥 , F푇) ≅ F푇 along the inclusion 푇 F푇. If the ∞-category F푇 admits fiber products – that is, precisely if 푇 is an orbital ∞-category in the sense of [4] – then the projection 푇 × 풪(F ) 푇 is a F푇 푇 cartesian fibration [9, Lm. 6.1.1.1]. Consequently, we may dualize to obtain the 푇-∞-category of finite 푇-sets F ≔ (푇 × 풪(F ))∨ 푇op. 푇 F푇 푇

Note that the fiber over an object 푉 of 푇 is equivalent to the overcategory (F푇)/푉. There are other, more sophisticated examples. Here are a few. 2.15. Example. Suppose 퐸 ⊇ 퐹 a profinite Galois extension of fields with Galois group 퐺.

Then we can define a 퐺-1-category Vect퐸⊇퐹 in the following manner. The objects are pairs (퐿, 푋) consisting of a subextension 퐸 ⊇ 퐿 ⊇ 퐹 that is finite over 퐹 and a finite-dimensional 퐿-vector space 푋. A (퐿, 푋) (퐿′, 푋′) is a field homomorphism 퐿 퐿′ over ′ ′ ′ 퐹 (but not necessarily under 퐸) along with an 퐿 -linear map 푋 ⊗퐿 퐿 푋 . op The functor (퐿, 푋) 퐺/Aut퐿(퐸) is the desired structure functor Vect퐸⊇퐹 O퐺 , which is easily seen to be a Grothendieck opfibration. This was our motivating example from the beginning of the Introduction. cn 2.16. Example. We consider the ∞-category CAlg of connective 퐸∞ rings and the huge ∞- relative the regular cardinal 휅1 cn Shvflat ⊂ Fun(CAlg , Kan(휅1)) of large sheaves on CAlgcn,op for the flat [10, Pr. 5.4]. We let

Mod QCoh

푞 푝

cn op CAlg Shvflat be the pullback square in which 푝 is the right Kan extension of 푞, as constructed in Pr. 2.9. The cn objects of QCoh can be thought of as pairs (푋, 푀) consisting of a 푋∶ CAlg Kan(휅1) for the flat topology and a quasicoherent module 푀 over 푋. cn,op Of course (QCoh, 푝) is a Shvflat-∞-category and (Mod, 푞) is a CAlg -∞-category, but normally it is awkward to work with such enormous bases. To pass to a smaller base, one can consider, for example, a connected, noetherian scheme ét 푋 and suppose 푥 a geometric point of 푋. Then if 휋1 (푋, 푥) is the étale of 푋, then by Grothendieck’s Galois duality [12, Exp. V, §7], the ∞-category FÉt(푋)conn of connected finite étale covers is equivalent to O ét ; then one can pull back 푝 along the 휋1 (푋,푥) forgetful functor conn O ét ≃ FÉt(푋) Shv . 휋1 (푋,푥) flat ét The result is a 휋1 (푋, 푥)-∞-category QCohFÉt(푋)conn . EXPOSÉ I – ELEMENTS OF PARAMETRIZED HIGHER CATEGORY THEORY 7

As a subexample, we may consider the case in which 푋 = Spec 퐹 with 푥 = Spec 훺 for a ét separable closure 훺 ⊃ 퐹 with absolute Galois group 퐺 = 휋1 (푋, 푥). Then we obtain a derived version of the 퐺-1-category above.

3. Parametrized opposites Forming the opposite of an ∞-category is a trivial matter: one simply precomposes the un- derlying simplicial set with the unique nontrivial involution op∶ 훥 훥. For parametrized ∞-categories, there is an added wrinkle: one must involve the dualization of [5]. 3.1. Definition. For any 푇-∞-category (퐶, 푝), the opposite 푇-∞-category is the 푇-∞-category op op ∨ op ∨ op op (퐶, 푝) ≔ ((퐶 ) ,(푝 ) ) = ((∨퐶) ,(∨푝) ), in the notation of [3]. For brevity, we may write 퐶vop for (퐶, 푝)op. (The “v” stands for vertical; we think of our construction as a fibrewise opposite construction.) Hence the objects of 퐶vop are precisely the objects of 퐶, and a map 푥 푦 of 퐶vop is a cospan

푢 푔 푓 푦 푥 of 퐶 in which 푝(푔) is a degenerate edge of 푇op, and 푓 is a 푝-cocartesian edge. It follows from [5] that the assignment (퐶, 푝) (퐶, 푝)op defines an auto-equivalence of the ∞-category Cat∞,푇.

4. Parametrized subcategories The notion of a subcategory of an ∞-category has a very rigid meaning. In effect, this notion is defined so that “lying in the subcategory” is a homotopy invariant notion for . We introduce the parametrized analogue of that notion. 4.1. Recollection. Recall [11, §1.2.11] that a subcategory of an ∞-category 퐶 is a simplicial subset 퐶′ ⊂ 퐶 that can be presented as a pullback

퐶′ 퐶

푁(ℎ퐶)′ 푁ℎ퐶 in which (ℎ퐶)′ ℎ퐶 is the inclusion of a subcategory. Note that this implies that the inclusion 퐶′ 퐶 is an inner fibration and that ℎ(퐶′) ≃ (ℎ퐶)′. Properties of subcategories are typically determined by their homotopy categories. For example, we will say that 퐶′ ⊂ 퐶 is stable under equivalences if (ℎ퐶)′ ⊂ ℎ퐶 is stable under isomorphisms; in this case the inclusion 퐶′ 퐶 is a categorical fibration. Similarly, we will say that 퐶′ ⊂ 퐶 is full if (ℎ퐶)′ ⊂ ℎ퐶 is full. 4.2. Definition. Suppose 푇 a ∞-category. A 푇-subcategory (퐶′, 푝′) of an 푇-∞-category (퐶, 푝) is a subcategory 퐶′ ⊂ 퐶 such that the restriction 푝′ ≔ 푝|퐶′ is a cocartesian fibration, and an edge of 퐶′ is 푝′-cocartesian if and only if it is 푝-cocartesian. We will say that (퐶′, 푝′) ⊂ (퐶, 푝) is stable under equivalences if a 푝-cocartesian edge 푥 푦 of 퐶 lies in 퐶′ just in case 푥 does. We will say that (퐶′, 푝′) ⊂ (퐶, 푝) is full just in ′ case, for any object 푉 ∈ 푇, the inclusion 퐶푉 퐶푉 is full. 8 CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH

4.3. Lemma. Suppose 푇 a ∞-category, suppose (퐶, 푝) an 푇-∞-category, and suppose 퐶′ ⊂ 퐶 a subcategory. Then 퐶′ is an 푇-subcategory stable under equivalences just in case the following conditions are satisfied. ▶ A 푝-cocartesian edge 푥 푦 of 퐶 lies in 퐶′ just in case 푥 does. ▶ For any commutative triangle

휂 푥 푦

푓 푔 푧

of 퐶 in which 휂 is 푝-cocartesian, if 푓 lies in 퐶′, then so does 푔. 4.4. Lemma. Suppose 푇 an ∞-category, suppose (퐶, 푝) a 푇-∞-category, and suppose (퐶′, 푝′) a 푇-subcategory thereof. Then (퐶′, 푝′) is full in (퐶, 푝) if and only if 퐶′ is full in 퐶.

