Lax Colimits and Free Fibrations in ∞-Categories

David Gepner^{1}, Rune Haugseng, and Thomas Nikolaus

Received: November 2, 2015 Revised: February 28, 2017

Communicated by Stefan Schwede

Abstract. We define and discuss lax and weighted colimits of diagrams in ∞-categories and show that the coCartesian fibration corresponding to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free Cartesian fibration on a functor of ∞-categories. As an application of these results, we prove that 2-representable functors are preserved under exponentiation, and also that the total space of a presentable Cartesian fibration between is presentable, generalizing a theorem of Makkai and Paré to the ∞-categories setting. Lastly, in an appendix, we observe that pseudofunctors between (2,1)-categories give rise to functors between ∞-categories via the Duskin nerve. setting and the Duskin nerve.

2010 Mathematics Subject Classification: 18D30, 18A30 Keywords and Phrases: Fibered categories, presentable categories, lax limits and colimits

Contents

1. Introduction 1226 2. Twisted Arrow ∞-Categories, (Co)ends, and Weighted (Co)limits 1229 3. CoCartesian Fibrations over a Simplex 1231 4. Free Fibrations 1235 5. Natural Transformations as an End 1239 6. Enhanced Mapping Functors 1241 7. Cartesian and CoCartesian Fibrations as Weighted Colimits 1244 8. 2-Representable Functors 1246 9. Some Cartesian Fibrations Identified 1247 10. Presentable Fibrations are Presentable 1253 Appendix A. Pseudofunctors and the Naturality of Unstraightening 1256 References 1265

1Supported by NSF grant DMS-1406529

1. Introduction

In the context of ordinary category theory, Grothendieck’s theory of fibrations [Gro63] can be used to give an alternative description of functors to the category Cat of categories. This has been useful, for example, in the theory of stacks in algebraic geometry, as the fibration setup is usually more flexible. When working with ∞-categories, however, the analogous notion of Cartesian fibrations is far more important: since defining a functor to the ∞-category Cat_{∞} of ∞-categories requires specifying an infinite amount of coherence data, it is in general not feasible to “write down” definitions of functors, so that manipulating Cartesian fibrations is often the only reasonable way to define key functors. For ordinary categories, the Grothendieck construction gives a simple description of the fibration classified by a functor F: C^{op} → Cat; this can also be described formally as a certain weighted colimit, namely the lax colimit of the functor F. For ∞-categories, on the other hand, the equivalence between Cartesian fibrations and functors has been proved by Lurie using the straightening functor, a certain left Quillen functor between model categories. This leaves the corresponding right adjoint, the unstraightening functor, quite inexplicit. One of our main goals in this paper is to show that Lurie’s unstraightening functor is a model for the ∞-categorical analogue of the Grothendieck construction. More precisely, we introduce ∞-categorical versions of lax and oplax limits and colimits and prove the following:

Theorem 1.1.

(i) Suppose F: 𝒞 → Cat_{∞} is a functor of ∞-categories, and ℰ → 𝒞 is a coCartesian fibration classified by F. Then ℰ is the oplax colimit of the functor F.

(ii) Suppose F: 𝒞^{op} → Cat_{∞} is a functor of ∞-categories, and ℰ → 𝒞 is a Cartesian fibration classified by F. Then ℰ is the lax colimit of the functor F.

To prove this we make use of an explicit description of the free Cartesian fibration on an arbitary functor of ∞-categories. More precisely, the ∞-category Cat_{∞/𝒞}^{cart} of Cartesian fibrations over 𝒞 is a subcategory of the slice ∞-category Cat_{∞/𝒞}, and we show that the inclusion admits a left adjoint given by a simple formula:

Theorem 1.2. Let 𝒞 be an ∞-category. For p: ℰ → 𝒞 any functor of ∞- categories, let F (p) denote the map ℰ ×_{𝒞^{{1}}} 𝒞^{Δ^{1}} → 𝒞^{{0}} (i.e. the pullback is along the map 𝒞^{Δ^{1}} → 𝒞 given by evaluation at 1 ∈ Δ^{1} and the projection is induced by evaluation at 0). Then F defines a functor Cat_{∞/𝒞} → Cat_{∞/𝒞}^{cart} , which is left adjoint to the forgetful functor Cat_{∞/𝒞}^{cart} → Cat_{∞/𝒞}.

