FIBRATIONS in ∞-CATEGORY THEORY Contents 1. Left, Right, and Kan Fibrations 2 2. Inner Fibrations and Isofibrations 5 3. Coca

Home , Fiber (mathematics), Homotopy category, Kan fibration

CLARK BARWICK AND JAY SHAH

Abstract. In this short expository note, we discuss, with plenty of examples, the bestiary of fibrations in quasicategory theory. We underscore the simplicity and clarity of the constructions these fibrations make available to end-users of higher category theory.

Contents 1. Left, right, and Kan fibrations 2 2. Inner fibrations and isofibrations 5 3. Cocartesian and cartesian fibrations 7 4. Flat inner fibrations 10 5. Categorical patterns 12 6. Constructing fibrations via additional functoriality 13 References 18

The theory of ∞-categories – as formalized in the model of quasicategories – offers two ways of specifying homotopy theories and functors between them. First, we may describe a homotopy theory via a homotopy-coherent universal property; this is a widely appreciated advantage, and it’s a feature that any sufficiently well-developed model of ∞-categories would have. For example, the ∞-category Top of spaces is the free ∞-category generated under (homotopy) colimits by a single object [7, Th. 5.1.5.6]. The second way of specifying ∞-categories seems to be less well-loved: this is the ability to perform completely explicit constructions with excellent formal properties. This allows one to avoid the intricate workarounds that many of us beleaguered homotopy theorists have been forced to deploy in order solve infinite hierarchies of homotopy coherence problems. This feature seems to be peculiar to the model of quasicategories, and the main instrument that makes these explicit constructions possible is the theory of fibrations of various sorts. In this étude, we study eight sorts of fibrations of quasicategories in use today – left, right, Kan, inner, iso (aka categorical), cocartesian, cartesian, and flat – and we discuss the beautifully explicit constructions they provide. In the end, ∞-category theory as practiced today combines these two assets, and the result is a powerful amalgam of universal characterizations and crashingly explicit constructions. We will here focus on the underappreciated latter feature, which provides incredibly concrete constructions to which we would not otherwise have access. We have thus written this under the assumption that readers are more or less familiar with the content of the first chapter of Lurie’s book [7].

We are thankful to Bob Bruner and Emily Riehl for their helpful comments on this paper. 1 2 CLARK BARWICK AND JAY SHAH

1. Left, right, and Kan fibrations The universal property of the ∞-category Top we offered above certainly characterizes it up to a contractible choice, but it doesn’t provide any simple way to specify a functor into Top. At first blush, this looks like very bad news: after all, even if 퐶 is an ordinary category, to specify a functor of ∞-categories 퐹∶ 퐶 Top, one has to specify an extraordinary amount of information: one has to give, for every object 푎 ∈ 퐶, a space 퐹(푎); for every morphism 푓∶ 푎 푏, a map 퐹(푓)∶ 퐹(푎) 퐹(푏); for every pair of composable morphisms 푓∶ 푎 푏 and 푔∶ 푏 푐, a homotopy 퐹(푔푓) ≃ 퐹(푔)퐹(푓); for every triple of composable morphisms, a homotopy of homotopies; etc., etc., ad infinitum. However, ordinary category theory suggests a way out: suppose 퐹∶ 퐶 Set a functor. One of the basic tricks of the trade in category theory is to build a category Tot 퐹, sometimes called the category of elements of 퐹. The objects of Tot 퐹 are pairs (푎, 푥), where 푎 ∈ 퐶 is an object and 푥 ∈ 퐹(푎) is an element; a morphism (푎, 푥) (푏, 푦) of Tot 퐹 is a morphism 푓∶ 푎 푏 such that 퐹(푓)(푥) = 푦. The category Tot 퐹, along with the projection 푝∶ Tot 퐹 퐶, is extremely useful for studying the functor 퐹. For example, the set of sections of 푝 is a limit of 퐹, and the set 휋0(Tot 퐹) of connected components is a colimit of 퐹. In fact, the assignment 퐹 Tot 퐹 is an equivalence of categories between the category of functors 퐶 Set and those functors 푋 퐶 such that for any morphism 푓∶ 푎 푏 of 퐶 and for any object 푥 ∈ 푋 with 푝(푥) = 푎, there exists a unique morphism 휙∶ 푥 푦 with 푝(휙) = 푓. (These functors are sometimes called discrete opfibrations.) In other words, functors 퐶 Set correspond to functors 푋 퐶 such that for any solid arrow commutative square

1 훬0 푁푋 푝 훥1 푁퐶, there exists a unique dotted lift. We may therefore hope that, instead of working with functors from 퐶 to Top, one might work with suitable ∞-categories over 퐶 instead. To make this work, we need to formulate the ∞-categorical version of this condition. To this end, we adopt the same attitude that permits us to arrive at the definition of an ∞-category: instead of demanding a single unique horn filler, we demand a whole hierarchy of horn fillers, none of which we require to be unique. The hierarchy ensures that the filler at any stage is unique up to a homotopy that is unique up to a homotopy, etc., etc., ad infinitum.

1.1. Definition. A left fibration is a map 푝∶ 푋 푆 of simplicial sets such that for any integer 푛 ≥ 1 and any 0 ≤ 푘 < 푛 and any solid arrow commutative square

푛 훬푘 푋 푝 훥푛 푆, there exists a dotted lift. FIBRATIONS IN ∞-CATEGORY THEORY 3

Dually, a right fibration is a map 푝∶ 푋 푆 of simplicial sets such that for any integer 푛 ≥ 1 and any 0 < 푘 ≤ 푛 and any solid arrow commutative square

푛 훬푘 푋 푝 훥푛 푆, there exists a dotted lift. Of course Kan fibration is a map of simplicial sets that is both a left and right fibration. To understand these notions, we should begin with some special cases. 1.2. Example. For any simplicial set 푋, the unique map 푋 훥0 is a left fibration if and only if 푋 is an ∞-groupoid (i.e., a Kan complex). Indeed, we see immediately that 푋 is an ∞-category, so to conclude that 푋 is an ∞-groupoid, it suffices to observe that the homotopy category ℎ푋 is a groupoid; this follows readily from the lifting condition for the 2 2 horn inclusion 훬0 훥 . Since pullbacks of left fibrations are again left fibrations, we conclude immediately that the fibers of a left fibration are ∞-groupoids.

