PARAMETRIZED HIGHER CATEGORY THEORY AND HIGHER ALGEBRA: EXPOSÉ I – ELEMENTS OF PARAMETRIZED HIGHER CATEGORY THEORY CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH Abstract. We introduce the basic elements of the theory of parametrized ∞-categories and functors between them. These notions are defined as suitable fibrations 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 functor ∞-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 Yoneda lemma. Contents 1. Parametrized ∞-categories 1 2. Examples of parametrized ∞-categories 3 3. Parametrized opposites 7 4. Parametrized subcategories 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 group. 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 homotopies that express the higher associativity of 휌. Following [6], we will want to encode this data very efficiently as a cocartesian fibration 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 homotopy) 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 CLARK BARWICK, 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 homotopy theory 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 nerve 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 HIGHER CATEGORY THEORY 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 subcategory 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 set (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 natural transformation 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.