Proof. Suppose 푋, 푌 ∈ 퐶′ are two objects, and write 푉 and 푊 for their images in 푇op. For any morphism 푓∶ 푉 푊 in 푇op, choose a 푝′-cocartesian edge 푋 푋′ lying over 푓 so 푓 that the fiber Map퐶′ (푋, 푌) of

Map퐶′ (푋, 푌) Map푇op (푉, 푊) 푓 ′ 푓 over 푓 is equivalent to Map ′ (푋 , 푌), and the fiber Map (푋, 푌) of 퐶푊 퐶

Map퐶(푋, 푌) Map푇op (푉, 푊) over 푓 is equivalent to Map푓 (푋′, 푌). We thus find that the inclusion map of fibers 퐶푊 푓 푓 Map퐶′ (푋, 푌) Map퐶(푋, 푌) is an equivalence for any vertex 푓 ∈ Map푇op (푉, 푊), whence Map퐶′ (푋, 푌) Map퐶(푋, 푌) is an equivalence as well.  4.5. Lemma. Suppose 푇 an ∞-category, suppose (퐶, 푝) an 푇-∞-category, and suppose 퐶′ ⊂ 퐶 a full subcategory. Then the following are equivalent. ▶ 퐶′ is a full 푇-subcategory of 퐶 that is stable under equivalences. ▶ For any 푝-cocartesian edge 푥 푦 of 퐶, if 푥 lies 퐶′ then so does 푦.

5. Constructing 푇-∞-categories via pairings One particularly powerful construction with cartesian and cocartesian fibrations comes from [9, §3.2.2]. 5.1. Recollection. Suppose 푝∶ 푋 푇op a cartesian fibration and 푞∶ 푌 푇op a cocarte- op sian fibration. Suppose X∶ 푇 Cat∞ a functor that classifies 푝 and Y∶ 푇 Cat∞ a functor that classifies 푞. Clearly one may define a functor op Fun(X, Y)∶ 푇 Cat∞ that carries a vertex 푠 of 푇op to the ∞-category Fun(X(푠), Y(푠)) and an edge 휂∶ 푠 푡 of 푇op to the functor Fun(X(푠), Y(푠)) Fun(X(푡), Y(푡)) given by the assignment 퐹 Y(휂) ∘ 퐹 ∘ X(휂). EXPOSÉ I – ELEMENTS OF PARAMETRIZED HIGHER CATEGORY THEORY 9

5.2. Construction. If one wishes to work instead with the fibrations directly (avoiding straightening and unstraightening), the following construction provides an elegant way of writing explicitly the cocartesian fibration classified by the functor Fun(X, Y). op Suppose 푝∶ 푋 푇 is a cartesian fibration classified by a functor X∶ 푇 Cat∞, and op op suppose 푞∶ 푌 푇 is a cocartesian fibration classified by a functor Y∶ 푇 Cat∞. ̃ ̃ op One defines a simplicial set Fun푇op (푋, 푌) and a map 푟∶ Fun푇op (푋, 푌) 푇 defined by the following universal property: for any map 휎∶ 퐾 푇op, one has a bijection ̃ Mor/푇op (퐾, Fun푇op (푋, 푌)) ≅ Mor/푇op (푋 ×푇op 퐾, 푌), functorial in 휎. It is then shown in [9, Cor. 3.2.2.13] (see also [6, Ex. 3.10]) that 푟 is a cocartesian fibration, and an edge 1 ̃ 푔∶ 훥 Fun푇op (푋, 푌) 1 is 푟-cocartesian just in case the induced map 푋 ×푇op 훥 푌 carries 푝-cartesian edges to 푞- ̃ cocartesian edges. The fiber of the map Fun푇op (푋, 푌) 푆 over a vertex 푠 is the ∞-category op Fun(푋푠, 푌푠), and for any edge 휂∶ 푠 푡 of 푇 , the functor 휂! ∶ 푇푠 푇푡 induced by 휂 is equivalent to the functor 퐹 Y(휂) ∘ 퐹 ∘ X(휂) described above.

6. A technical result: the strong pushforward Before we proceed to the three main results of this paper, we need a key technical result, Pr. 6.2 We will prove the existence of what we call the strong pushworward. In effect, this is an efficient way of selecting compatible families of cocartesian edges in a cocartesian fibration. Glasman has already constructed the strong pushforward in another context. In this section, we give a different, more general construction. 6.1. Notation. In this section, we fix a cocartesian fibration 푝∶ 푋 푆 of ∞-categories. We regard 풪(푆) ≔ Fun(훥1, 푆) as a cartesian fibration over 푆 via the source map. ∼ Write 푋 for the marked simplicial set (푋, 푖푋), where 푖푋 ⊂ 푋1 is the class of equivalences. 6.2. Proposition. The ∼ ♮ 휄∶ 푋 ♮푋 ×푆♯ 풪(푆) + of marked simplicial sets is a homotopy equivalence in 푠Set/푆. The point here is that we can functorially select a cocartesian arrow over every arrow of 푆 starting at a given point. 6.3. Lemma. Let 풪 cocart(푋) denote the full subcategory of 풪(푋) spanned by the 푝-cocartesian edges. Then the natural map cocart (휎, 푝)∶ 풪 (푋) 푋 ×푆 풪(푆) sending every cocartesian arrow 푓 to (푓(0), 푝(푓)) is a trivial Kan fibration. Proof. We need to show that for every solid arrow diagram