1.3. Example ([7, Cor. 2.1.2.2]). If 퐶 is an ∞-category and 푥 ∈ 퐶0 is an object, then recall that one can form the undercategory 퐶푥/ uniquely via the following functorial bijection: 0 Mor(퐾, 퐶푥/) ≅ Mor(훥 ⋆ 퐾, 퐶) ×Mor(훥0,퐶) {푥}. 0 The inclusion 퐾 훥 ⋆ 퐾 induces a forgetful functor 푝∶ 퐶푥/ 퐶. The key fact (due to Joyal) is that 푝 is a left fibration; in particular, 퐶푥/ is an ∞-category [7, Cor. 2.1.2.2]. The fiber of 푝 over a vertex 푦 ∈ 퐶0 is the ∞-groupoid whose 푛-simplices are maps 푛+1 {0} 푓∶ 훥 퐶 such that 푓(훥 ) = 푥 and 푓|훥{1,…,푛+1} is the constant map at 푦. In other words, 퐿 it is Hom퐶(푥, 푦) in the notation of [7, Rk. 1.2.2.5]. Dually, we can define the overcategory 퐶/푥 via 0 Mor(퐾, 퐶푥/) ≅ Mor(퐾 ⋆ 훥 , 퐶) ×Mor(훥0,퐶) {푥}, 0 and the inclusion 퐾 퐾 ⋆ 훥 induces a right fibration 푞∶ 퐶/푥 퐶. The fiber of 푞 over a 푅 vertex 푦 ∈ 퐶0 is then Hom퐶(푦, 푥). 1.3.1. Subexample. In the introduction, we mentioned that Top is the ∞-category that is freely generated under colimits by a single object ∗. This generator turns out to be the terminal object in Top. Let us write Top∗ for the overcategory Top∗/. The forgetful functor Top∗ Top is a left fibration. 퐿 The fiber over a vertex 푋 ∈ Top0 is the ∞-groupoid HomTop(∗, 푋), which we will want to think of as a model for 푋 itself. Here is the theorem that is going to make that possible: 1.4. Theorem (Joyal). Suppose 퐶 an ∞-category (or more generally, any simplicial set). For any functor 퐹∶ 퐶 Top, we may consider the left fibration

Top∗ ×Top,퐹 퐶 퐶. This defines an equivalence of ∞-categories Fun(퐶, Top) ∼ LFib(퐶), 4 CLARK BARWICK AND JAY SHAH where LFib(퐶) is the simplicial nerve of the full simplicial subcategory of 푠Set/퐶 spanned by the left fibrations. Dually, for any functor 퐺∶ 퐶 Topop, we may consider the right fibration op Top∗ ×Topop,퐺 퐶 퐶. This defines an equivalence of ∞-categories Fun(퐶op, Top) ∼ RFib(퐶), where RFib(퐶) is the simplicial nerve of the full simplicial subcategory of 푠Set/퐶 spanned by the right fibrations. A left fibration 푝∶ 푋 퐶 is said to be classified by 퐹∶ 퐶 Top just in case it is equivalent to the left fibration

Top∗ ×Top,퐹 퐶 퐶. Dually, a right fibration 푞∶ 푌 퐶 is said to be classified by 퐺∶ 퐶op Top just in case it is equivalent to the right fibration op Top∗ ×Topop,퐺 퐶 퐶. The proofs of Joyal’s theorem1 are all relatively involved, and they involve breaking this assertion up into several constituent parts. But rather than get distracted by these details (beautiful though they be!), let us instead swim in the waters of appreciation for this result as end-users. 1.5. Example. Even for 퐶 = 훥0, this theorem is nontrivial: it provides an equivalence ∼ Top Gpd∞, where of course Gpd∞ is the simplicial nerve of the full simplicial subcategory of 푠Set spanned by the Kan complexes. In other words, this result provides a concrete model for the ∞-category that was might have only been known through its universal property. But the deeper point here is that with the universal characterization of the introduction, it’s completely unclear how to specify a functor from an ∞-category 퐶 into Top. Even with the equivalence Top ≃ Gpd∞, we would still have to specify an infinite hierarchy of data to check this. However, with Joyal’s result in hand, our task becomes to construct a left fibration 푋 퐶. This is infinitely simpler: we are free to construct 푋 in a manner that will visibly map to our 퐶, and our only job is to check a bunch of horn-filling conditions. 1.6. Example. One may use Joyal’s theorem to find that if 푝∶ 푋 퐶 is a left fibration classified by a functor 퐹∶ 퐶 Top, then the colimit of 퐹 is weakly homotopy equivalent to

푋, and the the limit of 퐹 is weakly homotopy equivalent to the space Map퐶(퐶, 푋) of sections of 푝. 1.7. Example. One attitude toward ∞-categories is that they are meant to be categories “weakly enriched” in spaces. Whatever this means, it should at least entail corepresentable and representable functors 푥 op ℎ ∶ 퐶 Top and ℎ푥 ∶ 퐶 Top. But thanks to Joyal’s theorem, we already have these: the former is given by the left fibration 퐶푥/ 퐶, and the latter is given by the right fibration 퐶/푥 퐶. This also provides a recognition principle: a left fibration 푋 퐶 corresponds to a corepresentable functor if and only if 푋 admits an initial object; in this case, we call the left fibration

1We know four proofs: the original one due to Joyal, a modification thereof due to Lurie [7], a simplification due to Dugger and Spivak [3], and a recent extreme simplification due to Heuts and Moerdijk [4, 5]. FIBRATIONS IN ∞-CATEGORY THEORY 5 itself corepresentable. Dually, a right fibration 푋 퐶 corresponds to a representable functor if and only if 푋 admits an terminal object; in this case, we call the right fibration itself representable. 1.8. Example. In the same vein, we expect to have a functor op Map퐶 ∶ 퐶 × 퐶 Top for any ∞-category 퐶. From Joyal’s Theorem, our job becomes to construct a left fibration (푠, 푡)∶ 풪̃(퐶) 퐶op × 퐶. It turns out that this isn’t so difficult: define 풪̃(퐶) via the formula ̃ 푛,op 푛 풪(퐶)푛 ≔ Mor(훥 ⋆ 훥 , 퐶), and 푠 and 푡 are induced by the inclusions 훥푛,op 훥푛,op ⋆ 훥푛 and 훥푛 훥푛,op ⋆ 훥푛, respec- tively. So the claim is that (푠, 푡) is a left fibration. This isn’t a completely trivial matter, but there is a proof in [8], and another, slightly simpler, proof in [2]. The key point is to study the behavior of the left adjoint of the functor 풪̃on certain “left anodyne” monomorphisms. These two examples illustrate nicely a general principle about working “vertically” –i.e., with left and right fibrations – versus working “horizontally” – i.e., with functors to Top. It is easy to write down a left or right fibration, but it may not be easy to see that it is a left or right fibration. On the other hand, it is quite difficult even to write down a suitable functorto Top. So working vertically rather than horizontally relocates the difficulty in higher category theory from a struggle to make good definitions to a struggle to prove good properties. 1.9. Example. Suppose 퐶 an ∞-category. A Kan fibration to 퐶 is simultaneously a left fibration and a right fibration. So a Kan fibration must correspond to a both a covariant functor and a contravariant functor to Top, and one sees that the “pushforward” maps must be homotopy inverse to the “pullback” maps. That is, the following are equivalent for a map 푝∶ 푋 퐶 of simplicial sets: ▶ 푝 is a Kan fibration; ▶ 푝 is a left fibration, and the functor 퐶 Top that classifies it carries any morphism of 퐶 to an equivalence; ▶ 푝 is a right fibration, and the functor 퐶op Top that classifies it carries any morphism of 퐶 to an equivalence. From this point view, we see that when 퐶 is an ∞-groupoid, Kan fibrations 푋 퐶 are “essentially the same thing” as functors 퐶 Top, which are in turn indistinguishable from functors 퐶op Top.