Lax Colimits and Free Fibrations in ∞-Categories 1227

In the special case where p: ℰ → 𝒞 is a Cartesian fibration and 𝒞 is an ∞- category equipped with a “mapping ∞-category” functor MAP_{𝒞} : 𝒞^{op} × 𝒞 → Cat_{∞}, such as is the case when 𝒞 is the underlying ∞-category of an (∞, 2)- category, then it is natural to ask when p is classified by a functor of the form MAP_{𝒞} (−, X): 𝒞^{op} → Cat_{∞} for some object X of 𝒞. We say that p is 2-representable when this is the case. As an application of our theorems, we show that, if p: ℰ → 𝒞 is a 2-representable Cartesian fibration such that the mapping ∞-category functor MAP_{𝒞} is tensored and cotensored over Cat_{∞}, and 𝒟 is any ∞-category, then the exponential q: Fun(𝒟, ℰ) → Fun(𝒟, 𝒞) is itself 2-representable. This is relevant in the description of the functoriality of twisted cohomology theories as discussed in joint work of the third author with U. Bunke [BN14]. More precisely it describes a converse to the construction in Section 3 and Appendix A of this paper. The third main result of this paper provides a useful extension of the theory of presentable ∞-categories in the context of Cartesian fibrations, generalizing a theorem of Makkai and Paré [MP89] to the ∞-categorical context. More precisely, we show:

Theorem 1.3. Suppose p: ℰ → 𝒞 is a Cartesian and coCartesian fibration such that 𝒞 is presentable, the fibres ℰ_{x} are presentable for all x ∈ 𝒞, and the classifying functor F: 𝒞^{op} → Ĉat_{∞} preserves κ-filtered limits for some regular cardinal κ. Then the ∞-category ℰ is presentable, and the projection p is an accessible functor (i.e. it preserves λ-filtered colimits for some sufficiently large cardinal λ).

While the theory of accessible and presentable categories is already an important part of ordinary category theory, when working with ∞-categories the analogous notions turn out to be indispensable. Whereas, for example, it is often possible to give an explicit construction of colimits in an ordinary category, when working with ∞-categories we often have to conclude that colimits exist by applying general results on presentable ∞-categories. Similarly, while for ordinary categories one can frequently just write down an adjoint to a given functor, for ∞-categories an appeal to the adjoint functor theorem, which is most naturally considered in the presentable context, is often unavoidable. It is thus very useful to know that various ways of constructing ∞-categories give accessible or presentable ones; many such results are proved in [Lur09a, §5], and our result adds to these by giving a criterion for the source of a Cartesian fibration to be presentable.

1.1. Overview. In §2 we briefly review the definitions of twisted arrow ∞- categories and ∞-categorical ends and coends, and use these to define weighted (co)limits. Then in §3 we prove our main result for coCartesian fibrations over a simplex, using the mapping simplex defined in [Lur09a, §3.2.2]. Before we extend this result to general coCartesian fibrations we devote three sections to preliminary results: in §4 we give a description of the free Cartesian fibration, i.e. the left adjoint to the forgetful functor from Cartesian fibrations over 𝒞 to the slice ∞-category Cat_{∞/𝒞}; in §5 we prove that the space of natural transformations between two functors is given by an end (a result first proved by Glasman [Gla16, Proposition 2.3.]), and in §6 we prove that the straightening equivalence extends to an equivalence of the natural enrichments in Cat_{∞} of the two ∞-categories involved. §7 then contains the proof of our main result: Cartesian and coCartesian fibrations are given by weighted colimits of the classifying functors. In §8 we give a simple application of our results to functors that are representable via an enrichment in Cat_{∞}, and in §9 we apply them to identify the functor classifying a certain simple Cartesian fibration; this is a key step in our proof in §10 that the source of a presentable fibration is presentable. Finally, in appendix A we use Duskin’s nerve for strict (2,1)-categories to check that the pseudonaturality of the unstraightening functors on the level of model categories implies that they are natural on the level of ∞-categories.