휕훥푛 풪 cocart(푋) (휎, 푝) 푛 훥 푋 ×푆 풪(푆), 10 CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH there is a dotted . To do that we will use the formalism of marked simplicial set. Unwrap- ping the definitions we see that we need to find a lift on the following diagram of marked simplicial sets

푛 ♭ ((휕훥푛)♭×{0}) 푛 ♭ 1 ♯ ((훥 ) × {0}) ∪ ((휕훥 ) × (훥 ) ) ♮푋

(훥푛)♭ × (훥1)♯ 푆♯

By [9, Pr. 3.1.1.5(2′′)], the left vertical arrow is the opposite of a marked anodyne arrow, so by [9, Pr. 3.1.1.6] it lifts against all cocartesian fibrations.  cocart 6.4. Construction. Let us select a section 푐∶ 푋 ×퐵 풪(푆) 풪 (푋) of (휎, 푝). Then we can define the strong pushforward

푃푝 = 휏 ∘ 푐∶ 푋 ×푆 풪(푆) 푋 as the composition of 푐 with the target map from 풪 cocart(푋) to 푋.

6.5. Lemma. The strong pushforward 푃푝 is a marked homotopy inverse of 휄.

Proof. First we’ll show that 푃푝 is a marked map, and then we will produce the required marked homotopies 푃푝 ∘ 휄 ∼ id and 휄 ∘ 푃푝 ∼ id. ♮ ∼ First, we claim that the strong pushforward 푃푝 is a marked map from ♮푋 ×푆♯ 풪(푆) to 푋 . ♮ Indeed, choose a marked arrow in ♮푋 ×푆♯ 풪(푆) . This is the data of a 푝-cocartesian arrow 푓∶ 푥 푥′ in 푋 and a commutative square

푝(푥) 푠 푝(푓) 푟 푝(푥′) 푠′ in 푆 such that 푟 is an equivalence. The section 푐 carries this to a square

푥 푦 푓 푔 푥′ 푦′ in which the horizontal arrows are 푝-cocartesian. But by hypothesis 푓 is also 푝-cocartesian, whence 푔 is as well. Thus since 푔 lies above an equivalence, it follows that 푔 is an equivalence, as desired. ∼ Now we construct a marked homotopy between 푃푝 ∘ 휄 and the identity map from 푋 to itself. To do this, consider the map 푐 ∘ 휄∶ 푋 풪 푐표푐푎푟푡(푋) ⊆ 풪(푋). This corresponds to a map 푋 × 훥1 푋, which is easily seen to be a marked map 푋∼ × (훥1)♯ 푋∼. This defines our homotopy between the identity map and 푃푝 ∘ 휄. Finally, we construct a marked homotopy between 휄 ∘ 푃푝 and the identity map from ♮ ♮푋 ×푆♯ 풪(푆) to itself. We will construct a functor 1 1 1 푋 ×푆 풪(푆) Fun(훥 , 푋 ×푆 풪(푆)) ≅ Fun(훥 , 푋) ×Fun(훥1,푆) Fun(훥 , 풪(푆)) EXPOSÉ I – ELEMENTS OF PARAMETRIZED HIGHER CATEGORY THEORY 11 realizing the required homotopy. The first component is just given by 푐 via

푐표푐푎푟푡 푐∶ 푋 ×푆 풪(푆) 풪 (푋) ⊆ 풪(푋). The second component is obtained by composing the second projection with a map 풪(푆) Fun(훥1, 풪(푆)) ≅ Fun(훥1 × 훥1, 푆) induced by the map min∶ 훥1 × 훥1 훥1, which sends (0, 0) to 0 and the other objects to 1. Our functor thus sends an object (푥, 푓∶ 푝(푥) 푏) ∈ 푋 ×푆 풪(푆) to the pair

푓 푝(푥) 푠 (푐(푥)∶ 푥 푦, 푓 ) . 푠 푠

One verifies easily that this defines a marked map

∼ ♮ 1 ♯ ∼ ♮ (푋 ×푆♯ 풪(푆) ) × (훥 ) 푋 ×푆♯ 풪(푆) and therefore the desired homotopy. 

7. 푇-objects in ∞-categories We can talk about 퐺-objects in an arbitrary ∞-category in in exactly the manner pro- posed by Elmendorf: the ∞-category of 퐺-objects in an ∞-category 퐷 can simply be defined op as the ∞-category of functors O퐺 퐷. Similarly, the ∞-category of 푇-objects in an ∞- category 퐷 can simply be defined as the ∞-category of functors 푇op 퐷. We will go further in this section, and define a 퐺-∞-category of 퐺-objects in an ∞- category, whose 퐻-fixed point ∞-category is the ∞-category of 퐻-objects. Similarly, we define a 퐺-∞-category of 퐺-objects in an ∞-category, whose fibre over an object 푉 is the ∞- category of 푇/푉-objects. Furthermore, we will prove Th. 7.8 that this 푇-∞-category enjoys a useful universal property: if effect, it is the right adjoint to the Grothendieck construction (퐶, 푝) 퐶 (i.e., the oplax colimit). Put differently, the 푇-∞-category of 푇-objects in 퐷 is the cofree 푇-∞-category cogener- ated by 퐷. This fact, which appears to be quite well known in ordinary category theory, is new in the ∞-categorical context.

7.1. Definition. Suppose 퐷 an ∞-category. The category 퐷퐺 of 퐺-objects in 퐷 is simply the op functor ∞-category Fun(O퐺 , 퐷). More generally, for any ∞-category 푇, the category 퐷푇 of 푇-objects in 퐷 is the functor ∞-category Fun(푇op, 퐷).

7.2. Example. When 퐷 is Cat∞ or Top, the straightening/unstraightening equivalences justify this notation. Indeed, they can be rewritten in the following manner:

Cat∞,푇 ≃ (Cat∞)푇 and Top푇 ≃ (Top)푇.

7.3. Warning. Note, however, that the category F퐺 of finite 퐺-sets is not the ∞-category of 퐺-objects in the category of finite sets.