2. Inner fibrations and isofibrations Inner fibrations are tricky to motivate from a 1-categorical standpoint, because the nerve of any functor is automatically an inner fibration. We will discuss here a reasonable way of thinking about inner fibrations, but we do not know a reference for complete proofs, yet. 2.1. Definition. An inner fibration is a map 푝∶ 푋 푆 of simplicial sets such that for any integer 푛 ≥ 2 and any 0 < 푘 < 푛 and any solid arrow commutative square

푛 훬푘 푋 푝 훥푛 푆, there exists a dotted lift. 6 CLARK BARWICK AND JAY SHAH

2.2. Example. Of course a simplicial set 푋 is an ∞-category just in case the canonical map 푋 훥0 is an inner fibration. Consequently, any fiber of an inner fibration isan ∞-category.

2.3. Example. If 푋 is an ∞-category and 퐷 is an ordinary category, then it’s easy to see that any map 푋 푁퐷 is an inner fibration. On the other hand, a map 푝∶ 푋 푆 is an inner fibration if and only if, for any 푛-simplex 휎 ∈ 푆푛, the pullback 푛 푛 푋 ×푆,휎 훥 훥 is an inner fibration. Consequently, we see that 푝 is an inner fibration if and only if, for any 푛 푛-simplex 휎 ∈ 푆푛, the pullback 푋 ×푆,휎 훥 is an ∞-category.

So in a strong sense, we’ll understand the “meaning” of inner fibrations one we understand the “meaning” of functors from ∞-categories to 훥푛.

2.4. Example. When 푛 = 1, we have the following. For any ∞-categories 퐶0 and 퐶1, there is an equivalence of ∞-categories

∼ op {퐶 } × Cat 1 × {퐶 } Fun(퐶 × 퐶 , Top). 0 Cat∞/훥{0} ∞/훥 Cat∞/훥{1} 1 0 1

A proof of this fact doesn’t seem to be contained in the literature yet, but we will nevertheless take it as given. Now the ∞-category on the right of this equivalence can also be identified with the ∞-category

퐿 Fun (푃(퐶1), 푃(퐶0))

op op of colimit-preserving functors between 푃(퐶1) = Fun(퐶1 , Top) and 푃(퐶0) = Fun(퐶0 , Top). Such a colimit-preserving functor is sometimes called a profunctor.

2.5. Example. When 푛 = 2, if 퐶 is an ∞-category, and 퐶 훥2 is a functor, then we have three fibers 퐶0, 퐶1, and 퐶2 and three colimit-preserving functors 퐹∶ 푃(퐶2) 푃(퐶1), 퐺∶ 푃(퐶1) 푃(퐶0), and 퐻∶ 푃(퐶2) 푃(퐶0). Furthermore, there is natural transformation 훼∶ 퐺 ∘ 퐹 퐻.

푛 In the general case, a functor 퐶 훥 amounts to the choice of ∞-categories 퐶0, 퐶1,…, 퐶푛 and a lax-commutative diagram of colimit-preserving functors among the various ∞-categories 푃(퐶푖). What we would like to say now is that the ∞-category of inner fibrations 푋 푆 is equivalent to the ∞-category of lax functors from 푆op to a suitable “double ∞-category” of ∞-categories and profunctors. We do not know, however, how to make such an assertion precise. In any case, we could ask for more restrictive hypotheses. We could, for example, ask for fibrations 푋 푆 that are classified by functors from 푆 to an ∞-category of profunctors (so that all the 2-morphisms that appear are equivalences); this is covered by the notion of flatness, which will discuss in the section after next. More restrictively, we can ask for fibrations 푋 푆 that are classified by functors from 푆 to Cat∞ itself; these are cocartesian fibrations, which we will discuss in the next section. For future reference, let’s specify an extremely well-behaved class of inner fibrations. FIBRATIONS IN ∞-CATEGORY THEORY 7

2.6. Definition. Suppose 퐶 an ∞-category. Then an isofibration (aka a categorical fibration2) 푝∶ 푋 퐶 is an inner fibration such that for any object 푥 ∈ 푋0 and any equivalence 푓∶ 푝(푥) 푏 of 퐶, there exists an equivalence 휙∶ 푥 푦 of 푋 such that 푝(휙) = 푓. We shall revisit this notion in greater detail in a moment, but for now, let us simply com- ment that an isofibration 푋 퐶 is an inner fibration whose fibers vary functorially in the equivalences of 퐶. For more general bases, this definition won’t do, of course, but we won’t have any use for isofibrations whose targets are not ∞-categories.

3. Cocartesian and cartesian fibrations If 퐹∶ 퐶 Cat is an (honest) diagram of ordinary categories, then one can generalize the category of elements construction as follows: form the category 푋 whose objects are pairs (푐, 푥) consisting of an object 푐 ∈ 퐶 and an object 푥 ∈ 퐹(푐), in which a morphism (푓, 휙)∶ (푑, 푦) (푐, 푥) is a morphism 푓∶ 푑 푐 of 퐶 and a morphism 휙∶ 푋(푓)(푦) 푥 of 퐹(푐). This is called the Grothendieck construction, and there is an obvious forgetful functor 푝∶ 푋 퐶. One may now attempt to reverse-engineer the Grothendieck construction by trying to extract the salient features of the forgetful functor 푝. What we may notice is that for any morphism 푓∶ 푑 푐 of 퐶 and any object 푦 ∈ 퐹(푑) there is a special morphism 훷 = (푓, 휙)∶ (푑, 푦) (푐, 푋(푓)(푦)) of 푋 in which 휙∶ 퐹(푓)(푦) 퐹(푓)(푦) is simply the identity morphism. This morphism is initial among all the morphisms 훹 of 푋 such that 푝(훹) = 푓; that is, for any morphism 훹 of 푋 such that 푝(훹) = 푓, there exists a morphism 퐼 of 푋 such that 푝(퐼) = id푐 such that 훹 = 퐼 ∘ 훷. We call morphisms of 푋 that are initial in this sense 푝-cocartesian. Since a 푝-cocartesian edge lying over a morphism 푑 푐 is defined by a universal property, it is uniquely specified up to a unique isomorphism lying over id푐. The key condition that we are looking for is then that for any morphism of 퐶 and any lift of its source, there is a 푝-cocartesian morphism with that source lying over it. A functor 푝 satisfying this condition is called a Grothendieck opfibration. Now for any Grothendieck opfibration 푝∶ 푋 퐶, let us attempt to extract a functor 퐹∶ 퐶 Cat that gives rise to it in this way. We proceed in the following manner. To any object 푎 ∈ 퐶 assign the fiber 푋푎 of 푝 over 푎. To any morphism 푓∶ 푎 푏 assign a functor 퐹(푓)∶ 푋푎 푋푏 that carries any object 푥 ∈ 푋푎 to the target 퐹(푓)(푥) ∈ 푋푏 of “the” 푞-cocartesian edge lying over 푓. Right away, we have a problem: 푞-cocartesian edges are only unique up to isomorphism. So these functors cannot be strictly compatible with composition; rather, one will obtain natural isomorphisms 퐹(푔 ∘ 푓) ≃ 퐹(푔) ∘ 퐹(푓)