It turns out that the ∞-category 퐷퐺 is in fact the fiber over 퐺/퐺 of a 퐺-∞-category. We proceed to define this 퐺-∞-category. 12 CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH

7.4. Definition. Suppose 푇 an ∞-category, and suppose 퐷 an ∞-category. The source functor Fun(훥1, 푇op) Fun(훥{0}, 푇op) ≅ 푇op op is a cartesian fibration. We therefore use Cnstr. 5.2 to define a simplicial set 퐷푇 over 푇 thus: ̃ 1 op op 퐷푇 ≔ Fun푇op (Fun(훥 , 푇 ), 퐷 × 푇 ). op The structure morphism 퐷푇 푇 is thus a cocartesian fibration, whence it exhibits 퐷푇 as a 푇-∞-category. We’ll refer to 퐷푇 as the 푇-∞-category of 푇-objects in 퐷. When 푇 = O퐺, we will write 퐷퐺 for 퐷푇.

7.5. Example. Suppose 푇 an ∞-category. Taking 퐷 = Cat∞ in Df. 7.4, we obtain the 푇-∞-category of 푇-∞-categories Cat∞,푇. Similarly, we obtain the 푇-∞-category of 푇-spaces Top . 푇

7.6. Suppose 푇 an orbital ∞-category, and suppose 퐷 an ∞-category. The fiber (퐷푇)푉 of 퐷푇 over an object 푉 ∈ 푇 is the ∞-category of functors 푉 퐷, giving an equivalence (퐷 ) ≃ 퐷 . 푇 푉 푇/푉 In particular, for any 퐺-orbit 퐺/퐻, the fiber (퐷퐺)(퐺/퐻) of 퐷퐺 is the ∞-category of functors 퐺/퐻 퐷.

An equivalence O퐻 ≃ (O퐺)/(퐺/퐻) specifies an equivalence (퐷퐺)(퐺/퐻) ≃ 퐷퐻. 7.7. Amusingly, one recovers the Bockstein map in an unexpected manner here. Indeed, op suppose 퐺 a profinite group and 퐻 ≤ 퐺 an open subgroup. Then the full subcategory of O퐺 spanned by 퐺/퐻 is the 퐵(푁퐺퐻/퐻). If we restrict the cocartesian fibration op 퐷퐺 O퐺 to this full subcategory, we obtain, for any ∞-category 퐷, an action of 푁퐺퐻/퐻 on the fibre (퐷퐺)(퐺/퐻). By the Yoneda lemma, this action is induced by an (푁퐺퐻/퐻)-action on 퐺/퐻 itself; this action is given by restricting the cartesian fibration 1 op {0} op op Fun(훥 , O퐺 ) Fun(훥 , O퐺 ) ≃ O퐺 to 퐵(푁퐺퐻/퐻). The full subcategory of the fiber product 1 op Fun(훥 , O ) × op 퐵(푁 퐻/퐻) 퐺 O퐺 퐺 spanned by the object 퐺/퐻 퐺/1 is equivalent to the classifying space 퐵푁퐺퐻, and if we restrict to this full subcategory, the resulting right fibration 퐵푁퐺퐻 퐵(푁퐺퐻/퐻) is induced by the quotient homomorphism 푁퐺퐻 푁퐺퐻/퐻. Now under the equivalence Top ≃ Fun(퐵(푁 퐻/퐻), Top), /퐵(푁퐺퐻/퐻) 퐺 op this assertion corresponds to the observation that restricting the action of 퐺/퐻 on O퐻 to the full subcategory 퐵퐻 spanned by the object 퐻/1 recovers the Bockstein map

퐵(푁퐺퐻/퐻) 퐵 Aut(퐵퐻). We now turn to the universal property of the 퐺-∞-category of 퐺-objects (or, more gen- erally, the 푇-∞-category of 푇-objects). In effect, we will show that the the 퐺-∞-category of 퐺-objects in an ∞-category 퐷 is the cofree 퐺-∞-category cogenerated by 퐷. EXPOSÉ I – ELEMENTS OF PARAMETRIZED HIGHER CATEGORY THEORY 13

7.8. Theorem. Suppose 푇 an ∞-category, 퐶 a 푇-∞-category, and 퐷 an ∞-category. Then there is a natural equivalence

Fun푇op (퐶, 퐷푇) ≃ Fun(퐶, 퐷).

Proof. Since the cocartesian arrows in 퐷푇 are those for which the map 1 1 op 훥 ×푇op Fun(훥 , 푇 ) 퐷 sends cartesian arrows to equivalences, we can identify Fun푇op (퐶, 퐷푇) with the ∞-category ♭ 1 op ♮ ∼ Map (♮퐶 ×(푇op)♯ Fun(훥 , 푇 ) , 퐷 ). Note that the map 푇op Fun(훥1, 푇op) sending every object to its identity arrow pro- duces a natural cofibration of marked simplicial sets ∼ 1 op ♮ 휄∶ 퐶 ♮퐶 ×(푇op)♯ Fun(훥 , 푇 ) . This in turns yields a natural map

⋆ ♭ 1 op ♮ ∼ 휄 ∶ Fun푇op (퐶, 퐷푇) ≅ Map (♮퐶 ×(푇op)♯ Fun(훥 , 푇 ) , 퐷 ) Map♭(퐶∼, 퐷∼) ≅ Fun(퐶, 퐷). The map 휄 is a trivial cofibration for the cartesian model structure on the category of marked simplicial sets (see Pr. 6.2), whence 휄⋆ is a trivial fibration. This completes the proof.  If now 퐷 is a presentable ∞-category such that finite products in 퐷 preserve colimits separately in each variable, then 퐷퐺 admits internal Homs. We now show that these internal Homs admit a convenient description. A technical lemma is required. 7.9. Lemma. Suppose 푇 an ∞-category, and suppose 퐷 a presentable ∞-category such that finite products in 퐷 preserve colimits separately in each variable. For any object 푉 ∈ 푇, let op 푖푉 ∶ 푉 푇 denote the structure morphism for the 푇-space represented by 푉, and consider the adjunction 푖 ∶ 퐷 퐷 ∶ 푖⋆ . 푉,! 푇/푉 푇 푉 For any 퐺-object 푋 of 퐷, the morphism ⋆ 푖푉,!푖푉푋 푋 × 푖푉,!(∗) of 퐷푇 given by the counit of the adjunction on the first factor and 푖푉,! of the projection from ⋆ 푖푉푋 to the terminal object 푡 on the second is an equivalence. Proof. For any object 푊 ∈ 푇, the claim is that the natural transformation from the composite