2Emily Riehl makes the clearly compelling case that “isofibration” is preferable terminology, because it actually suggests what kind of lifting property it will have, whereas the word “categorical” is unhelpful in this regard. She also tells us that “isofibration” is a standard term in 1-category theory, and that the nerve of functor is an isofibration iff the functor is an isofibration. We join her in her view that “isofibration” is better. 8 CLARK BARWICK AND JAY SHAH that will satisfy a secondary layer of coherences that make 퐹 into a pseudofunctor. Fortunately, one can rectify this pseudofunctor to an equivalent honest functor, which in turn gives rise to 푝, up to equivalence. As we have seen in our discussion of left and right fibrations, there are genuine advantages in homotopy theory to working with fibrations instead of functors. Consequently, we define a class of fibrations that is a natural generalization of the class of Grothendieck opfibrations. 3.1. Definition. If 푝∶ 푋 푆 is an inner fibration of simplicial sets, then an edge 푓∶ 훥1 푋 is 푝-cocartesian just in case, for each integer 푛 ≥ 2, any extension

푓 훥{0,1} 푋,

퐹 푛 훬0 and any solid arrow commutative diagram

푛 퐹 훬0 푋 푝 퐹 훥푛 푆, a dotted lift exists. We say that 푝 is a cocartesian fibration if, for any edge 휂∶ 푠 푡 of 푆 and for every vertex 푥 ∈ 푋0 such that 푝(푥) = 푠, there exists a 푝-cocartesian edge 푓∶ 푥 푦 such that 휂 = 푝(푓). Cartesian edges and cartesian fibrations are defined dually, so that an edge of 푋 is 푝- cartesian just in case the corresponding edge of 푋op is cocartesian for the inner fibration 푝op ∶ 푋op 푆op, and 푝 is a cartesian fibration just in case 푝op is a cocartesian fibration. 3.2. Example ([7, Rk 2.4.2.2]). A functor 푝∶ 퐷 퐶 between ordinary categories is a Grothendieck opfibration if and only if the induced functor 푁(푝)∶ 푁퐷 푁퐶 on nerves is a cocartesian fibration. 3.3. Example. Any left fibration is a cocartesian fibration, and a cocartesian fibration isa left fibration just in case its fibers are ∞-groupoids. Dually, of course, the class of right fibrations coincides with the class of cartesian fibrations whose fibers are ∞-groupoids. 3.4. Example. Suppose 퐶 an ∞-category and 푝∶ 푋 퐶 an inner fibration. Then for any morphism 휂 of 푋, the following are equivalent. ▶ 휂 is an equivalence of 푋; ▶ 휂 is 푝-cocartesian, and 푝(휂) is an equivalence of 퐶; ▶ 휂 is 푝-cartesian, and 푝(휂) is an equivalence of 퐶. It follows readily that if 푝 is a cocartesian or cartesian fibration, then it is an isofibration. Conversely, if 퐶 is an ∞-groupoid, then the following are equivalent. ▶ 푝 is an isofibration; ▶ 푝 is a cocartesian fibration; ▶ 푝 is a cartesian fibration. 3.5. Example ([7, Cor. 2.4.7.12]). For any ∞-category 퐶, we write 풪(퐶) ≔ Fun(훥1, 퐶). Evaluation at 0 defines a cartesian fibration 푠∶ 풪(퐶) 퐶, and evaluation at 1 defines a cocartesian fibration 푡∶ 풪(퐶) 퐶. FIBRATIONS IN ∞-CATEGORY THEORY 9

One can ask whether the functor 푠∶ 풪(퐶) 퐶 is also a cocartesian fibration. One may observe [7, Lm. 6.1.1.1] that an edge 훥1 풪(퐶) is 푠-cocartesian just in case the correpond- 2 B 1 1 ing diagram (훬0) ≅ 훥 × 훥 퐶 is a pushout square. rep 3.6. Example. Consider the full subcategory RFib ⊂ 풪(Cat∞) spanned by the repre- rep sentable right fibrations. The restriction of the functor 푡∶ 풪(Cat∞) Cat∞ to RFib is again a cocartesian fibration.

In the following, we will denote by Cat∞ the simplicial nerve of the (fibrant) simplicial category whose objects are ∞-categories, in which Map(퐶, 퐷) is the maximal ∞-groupoid contained in Fun(퐶, 퐷). Similarly, for any ∞-category 퐶, we will denote by Cocart(퐶) (re- spectively, Cart(퐶)) the simplicial nerve of the (fibrant) simplicial category whose objects are cocartesian (resp., cartesian) fibrations 푋 퐶, in which Map(푋, 푌) is the ∞-groupoid whose 푛-simplices are functors 푋 × 훥푛 푌 over 퐶 that carry any edge (푓, 휏) in which 푓 is cocartesian (resp. cartesian) to a cocartesian edge (resp., a cartesian edge).

3.7. Theorem. Suppose 퐶 an ∞-category. For any functor 퐹∶ 퐶 Cat∞, we may consider the cocartesian fibration RFibrep × 퐶 퐶. Cat∞,퐹 This defines an equivalence of categories ∼ Fun(퐶, Cat∞) Cocart(퐶). op Dually, for any functor 퐺∶ 퐶 Cat∞, we may consider the cartesian fibration

rep,op op RFib ×Cat∞,퐺 퐶 퐶. This defines an equivalence of categories op ∼ Fun(퐶 , Cat∞) Cart(퐶). A cocartesian fibration 푝∶ 푋 퐶 is said to be classified by 퐹 just in case it is equivalent to the cocartesian fibration RFibrep × 퐶 퐶. Cat∞,퐹 Dually, a cartesian fibration 푞∶ 푌 퐶 is said to be classified by 퐹 just in case it is equivalent to the cocartesian fibration

rep,op op RFib ×Cat∞,퐺 퐶 퐶. 3.8. Example. Suppose 퐶 an ∞-category, and suppose 푋 퐶 an isofibration. If 휄퐶 ⊆ 퐶 is the largest ∞-groupoid contained in 퐶, then the pulled back isofibration