푖 op op op 푉 op 푋 (푇 )푉/ ×푇op (푇 )/푊 (푇 )푉/ 푇 퐷 op op to the constant functor (푇 )푉/ ×푇op (푇 )/푊 퐷 at the object 푋(푊) induces an equiva- lence on colimits. op op Observe that the full subcategory 퐾 of (푇 )푉/ ×푇op (푇 )/푊 spanned by objects of the form (휋∶ 푉 푈, 휌∶ 푈 ∼ 푊) where 휌 is an isomorphism is cofinal. This full subcategory 퐾 is an ∞-groupoid, and the result follows from the identification 퐾 ⊗ 푋(푊) ≃ (퐾 ⊗ ∗) × 푋(푊), 14 CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH since the product preserves colimits separately in each variable.  7.10. Notation. Suppose 푇 an ∞-category, and suppose 퐷 a presentable ∞-category such that finite products in 퐷 preserve colimits separately in each variable. Denote by 퐹푇(−, −) the internal mapping object in 퐷푇, so that 퐹푇(푋, −) is right adjoint to 푋 × − for any object 푋 ∈ 퐷 . Denote by 퐹 (−, −) the internal mapping object in 퐷 . 푇 퐷푇 푇 7.11. Proposition. Suppose 푇 an ∞-category, and suppose 퐷 a presentable ∞-category such that finite products in 퐷 preserve colimits separately in each variable. For each object 푉 ∈ 푇 and each pair (푋, 푌) of 푇-objects of 퐷, we have an equivalence ⋆ ⋆ 퐹푇(푋, 푌)(푉) ≃ 퐹퐷 (푖푉푋, 푖푉푌). 푇/푉 op op Proof. Since 푉 is an initial object of (푇 )푉/, the constant functor ∗∶ 푇 퐷 is left Kan extended from {푉}, and we have ⋆ 퐹퐷 (∗, 푖푉푍) ≃ 푍(푉). 푇/푉 for 푍 ∈ 퐷푇. So ⋆ 퐹푇(푋, 푌)(푉) ≃ 퐹퐷 (∗, 푖푉퐹푇(푋, 푌)) 푇/푉 ≃ 퐹 (푖 (∗), 퐹 (푋, 푌)) 퐷푇 푉,! 푇 ≃ 퐹 (푋 × 푖 (∗), 푌) 퐷푇 푉,! ≃ 퐹 (푖 푖⋆ 푋, 푌) (by 7.9) 퐷푇 푉,! 푉 ⋆ ⋆ ≃ 퐹퐷 (푖푉푋, 푖푉푌), 푇/푉 and this completes the proof.  8. Parametrized fibrations The universal property of 푇-objects in an ∞-category can be used to formulate an equi- variant version of the usual straightening/unstraightening constructions. 8.1. Observe that if 퐶 푇op is a 푇-∞-category, and if 퐷 퐶 is cocartesian fibration, then 퐷 automatically inherits the structure of a 푇-∞-category. 8.2. Definition. Suppose 푇 an ∞-category, and suppose 퐶 and 퐷 two 푇-∞-categories. Then a cocartesian 푇-fibration 퐶 퐷 is nothing more than a 푇-functor that is also a cocartesian fibration. Likewise, a left 푇-fibration 퐶 퐷 is nothing more than a 푇-functor that is also a left fibration. Combining Th. 7.8 with the usual straightening/unstraightening equivalence, we obtain 8.3. Proposition. Suppose 푇 an ∞-category, and suppose 퐶 a 푇-∞-category. Then there are equivalences of ∞-categories cocart left Fun op (퐶, Cat ) ≃ Cat and Fun op (퐶, Top ) ≃ Cat , 푇 ∞,푇 ∞,/퐶 푇 푇 ∞,/퐶 cocart where Cat∞,/퐶 denotes the subcategory of Cat∞,/퐶 whose objects are cocartesian fibrations and left whose morphisms presreve cocartesian edges, whereas Cat∞,/퐶 denotes the full subcategory of Cat∞,/퐶spanned by the left fibrations. We may therefore speak of cocartesian 푇-fibrations over a 푇-∞-category 퐶 as being classified by a 푇-functor 퐶 Cat∞,푇 and, similarly, of left 푇-fibrations over a 푇-∞-category 퐶 as being classified by a 푇-functor 퐶 Top . 푇 EXPOSÉ I – ELEMENTS OF PARAMETRIZED HIGHER CATEGORY THEORY 15

9. Parametrized functor categories It is important to know that parametrized ∞-categories have an internal Hom. More pre- cisely, one wishes to know that for any two 푇-∞-categories 푋 and 푌, there is a 푇-∞-category

Fun푇op (푋, 푌) that enjoys the universal property

Fun푇op (푍 × 푋, 푌) ≃ Fun푇op (푍, Fun푇op (푋, 푌)) for any 푇-∞-category 푍. 9.1. Recollection. If 푝∶ 푋 푇op and 푞∶ 푌 푇op are two 푇-∞-categories that are clas- sified by functors op X, Y∶ 푇 Cat∞, then it is easy to see that the product functor X × Y classifies the cocartesian fibration op (푝, 푞)∶ 푋 ×푇op 푌 푇 . It’s also apparent that the product admits a right adjoint; indeed, the functor ∞-category op Fun(푇 , Cat∞) admits an internal Hom given by the functor

푉 Nat(H푉 × X, Y), op where H푉 ∶ 푇 Top ⊂ Cat∞ is the functor represented by 푠. What is less clear is how to write down the cocartesian fibration classified by this internal Hom functor in a manner that avoids straightening and unstraightening. It turns out that rather than use the 푇-space 푉 as a model for the functor H푉, it is more convenient to use the “alternative” undercategory (푇op)푉/ of [9, §4.2.1]. The forgetful functor 푢∶ (푇op)푉/ 푇op is still a left fibration that is classified by the corepresentable functor H푉, so we are led to construct a cocartesian fibration op Fun푇op (푋, 푌) 푇 whose fiber over an object 푉 ∈ 푇op is the ∞-category of functors (푇op)푉/ × 푋 푌 over 푇op that carry (푢, 푝)-cocartesian edges to 푞-cocartesian edges. This is exactly what we will do. 9.2. Definition. Now suppose 퐶 and 퐷 two 푇-∞-categories. Define a simplicial set ′ Fun푇op (퐶, 퐷) over 푇op via the following universal property. For any map 휂∶ 퐾 푇op, one demands a bijection ′ Mor op (퐾, Fun op (퐶, 퐷)) ≅ Mor op (퐾 ↓ op 퐶, 퐾 ↓ op 퐷), 푇 푇 퐾↓푇op 푇 푇 푇 natural in 휂. (Here we are using the notation of the lax pullback from Nt. 2.7.) Now let us define ′ Fun푇op (퐶, 퐷) ⊂ Fun푇op (퐶, 퐷) as the full subcategory spanned by those objects op 푉/ op 푉/ (푇 ) ×푇op 퐶 ≅ {푉} ↓푇op 퐶 {푉} ↓푇op 퐷 ≅ (푇 ) ×푇op 퐷 that carry cocartesian edges over (푇op)푉/ to cocartesian edges over (푇op)푉/.

9.3. We will soon show that Fun푇op (퐶, 퐷) is a 푇-∞-category. The fibre of this 푇-∞-category over any object 푉 ∈ 푇 is by construction equivalent to the ∞-category of 푉op-functors 퐶 × 푉 퐷 × 푉 16 CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH

9.4. Notation. If 푀 푆 and 푁 푆 are two maps of simplicial sets, and if 퐸 ⊂ 푀1 and 퐹 ⊂ 푁1 are collections of marked edges (which as usual we assume contain all degenerate edges), then we may write 1 ♯ (푀, 퐸) ↓푆 (푁, 퐹) ≔ (푀, 퐸) × Fun(훥 , 푆) × (푁, 퐹). Fun(훥{0},푆)♯ Fun(훥{1},푆)♯ 9.5. With this notation in hand, if 퐶 and 퐷 are two 푇-∞-categories, we may describe op op Fun푇op (퐶, 퐷) 푇 via the following universal property: for any map 퐾 푇 , we de- mand a bijection ♭ Mor/푇op (퐾, Fun푇op (퐶, 퐷)) ≅ Mor/푇op (퐾 ↓푇op ♮퐶, ♮퐷), functorial in 휎. 9.6. Proposition. Suppose 퐶 and 퐷 two 푇-∞-categories. Then the restriction op Fun푇op (퐶, 퐷) 푇 of the structure map above is a cocartesian fibration, and an edge 푒 is marked if and only if the corresponding map 1 1 훥 ↓푇op 퐶 훥 ↓푇op 퐷 1 op over 훥 ↓푇op 푇 carries cocartesian edges to cocartesian edges. op Proof. Let us write 푆 ≔ 푇 . The map 푟∶ Fun푆(퐶, 퐷) 푆 can be described as follows: for any map of simplicial sets 퐾 푆, one has a bijection ♭ Mor푆(퐾, Fun푆(퐶, 퐷)) ≅ Mor푆(퐾 ↓푆 ♮퐶, ♮퐷). We see that the data of the source map

푠∶ 푆 ↓푆 퐶 푆, the target map 푡∶ 푆 ↓푆 퐶 푆, ♯ and the marking on 푆 ↓푆 ♮퐶 together enjoy the conditions of [6, Cor. 6.2.1]. Indeed, it is a trivial matter to see that equivalences of 푆 ↓푆 퐶 are marked and that marked edges are closed under composition. By [9, Cor. 2.4.7.12], the source map 푠 is a cartesian fibration, and that a map is 푠-cartesian if and only if its projection to 퐶 is an equivalence. Consequently, since the marked edges are those whose projection to 퐶 is 푝-cocartesian, it follows that for any 2-simplex 푦 휙 휓

푥 휒 푧 in which 휓 is 푠-cartesian, the edge 휙 is marked just in case the edge 휒 is so. This proves that the conditions of [6, Cor. 6.2.1] indeed apply.

As a consequence, we deduce that 푟 is a cocartesian fibration, and that an edge of Fun푆(퐶, 퐷) 1 1 1 is 푟-cocartesian just in case the corresponding functor 훥 ↓푆 퐷 훥 ↓푆 퐷 over 훥 ↓푆 푆 carries cocartesian edges to cocartesian edges, as desired. 

We now set about proving that Fun푇op really is an internal hom. 9.7. Theorem. Suppose 퐶, 퐷, and 퐸 three 푇-∞-categories. (1) There is a natural equivalence of 푇-∞-categories ∼ Fun푇op (퐶, Fun푇op (퐷, 퐸)) Fun푇op (퐶 × 퐷, 퐸). EXPOSÉ I – ELEMENTS OF PARAMETRIZED HIGHER CATEGORY THEORY 17

(2) Accordingly, there is a natural equivalence of ∞-categories ∼ Fun푇op (퐶, Fun푇op (퐷, 퐸)) Fun푇op (퐶 × 퐷, 퐸). To prove this result, we will need a lemma. 9.8. Lemma. Let 푆 be an ∞-category, 휄∶ 푆 풪(푆) be the identity section and regard 풪(푆)♯ as a marked simplicial set over 푆 via the target map. Then (1) For every marked simplicial set 푋 푆 and cartesian fibration 퐶 푆, ♮ ♮ id푋 × 휄 × id퐶 ∶ 푋 ×푆 퐶 푋 ↓푆 퐶 + is a cocartesian equivalence in 푠Set/푆. (2) For every marked simplicial set 푋 푆 and cocartesian fibration 퐶 푆,

id퐶 × 휄 × id푋 ∶ ♮퐶 ×푆 푋 ♮퐶 ↓푆 푋 + is a homotopy equivalence in 푠Set/푆. Proof. For (1), using [6, Cor. 6.2.1] on the span 푆 퐶♮ 푆, we reduce to the case where ♯ 퐶 = 푆. By definition, 푋 푋 ↓푆 푆 is a cocartesian equivalence if and only if for every cocartesian fibration 푍 푆, the map ♯ ♯ ♯ Map푆(푋 ↓푆 푆 , ♮푍) Map푆(푋, ♮푍) is a trivial Kan fibration. In other words, for every of simplicial sets 퐴 퐵 and cocartesian fibration 푍 푆, we need to provide a dotted lift in the following commu- tative square 휙 ♯ (퐴♯×푋) ♯ ♯ (퐵 × 푋) ∪ ((퐴 × 푋) ↓푆 푆 ) ♮푍