푋 ×퐶 휄퐶 휄퐶 is both cocartesian and cartesian, and so it corresponds to a functor op 휄퐶 ≃ 휄퐶 Cat∞. This is the sense in which the fibers of an isofibration vary functorially in equivalences if 퐶.

op 3.9. Example. For any ∞-category 퐶, the functor 퐶 Cat∞ that classifies the cartesian fibration 푠∶ 풪(퐶) 퐶 is the functor that carries any object 푎 of 퐶 to the undercategory ⋆ 퐶푎/ and any morphism 푓∶ 푎 푏 to the forgetful functor 푓 ∶ 퐶푏/ 퐶푎/. If 퐶 admits all pushouts, then the cocartesian fibration 푠∶ 풪(퐶) 퐶 is classified by a functor 퐶 Cat∞ that carries any object 푎 of 퐶 to the undercategory 퐶푎/ and any morphism 푓∶ 푎 푏 to the functor 푓! ∶ 퐶푎/ 퐶푏/ that is given by pushout along 푓. 10 CLARK BARWICK AND JAY SHAH

One particularly powerful construction with cartesian and cocartesian fibrations comes from [7, §3.2.2]. We’ve come to call this the cartesian workhorse. 3.10. Example. Suppose 푝∶ 푋 퐵op a cartesian fibration and 푞∶ 푌 퐵op a cocartesian op fibration. Suppose 퐹∶ 퐵 Cat∞ a functor that classifies 푝 and 퐺∶ 퐵 Cat∞ a functor that classifies 푞. Clearly one may define a functor op Fun(퐹, 퐺)∶ 퐵 Cat∞ that carries a vertex 푠 of 퐵op to the ∞-category Fun(퐹(푠), 퐺(푠)) and an edge 휂∶ 푠 푡 of 퐵op to the functor Fun(퐹(푠), 퐺(푠)) Fun(퐹(푡), 퐺(푡)) given by the assignment 퐹 퐺(휂) ∘ 퐹 ∘ 퐹(휂). 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(퐹, 퐺). Suppose 푝∶ 푋 퐵op is a cartesian fibration classified by a functor

퐹∶ 퐵 Cat∞, and suppose 푞∶ 푌 퐵op is a cocartesian fibration classified by a functor op 퐺∶ 퐵 Cat∞. ̃ ̃ op One defines a simplicial set Fun퐵(푋, 푌) and a map 푟∶ Fun퐵(푋, 푌) 퐵 defined by the following universal property: for any map 휎∶ 퐾 퐵op, one has a bijection ̃ Mor/퐵op (퐾, Fun퐵(푋, 푌)) ≅ Mor/퐵op (푋 ×퐵op 퐾, 푌), functorial in 휎. It is then shown in [7, Cor. 3.2.2.13] (but see 6.3 below for a better proof) that 푟 is a cocartesian fibration, and an edge 1 ̃ 푔∶ 훥 Fun퐵(푋, 푌) 1 is 푟-cocartesian just in case the induced map 푋 ×퐵op 훥 푌 carries 푝-cartesian edges to ̃ 푞-cocartesian edges. The fiber of the map Fun퐵(푋, 푌) 푆 over a vertex 푠 is the ∞-category op Fun(푋푠, 푌푠), and for any edge 휂∶ 푠 푡 of 퐵 , the functor 휂! ∶ 푇푠 푇푡 induced by 휂 is equivalent to the functor 퐹 퐺(휂) ∘ 퐹 ∘ 퐹(휂) described above.

4. Flat inner fibrations We have observed that if 퐶 is an ∞-category, and if 퐶 훥1 is any functor with fibers 퐶0 and 퐶1, then there is a corresponding profunctor from 퐶1 to 퐶0, i.e., a colimit-preserving 1 functor 푃(퐶1) 푃(퐶0). Furthermore, the passage from ∞-categories over 훥 to profunctors is even in some sense an equivalence. So to make this precise, let Prof denote the full subcategory of the ∞-category Pr퐿 of presentable ∞-categories and left adjoints spanned by those ∞-categories of the form 푃(퐶). We can almost – but not quite – construct an equivalence between the ∞-categories Cat∞/훥1 and the ∞-category Fun(훥1, Prof). The trouble here is that there are strictly more equivalences in Prof than there are in Cat∞; two ∞-categories are equivalent in Prof if and only if they have equivalent idempotent completions. So that suggests the fix for this problem: we employ a pullback that will retain the data of the ∞-categories that are the source and target of our profunctor. FIBRATIONS IN ∞-CATEGORY THEORY 11

4.1. Proposition. There is an equivalence of ∞-categories 1 1 Cat∞/훥1 ≃ Fun(훥 , Prof) ×Fun(휕훥1,Prof) Fun(휕훥 , Cat∞). We do not know of a reference for this result, yet, but we expect this to appear in a future work of P. Haine. When we pass to ∞-categories over 훥2, we have a more complicated problem: a func- 2 tor 퐶 훥 only specifies a lax commutative diagram of profunctors: three fibers 퐶0, 퐶1, and 퐶2; three colimit-preserving functors 퐹∶ 푃(퐶0) 푃(퐶1), 퐺∶ 푃(퐶1) 푃(퐶2), and 퐻∶ 푃(퐶0) 푃(퐶2); and a natural transformation 훼∶ 퐺 ∘ 퐹 퐻. In order to ensure that 훼 be a natural equivalence, we need a condition on our fibration. This is where flatness comes in. 4.2. Definition. An inner fibration 푝∶ 푋 푆 is said to be flat just in case, for any inner anodyne map 퐾 퐿 and any map 퐿 푆, the pullback

푋 ×푆 퐾 푋 ×푆 퐿 is a categorical equivalence. We will focus mostly on flat categorical fibrations. Let us see right away that some familiar examples and constructions yield flat inner fibrations. 4.3. Example (Lurie, [9, Ex. B.3.11]). Cocartesian and cartesian fibrations are flat isofibrations. In particular, if 퐶 is an ∞-groupoid, then any isofibration 푋 퐶 is flat. 4.4. Example (Lurie, [9, Pr. B.3.13]). If 푝∶ 푋 푆 is a flat inner fibration, then for any vertex 푥 ∈ 푋0, the inner fibrations 푋푥/ 푆푝(푥)/ and 푋/푥 푆/푝(푥) are flat as well. It is not necessary to test flatness with all inner anodyne maps; in fact, one can make do with the inner horn of a 2-simplex: 4.5. Proposition (Lurie, [9, Pr. B.3.14]). An inner fibration 푝∶ 푋 푆 is flat if and only if, for any 2-simplex 휎 ∈ 푆2, the pullback 2 2 푋 × 2 훬 푋 × 훥 푝,푆,(휎|훬1) 1 푝,푆,휎 is a categorical equivalence. 4.6. Proposition (Lurie, [9, Pr. B.3.2, Rk. B.3.9]). An inner fibration 푋 푆 is flat just in case, for any 2-simplex 푣

푢 푤 푓 of 푆 any for any edge 푥 푦 lying over 푓, the simplicial set 푋푥//푦 ×푆 {푣} is weakly contractible. 4.7. Note that the condition of the previous result is vacuous if the 2-simplex is degenerate. Consequently, if 푆 is 1-skeletal, then any inner fibration 푋 푆 is flat. 4.8. Proposition. Suppose 퐶 an ∞-category. Then there is an equivalence ∼ op op Flat(퐶) Fun(퐶 , Prof) ×Fun(휄퐶op,Prof) Fun(휄퐶 , Cat∞), where Flat(퐶) is an ∞-category of flat isofibrations 푋 퐶. Once again, we do not know a reference for this in the literature yet, but we expect that this will be shown in future work of P. Haine. 12 CLARK BARWICK AND JAY SHAH

5. Categorical patterns One of the primary appeals that flat inner fibrations have to offer is a collection of right Quillen functors for certain combinatorial model structures defined by means of categorical patterns. 5.1. Definition. A categorical pattern on a simplicial set 푆 is a triple (푀, 푇, 푃) consisting of the following:

▶ a set 푀 ⊂ 푆1 of marked edges that contains all the degenerate edges, ▶ a set 푇 ⊂ 푆2 of scaled 2-simplices that contains all the degenerate 2-simplices, and C C C ▶ a set 푃 of maps 푓훼 ∶ 퐾훼 푆 such that 푓((퐾훼 )1) ⊂ 푀 and 푓((퐾훼 )2) ⊂ 푇. 5.2. Notation. For the purposes of this appendix, if 푝∶ 푋 푆 and 푓∶ 퐾 푆 are maps of simplicial sets, then let us write

푝푓 ∶ 푋 ×푆 퐾 퐾 for the pullback of 푝 along 푓. 5.3. Definition. Suppose (푀, 푇, 푃) a categorical pattern on a simplicial set 푆. Then a marked map 푝∶ (푋, 퐸) (푆, 푀) is said to be (푀, 푇, 푃)-fibered if the following conditions obtain. ▶ The map 푝∶ 푋 푆 is an inner fibration. ▶ For every marked edge 휂 of 푆, the pullback 푝휂 is a cocartesian fibration. ▶ An edge 휖 of 푋 is marked just in case it is 푝푝(휖)-cocartesian. ▶ For any commutative square

휖 훥{0,1} 푋 푝 2 훥 휎 푆

in which 휎 is scaled, if 휖 is marked, then it is 푝휎-cocartesian. C ▶ For every element 푓훼 ∶ 퐾훼 푆 of 푃, the cocartesian fibration 푝 ∶ 푋 × 퐾C 퐾C 푓훼 푆 훼 훼 C is classified by a limit diagram 퐾훼 Cat∞. ▶ For every element 푓 ∶ 퐾C 푆 of 푃, any cocartesian section of 푝 is a 푝-limit 훼 훼 푓훼 diagram in 푋.

♯ 5.4. Example. For any simplicial set 푆, the (푆1, 푆2, ∅)-fibered maps (푋, 퐸) 푆 are pre- ♯ cisely those maps of the form ♮푋 푆 , where the underlying map of simplicial sets 푋 푆 is a cocartesian fibration. ♯ In particular, if 퐵 is an orbital ∞-category, then an (푆1, 푆2, ∅)-fibered maps (푋, 퐸) 푆 is essentially the same thing as a 퐵-∞-category. 5.5. Example. Suppose 훷 a perfect operator category [1, §6], and consider the following categorical pattern

(Ne, 푁(훬(훷))2, 푃) on the nerve 푁훬(훷) of the Leinster category [1, §7]. Here the class Ne ⊂ 푁(훬(훷))1 consists 2 of all the inert morphisms of 푁훬(훷). The class 푃 is the set of maps 훬0 푁훬(훷) given by diagrams 퐼 퐽 퐼′ of 훬(훷) in which both 퐽 퐼 and 퐽 퐼′ are inert, and ′ ′ |퐽| = |퐽 ×푇퐼 퐼| ⊔ |퐽 ×푇퐼′ 퐼 | ≅ |퐼| ⊔ |퐼 |. FIBRATIONS IN ∞-CATEGORY THEORY 13

† Then a marked map (푋, 퐸) (푁훬(훷), 푁훬 (훷)) is (Ne, 푁(훬(훷))2, 푃)-fibered just in case the underlying map of simplicial sets 푋 푁훬(훷) is a ∞-operad over 훷, and 퐸 is the collection of cocartesian edges over the inert edges [1, §8]. In particular, when 훷 = F, the category of finite sets, this recovers the collection of ∞-operads; cf. [9, Pr. 2.4.1.6]. 5.6. Example. If 훷 is a perfect operator category, we can contemplate another categorical pattern

(푁(훬(훷))1, 푁(훬(훷))2, 푃) on 훬(훷). Here 푃 is as in the previous example. † Now a marked map (푋, 퐸) (푁훬(훷), 푁훬 (훷)) is (푁(훬(훷))1, 푁(훬(훷))2, 푃)-fibered just in case the underlying map of simplicial sets 푋 푁훬(훷) is a 훷-monoidal ∞-category, and 퐸 is the collection of all cocartesian edges. 5.7. Theorem (Lurie, [9, Th. B.0.20]). Suppose (푀, 푇, 푃) a categorical pattern on a simplicial + set 푆. Then there exists a left proper, combinatorial, simplicial model structure on 푠Set/(푆,푀) in which the cofibrations are monomorphisms, and a marked map 푝∶ (푋, 퐸) (푆, 푀) is fibrant just in case it is (푀, 푇, 푃)-fibered. + We will denote this model category 푠Set/(푆,푀,푇,푃). We also have a characterization of the fibrations between fibrant objects in this model structure: 5.8. Proposition (Lurie, [9, Th. B.2.7]). Suppose 푆 an ∞-category, and suppose (푀, 푇, 푃) a categorical pattern on 푆 such that every equivalence of 푆 is marked, and every 2-simplex 2 {0,1} + 훥 푆 whose restriction to 훥 is an equivalence is scaled. If (푌, 퐹) is fibrant in 푠Set/(푆,푀,푇,푃), + then a map 푓∶ (푋, 퐸) (푌, 퐹) over (푆, 푀) is a fibration in 푠Set/(푆,푀,푇,푃) if and only if (푋, 퐸) is fibrant, and the underlying map 푓∶ 푋 푌 is an isofibration. 5.9. Example. If (푆, 푀) is a marked simplicial set, then a typical categorical pattern on a + simplicial set 푆 is simply (푀, 푆2, ∅). We call the resulting model structure on 푠Set/(푆,푀) the cocartesian model structure. + Indeed, the cocartesian model structure on 푠Set/푆♯ is precisely the cocartesian model structure described by Lurie. Furthermore, in light of the previous proposition, if 푝∶ 퐶 푆 is a cocartesian fibration, then we claim that the cocartesian model structure on 푠Set+ is /♮퐶 + created by the forgetful functor to the cocartesian model structure 푠Set/푆 – i.e., by regarding 푠Set+ as (푠Set+ ) . (Note that a model structure is uniquely charcterized by its cofibrations /♮퐶 /푆 /푝 and fibrant objects.)