♯ ♯ ♯ (퐵 × 푋) ↓푆 푆 푆 . ♯ 1 ♯ ♯ ♯ ♯ Define ℎ0 ∶ 풪(푆) ×(훥 ) 풪(푆) as the adjoint to the map 풪(푆) 풪(풪(푆)) obtained by precomposing by the functor min∶ 훥1 × 훥1 훥1, so that (1, 1) is the unique vertex sent to 1. Precomposing 휙 by id퐴♯×푋 × ℎ0, define a homotopy ♯ ♯ 1 ♯ ℎ∶ (퐴 × 푋) ↓푆 푆 × (훥 ) ♮푍 from 휙| ♯ ∘ pr ♯ to 휙| ♯ ♯ . Using ℎ and 휙| ♯ , define a map 퐴 ×푋 퐴 ×푋 (퐴 ×푋)↓푆푆 퐵 ×푋 ♯ (퐴♯×푋) ♯ ♯ 1 ♯ 휓∶ (퐵 × 푋) ∪ ((퐴 × 푋) ↓푆 푆 ) Fun((훥 ) , ♮푍) such that 휓| ♯ is adjoint to 휙| ♯ ∘ pr ♯ and 휓| ♯ ♯ is adjoint to ℎ. Then we may 퐵 ×푋 퐵 ×푋 퐵 ×푋 (퐴 ×푋)↓푆푆 1 ♯ ♯ factor the above square through the trivial fibration Fun((훥 ) , ♮푍) ♮푍 ↓푆 푆 of Lm. 6.3 to obtain the commutative rectangle

휓 푒 ♯ (퐴♯×푋) ♯ ♯ 1 ♯ 1 (퐵 × 푋) ∪ ((퐴 × 푋) ↓푆 푆 ) Fun((훥 ) , ♮푍) ♮푍 ∼ 휓̃ ♯ ♯ ♯ ♯ (퐵 × 푋) ↓푆 푆 ♮푍 ↓푆 푆 푆 푒1 휙|퐵♯×푋 × id

The dotted lift 휓̃ exists, and 푒1 ∘ 휓̃ is our desired lift. The latter statement follows readily from the proof of Lm. 6.5, once one notes that the homotopy inverse and the relevant homotopies all respect the markings in (2).  Armed with this, the result is no trouble to prove. 18 CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH

Proof of Th. 9.7. Set 푆 ≔ 푇op. For (1), consider the

♯ 푆 ↓푆 ♮(퐶 × 퐷)

푓 ♯ ♯ ♯ 푆 ↓푆 ♮퐶 ↓푆 ♮퐷 푆 ↓푆 ♮퐷 푆

♯ ♯ 푆 ↓푆 ♮퐶 푆

푆♯ where the square is a pullback square. By Lm. 9.8(2), the map

♮(풪(푆) ×푆 퐶) ×푆 ♮퐷 ♮(풪(푆) ×푆 퐶) ↓푆 ♮퐷

+ is a homotopy equivalence in 푠Set/푆 (via the target map to 푆), hence so is 푓, because the additional marked edges do not obstruct the data of the homotopy equivalence. Invoking [6, Lm. 6.6] and [6, Lm. 6.7], we conclude. For (2), we may simply repeat the same argument with the leftmost copy of 푆 removed. 

9.9. Example. One of the most important examples, not surprisingly, is the 푇-presheaf 푇-∞-category vop P(퐶) ≔ Fun op (퐶 , Top ), 푇 푇 defined for any 푇-∞-category 퐶. In the next section, we will provide a Yoneda embedding, and we will show that it is fully faithful.

9.10. Warning. The construction Fun푇op (−, −) does not make homotopical sense when the + first variable is not fibrant, so it does not yield a Quillen bifunctor on 푠Set/푇op . Nevertheless, we can say the following about varying the first variable.

9.11. Proposition. Let 퐶, 퐷, and 퐸 be 푇-∞-categories, and let 퐹∶ 퐶 퐷 be a 푇-functor. Denote by ⋆ 퐹 ∶ Fun푇op (퐷, 퐸) Fun푇op (퐶, 퐸) denote the 푇-functor induced by 퐹. (1) If 퐹 is an equivalence, then so is 퐹⋆. ⋆ + (2) If 퐹 is a cofibration, then 퐹 is a fibration in 푠Set/푆.

op + Proof. Let 푆 = 푇 . For (1), since 퐹∶ ♮퐶 ♮퐷 is a homotopy equivalence in 푠Set/푆, so is ♯ ♯ ⋆ 푆 ↓푆 ♮퐶 푆 ↓푆 ♮퐷. Thus it follows from [6, Lm. 6.7] that 퐹 is an equivalence. + For (2), we need to verify that for any trivial cofibration 퐴 퐵 in 푠Set/푆, the map

(퐴↓푆♮퐶) (퐴 ↓푆 ♮퐷) ∪ (퐵 ↓푆 ♮퐶) 퐵 ↓푆 ♮퐷

+ is a trivial cofibration in 푠Set/푆. By the proof of Pr. 9.6, − ↓푆 ♮퐶 preserves trivial and ditto for ♮퐷. The result then follows.  EXPOSÉ I – ELEMENTS OF PARAMETRIZED HIGHER CATEGORY THEORY 19

10. The parametrized Yoneda embedding In [5, §5], we have already constructed a parametrized form of the twisted arrow ∞- category of a 푇-∞-category. Let us recall elements of this construction. 10.1. Recollection. Suppose 푇 an ∞-category, and suppose (퐶, 푝) a 푇-∞-category. In [5, 5.5] we define 풪̃(퐶/푇op) so that the objects are morphisms 푓∶ 푢 푣 of 퐶 such that 푝(푓) is an identity morphism in 푇, and a morphism 푓 푔 is a commutative diagram

푢 휙 휓 푥 푤 푓 푔

푣 푦 휉 in which 휙 is 푝-cocartesian, and 푝(휓) is an identity morphism. Composition is performed by forming the pushout along the top and simple composition along the bottom. The functor 푀∶ 풪̃(퐶/푇op) 퐶vop × 퐶 will carry an object 푓 ∈ 풪̃(퐶/푇op) as above to the pair of objects (푢, 푣) ∈ 퐶vop × 퐶, and it will carry a morphism 푓 푔 as above to the pair of morphisms

푤 휉 ( 휙 휓 , 푣 푦) ∈ 퐶vop × 퐶. 푢 푥

We demonstrate that 푀 is a left 푇-fibration. 10.2. Definition. In light of Pr. 8.3, the left 푇-fibration 푀 is classified by a 푇-functor Mor ∶ 퐶vop × 퐶 Top , 퐶 푇 and therefore by Th. 9.7 also a 푇-functor 푗∶ 퐶 P(퐶), called the parametrized Yoneda embedding. In order to justify this bit of terminology, we should actually demonstrate that 푗 is, in fact, fully faithful. To verify this, we work fibrewise: for any object 푉 ∈ 푇, we look at the functor on fibers 푗 ∶ 퐶 Fun (퐶vop × 푉, Top ). 푉 푉 푉 푉op

Our universal property of 푇-spaces (Th. 7.8) now permits one to regard the target of 푗푉 as vop the ∞-category Fun(퐶 ×푇op 푉, Top). Unwinding the definitions above, it is easy to see vop op that 푗푉 factors up to homotopy as the inclusion 퐶푉 (퐶 ×푇op 푉) followed by Yoneda embedding vop op vop (퐶 ×푇op 푉) Fun(퐶 ×푇op 푉, Top).