6. Constructing fibrations via additional functoriality Now we can use the theory of flat isofibrations to describe some extra functoriality of these categorical pattern model structures in 푆. We will use this extra direction of functoriality to perform some beautifully explicit constructions of fibrations of various kinds. The main result is a tad involved, but the constructive power it offers is worth it in the end. 6.1. Notation. Let us begin with the observation that, given a map of marked simplicial sets 휋∶ (푆, 푀) (푇, 푁), there is a string of adjoints ⋆ 휋! ⊣ 휋 ⊣ 휋⋆, where the far left adjoint + + 휋! ∶ 푠Set/(푆,푀) 푠Set/(푇,푁) 14 CLARK BARWICK AND JAY SHAH is simply composition with 휋; the middle functor ⋆ + + 휋 ∶ 푠Set/(푇,푁) 푠Set/(푆,푀) is given by the assignment (푌, 퐹) (푌, 퐹) ×(푇,푁) (푆, 푀); and the far right adjoint + + 휋⋆ ∶ 푠Set/(푆,푀) 푠Set/(푇,푁) is given by a “space of sections.” That is, an 푛-simplex of 휋⋆(푋, 퐸) is a pair (휎, 푓) consisting of an 푛-simplex 휎 ∈ 푇푛 along with a marked map 푛 ♭ (훥 ) ×(푇,푁) (푆, 푀) (푋, 퐸) over (푆, 푀); an edge (휂, 푓) of 휋⋆(푋, 퐸) is marked just in case 휂 is marked, and 푓 carries any 1 edge of 훥 ×푇 푆 that projects to a marked edge of 푆 to a marked edge of 푋. 6.2. Theorem (Lurie, [9, Th. B.4.2]). Suppose 푆, 푆′, and 푋 three ∞-categories; suppose (푀, 푇, 푃) a categorical pattern on 푆; suppose (푀′, 푇′, 푃′) a categorical pattern on 푆′; and suppose 퐸 a collection of marked edges on 푋. Assume these data satisfy the following conditions. ▶ Any equivalence of either 푆 or 푋 is marked. ▶ The marked edges in 푆 and 푋 are each closed under composition. ▶ Every 2-simplex 훥2 푆 whose restriction to 훥{0,1} is an equivalence is scaled. Suppose 휋∶ (푋, 퐸) (푆, 푀) a map of marked simplicial sets satisfying the following conditions. ▶ The functor 휋∶ 푋 푆 is a flat isofibration. ▶ For any marked edge 휂 of 푆, the pullback 1 1 휋휂 ∶ 푋 ×푆 훥 훥 is a cartesian fibration. C ▶ For any element 푓훼 ∶ 퐾훼 푆 of 푃, the simplicial set 퐾훼 is an ∞-category, and the pullback 휋 ∶ 푋 × 퐾C 퐾C 푓훼 푆 훼 훼 is a cocartesian fibration. ▶ Suppose 푦 휙 휓

푥 휒 푧 a 2-simplex of 푋 in which 휋(휙) is an equivalence, 휓 is locally 휋-cartesian, and 휋(휓) is marked. Then the edge 휙 is marked just in case 휒 is. C ▶ Suppose 푓훼 ∶ 퐾훼 푆 an element of 푃, and suppose

푦 휙 휓

푥 휒 푧

a 2-simplex of 푋 × 퐾C in which 휙 is 휋 -cocartesian and 휋 (휓) is an equivalence. 푆 훼 푓훼 푓훼 Then the image of 휓 in 푋 is marked if and only if the image of 휒 is. FIBRATIONS IN ∞-CATEGORY THEORY 15

Finally, suppose 휌∶ (푋, 퐸) (푆′, 푀′) a map of marked simplicial sets satisfying the following conditions. ▶ Any 2-simplex of 푋 that lies over a scaled 2-simplex of 푆 also lies over a scaled 2-simplex of 푆′. ▶ For any element 푓 ∶ 퐾C 푆 of 푃 and any cocartesian section 푠 of 휋 , the composite 훼 훼 푓훼 C 푠 C 휌 ′ 퐾훼 푋 ×푆 퐾훼 푋 푆 lies in 푃′. Then the adjunction ⋆ + + ⋆ 휌! ∘ 휋 ∶ 푠Set/(푆,푀,푇,푃) 푠Set/(푆′,푀′,푇′,푃′) ∶ 휋⋆ ∘ 휌 is a Quillen adjunction. Straight away, let us put this result to work. 6.2.1. Corollary. Suppose 휋∶ 푋 푆 and 휌∶ 푋 푇 two functors of ∞-categories. Suppose 퐸 ⊂ 푋1 a collection of marked edges on 푋, and assume the following. ▶ Any equivalence of 푋 is marked, and the marked edges are closed under composition. ▶ The functor 휋∶ 푋 푆 is a flat locally cartesian fibration. ▶ Suppose 푦 휙 휓

푥 휒 푧 a 2-simplex of 푋 in which 휋(휙) is an equivalence, and 휓 is locally 휋-cartesian. Then the edge 휙 is marked just in case 휒 is. ⋆ Then the functor 휋⋆ ∘ 휌 is a right Quillen functor + + 푠Set/푇 푠Set/푆 for the cocartesian model structure. In particular, if 푌 푇 is a cocartesian fibration, then the map 푟∶ 푍 푆 given by the universal property ♭ Mor푆(퐾, 푍) ≅ Mor푇(퐾 ×푆♯ (푋, 퐸), ♮푌) 1 is a cocartesian fibration, and an edge of 푍 is 푟-cocartesian just in case the map 훥 ×푆 푋 푌 carries any edge whose projection to 푋 is marked to a 푞-cocartesian edge. We will speak of applying Cor. 6.2.1 to the span 푆 휋 (푋, 퐸) 휌 푇 to obtain a Quillen adjunction ⋆ + + ⋆ 휌! ∘ 휋 ∶ 푠Set/푆 푠Set/푇 ∶ 휋⋆ ∘ 휌 . 6.3. Example. Suppose 푝∶ 푋 푆 is a cartesian fibration, and suppose 푞∶ 푌 푆 is a ̃ cocartesian fibration. The construction of 3.10 gives a simplicial set Fun푆(푋, 푌) over 푆 given by the universal property ̃ Mor푆(퐾, Fun푆(푋, 푌)) ≅ Mor푆(퐾 ×푆 푋, 푌). ̃ By the previous result, the functor Fun푆(푋, 푌) 푆 is a cocartesian fibration. Consequently, one finds that [7, Cor. 3.2.2.13] is a very elementary, very special case of the previous result. 16 CLARK BARWICK AND JAY SHAH