This already proves that 푗푉 is fully faithful, but in fact we also can conclude the following.

10.3. Proposition. Given 푇, 퐶, and 푉 as above, for any object 푋 ∈ 퐶푉, and for any 푇-functor 퐹∶ 퐶vop × 푉 Top 푉 we have an equivalence

퐹(푋) ≃ MapFun (퐶vop × 푉,Top )(푗푉(푋), 퐹). 푉 푉op 20 CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH

In any case, we immediately deduce the following. 10.4. Theorem (Parametrized Yoneda lemma). For any ∞-category 푇, and for any 푇-∞-cate- gory 퐶, the parametrized Yoneda embedding 푗∶ 퐶 P(퐶) is fully faithful.

Appendix A. Notational glossary

퐶∼ the marked simplicial set (퐶, 푖퐶) attached to an ∞-category 퐶 in which ∼ ♮ 푖퐶 ⊂ 퐶1 is the class of equivalences; equivalently, 퐶 = 퐶 = ♮퐶 for the cartesian and cocartesian fibration 퐶 훥0. vop vop op ∨ op 퐶 the opposite 푇-∞-category 퐶 = (퐶 ) = (∨퐶) of a 푇-∞-category 퐶; see Df. 3.1. Cat∞ the ∞-category of ∞-categories, i.e., the simplicial nerve of the simplicial category of fibrant marked simplicial sets; see [9, Df. 3.0.0.1]. Cat∞,푇 the ∞-category of 푇-∞-categories, i.e., the simplicial nerve of the simpli- cial category of fibrant marked simplicial sets over 푇op in the cocartesian model structure; see Nt. 1.4. F퐺 the nerve of the ordinary category of all finite, continuous 퐺-sets and 퐺-equivariant maps; see Nt. 2.10. op F푇 the ∞-category of finite 푇-sets, i.e., the full subcategory of Fun(푇 , Top) spanned by the finite corpoducts of representables; see Nt. 2.10.

F푇 the 푇-∞-category of finite 푇-sets; see Ex. 2.14. Fun푇op the ∞-category of functors between two ∞-categories; see Nt. 1.4. ̃ op Fun푇op the 푇-∞-category of functors from the left of a 푇 -∞-category to a 푇-∞-category; see Cnstr. 5.2.

Fun푇op the 푇-∞-category of functors between two 푇-∞-categories; see Df. 9.2. 퐺 a profinite group.

Map푇op the space of 푇-functors between two 푇-∞-categories; see Nt. 1.4. 푀 ↓푆 푁 the lax pullback of two maps 푀 푆 and 푁 푆; see Nt. 2.7. 풪(퐶) the arrow ∞-category Fun(훥1, 퐶) of an ∞-category 퐶. 풪̃(퐶) the twisted arrow ∞-category of an ∞-category 퐶, so that the 푛-simplices of 풪̃(퐶) are functors (훥푛)op ⋆ 훥푛 퐶; see [1, Nt. 2.3], [5, Df. 2.1], and [3, §1]. O퐺 the full subcategory of F퐺 spanned by the finite 퐺-orbits, i.e., the finite transitive (continuous) 퐺-sets; see Df. 1.1. Orbit(푈) the set of transitive 퐺-subsets 푊 ⊂ 푈 of a 퐺-set 푈, or, more generally, the set of representable summands of a finite 푇-set 푈; see Nt. 2.10. P(퐶) the 푇-∞-category of 푇-presheaves on a 푇-∞-category 퐶; see Ex. 9.9. Top the ∞-category of spaces, i.e., the ∞-category freely generated under col- imits by one object, or, equivalently, the simplicial nerve of the fibrant simplicial category of Kan complexes, or, again equivalently, the full sub- category of Cat∞ spanned by those objects in which every edge is marked; see [9, Df. 1.2.16.1]. 푈 the discrete 푇-space attached to a finite 푇-set 푈; see Ex. 2.4 and Ex. 2.11. EXPOSÉ I – ELEMENTS OF PARAMETRIZED HIGHER CATEGORY THEORY 21

푋♮ the marked simplicial set (푋, 푖푆푋) attached to a cartesian fibration 푆 푝∶ 푋 푆 in which 푖 푋 ⊂ 푋1 is the class of 푝-cartesian edges; see [9, Df. 3.1.1.8]. 푋∨ the right dual cocartesian fibration 푝∨ ∶ 푋∨ 푆op of a cartesian fibra- tion 푝∶ 푋 푆, i.e., the simplicial set whose 푛-simplices are functors ̃ 푛 op 푥∶ 풪(훥 ) 푋 such that any morphism of the form 푥푖푘 푥푗푘 lies over an identity in 푆, and any morphism of the form 푥푖푗 푥푖푘 is 푝- cartesian; see [5] and [3, Cnstr. 2.6]. ♮푌 the marked simplicial set (푌, 푖푇푌) attached to a cocartesian fibration 푞∶ 푌 푇 in which 푖푇푌 ⊂ 푌1 is the class of 푞-cocartesian edges; see [9, Df. 3.1.1.8]. op ∨ op op ∨ op op ∨푌 the dual cartesian fibration ∨푞 ≔ ((푞 ) ) ∶ ∨푌 ≔ ((푋 ) ) 푇 of a cocartesian fibration 푞∶ 푌 푇; see [5] and [3, Cnstr. 2.6].

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Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA E-mail address: [email protected]

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA E-mail address: [email protected]

University of Minnesota, School of Mathematics, Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455, USA E-mail address: [email protected]

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA E-mail address: [email protected]

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA E-mail address: [email protected]