Another compelling example can be obtained as follows. Suppose 휙∶ 퐴 퐵 is a functor of ∞-categories. Of course if one has a cocartesian fibration 푌 퐵 classified by a functor Y, then one may pull back to obtain the cocartesian fibration 퐴 ×퐵 푌 퐴 that is classified by the functor Y ∘ 휙. In the other direction, however, if one has a cocartesian fibration 푋 퐴 that is classified by a functor X, then how might one write explicitly the cocartesian fibration 푍 퐵 that is classified by the right Kan extension? It turns out that this is just the sort of situation that Cor. 6.2.1 can handle gracefully. We’ll need a spot of notation: 6.4. 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 simplicial set 푀 ↓푆 푁 is a model for the lax pullback of 푓 along 푔. 6.5. Proposition. If 휙∶ 퐴 퐵 is a functor of ∞-categories, and if 푝∶ 푋 퐴 is a cocartesian fibration, then form the simplicial set 푌 over 퐵 given by the following universal property: for any map 휂∶ 퐾 푍, we demand a bijection

Mor/퐵(퐾, 푌) ≅ Mor/퐴(퐾 ↓퐵 퐴, 푋), functorial in 휂. Then the projection 푞∶ 푌 퐵 is a cocartesian fibration, and if 푝 is classified by a functor X, then 푞 is classified by the right Kan extension of X along 휙.

Proof. As we have seen, we need conditions on 휙 to ensure that the adjunction ∗ + + 휙 ∶ 푠Set/퐵 푠Set/퐴 ∶ 휙∗ is Quillen. However, we can rectify this situation by applying Cor. 6.2.1 to the span ev ∘ pr pr 퐵 0 풪(퐵) (풪(퐵) × 퐴)♯ 퐴 퐴 ev1,퐵,휙 in view of the following two facts, which we leave to the reader to verify:

▶ the functor ev0 ∘pr풪(퐵) is a cartesian fibration; + ▶ for all 퐶 퐵 fibrant in 푠Set/퐵, the identity section 퐵 풪(퐵) induces a functor ♯ ♯ ♯ 퐶 ×퐵 퐴 퐶 ×퐵 풪(퐵) ×퐵 퐴 + which is a cocartesian equivalence in 푠Set/퐴. We thereby obtain a Quillen adjunction ∗ + + ∗ (pr퐴)! ∘ (ev0 ∘pr풪(퐵)) ∶ 푠Set/퐵 푠Set/퐴 ∶ (ev0 ∘pr풪(퐵))∗ ∘ (pr퐴) which models the adjunction of ∞-categories ∗ 휙 ∶ Fun(퐵, Cat∞) Fun(퐴, Cat∞) ∶ 휙∗, as desired. This result illustrates the basic principle that replacing strict pullbacks by lax pullbacks allows one to make homotopically meaningful constructions. Of course, any suitable theory of ∞-categories would allow one to produce the right adjoint 휙∗ less explicitly, by recourse to the adjoint functor theorem. This example rather illustrates, yet again, how the model of quasicategories allows one to control common categorical constructions in an utterly transparent manner. FIBRATIONS IN ∞-CATEGORY THEORY 17

Let us end our study with some simple but gratifying observations concerning the inter- action of Th. 6.2 with compositions and homotopy equivalences of spans. First the compositions: 6.6. Lemma. Suppose we have spans of marked simplicial sets

휋0 휌0 퐶0 퐷0 퐶1 and 휋1 휌1 퐶1 퐷1 퐶2 and categorical patterns (푀푖, 푇푖, ∅) on 퐶푖 such that the hypotheses of Th. 6.2 are satisfied for each span. Then the hypotheses of Th. 6.2 are satisfied for the span pr pr 퐷 0 퐷 × 퐷 1 퐷 0 0 퐶1 1 1 with the categorical patterns (푀 , 휋−1(푇 ), ∅) on each 퐷 . Consequently, we obtain a Quillen 퐷푖 푖 푖 푖 adjunction (휌 ∘ pr ) ∘ (휋 ∘ pr )⋆ ∶ 푠Set+ 푠Set+ ∶ (휋 ∘ pr ) ∘ (휌 ∘ pr )⋆, 1 1 ! 0 0 /(퐶0,푀0,푇0,∅) /(퐶2,푀2,푇2,∅) 0 0 ⋆ 1 1 which is the composite of the Quillen adjunction from 푠Set+ to 푠Set+ with /(퐶0,푀0,푇0,∅) /(퐶1,푀1,푇1,∅) the one from 푠Set+ to 푠Set+ . /(퐶1,푀1,푇1,∅) /(퐶2,푀2,푇2,∅) Proof. The proof is by inspection. However, one should beware that the “long” span 퐶 퐷 × 퐷 퐷 0 0 퐶1 1 1 can fail to satisfy the hypotheses of Th. 6.2, because the composition of locally cartesian fibrations may fail to be again be locally cartesian; this explains the roundabout formulation ∗ of the statement. Finally, observe that if we employ the base-change isomorphism 휌0 휋1,∗ ≅ ∗ pr0,∗ ∘ pr1 , then we obtain our Quillen adjunction as the composite of the two given Quillen adjunctions. Now let us see that Th. 6.2 is compatible with homotopy equivalences of spans: 6.7. Lemma. Suppose a morphism of spans of marked simplicial sets

퐾 휋 휌 푓

퐶 퐿 퐶′ 휋′ 휌′

∗ ′ ′ ∗ where 휌!휋 and (휌 )!(휋 ) are left Quillen with respect to the model structures given by cate- ′ gorical patterns 푃퐶 and 푃퐶′ on 퐶 and 퐶 . Suppose moreover that 푓 is a homotopy equivalence in 푠Set+ – i.e., suppose that there exists a homotopy inverse 푔 and homotopies /푃퐶′ ℎ∶ id ≃ 푔 ∘ 푓 and 푘∶ id ≃ 푓 ∘ 푔. ∗ ′ ′ ∗ Then the natural transformation 휌!휋 (휌 )!(휋 ) induced by 푓 is a weak equivalence on ′ ′ ∗ ∗ all objects, and, consequently, the adjoint natural transformation (휋 )∗(휌 ) 휋∗휌 is a weak equivalence on all fibrant objects. Proof. The homotopies ℎ and 푘 pull back to show that for all 푋 퐶, the map

id푋 ×퐶 푓∶ 푋 ×퐶 퐾 푋 ×퐶 퐿 is a homotopy equivalence with inverse id푋 ×퐶 푔. The last statement now follows from [6, 1.4.4(b)]. 18 CLARK BARWICK AND JAY SHAH

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