1.2. Notation. Much of this paper is based on work of Lurie in [Lur09a, Lur14]; we have generally kept his notation and terminology. In particular, by an ∞-category we mean an (∞, 1)-category or more specifically a quasicategory. We also use the following conventions, some of which differ from those of Lurie:

• Generic categories are generally denoted by single capital bold-face letters (A, B, C) and generic ∞-categories by single caligraphic letters (𝒜, ℬ, 𝒞). Specific categories and ∞-categories both get names in the normal text font.

• If 𝒞 is an ∞-category, we write ι𝒞 for the interior or underlying space of 𝒞, i.e. the largest subspace of 𝒞 that is a Kan complex.

• If f: 𝒞 → 𝒟 is left adjoint to a functor g: 𝒟 → 𝒞, we will refer to the adjunction as f ⊣ g.

• We write Pr^{L} for the ∞-category of presentable ∞-categories and functors that are left adjoints, i.e. colimit-preserving functors, and Pr^{R} for the ∞- category of presentable ∞-categories and functors that are right adjoints, i.e. accessible functors that preserve all small limits.

• If 𝒞 and 𝒟 are ∞-categories, we will denote the ∞-category of functors 𝒞 → 𝒟 by both Fun(𝒞, 𝒟) and 𝒟^{𝒞}.

• If S is a simplicial set, we write

St_{S}^{+} : (Set_{Δ}^{+})_{/S^{♯}} ⇄ Fun(ℭ(S)^{op}, Set_{Δ}^{+}) : Un_{S}^{+}

for the marked (un)straightening Quillen equivalence, as defined in [Lur09a, §3.2].

• We write Cat_{∞/𝒞}^{cart} for the subcategory of Cat_{∞/𝒞} consisting of Cartesian fibrations over 𝒞, with morphisms the functors that preserve Cartesian edges, Map_{𝒞}^{Cart}(–, –) for the mapping spaces in Cat_{∞/𝒞}^{cart} , and Fun_{𝒞}^{cart}(–, –) for the ∞-category of functors that preserve Cartesian edges, defined as a full subcategory of the ∞-category Fun_{𝒞} (–, –) of functors over 𝒞. Similarly, we write Cat_{∞/𝒞}^{cocart} for the ∞-category of coCartesian fibrations over 𝒞, Map_{𝒞}^{cocart}(–, –) for the mapping spaces in Cat_{∞/𝒞}^{cocart}, and Fun_{𝒞}^{cocart}(–, –) for the full subcategory of Fun_{𝒞} (–, –) spanned by the functors that preserve coCartesian edges.

• If 𝒞 is an ∞-category, we write

St_{𝒞} : Cat_{∞/𝒞}^{cart} ⇄ Fun(𝒞^{op}, Cat_{∞}): Un_{𝒞}

for the adjoint equivalence of ∞-categories induced by the (un)straightening Quillen equivalence via [Lur09a, Proposition 5.2.4.6].

• If S is a simplicial set, we write

St_{S}^{+,co} : (Set_{Δ}^{+})_{/S^{♯}} ⇄ Fun(ℭ(S), Set_{Δ}^{+}) : Un_{S}^{+,co}

for the coCartesian marked (un)straightening Quillen equivalence, given by St_{S}^{+,co}(X) := (St_{S^{op}}^{+}(X^{op}))^{op}.

• If 𝒞 is an ∞-category, we write

St_{𝒞}^{co}: Cat_{∞/𝒞}^{cocart} ⇄ Fun(𝒞, Cat_{∞}): Un_{𝒞}^{co}

for the adjoint equivalence of ∞-categories induced by the coCartesian (un)straightening Quillen equivalence.

• If 𝒞 is an ∞-category, we denote the Yoneda embedding for 𝒞 by

y_{𝒞} : 𝒞 → 𝒫(𝒞),

where 𝒫(𝒞) is the presheaf ∞-category Fun(𝒞^{op}, 𝒮) with 𝒮 the ∞-category of spaces.

1.3. Acknowledgments. David: Thanks to Joachim Kock for helpful discussions regarding free fibrations and lax colimits. Rune: I thank Clark Barwick for helpful discussions of the presentability result and Michael Shulman for telling me about [MP89, Theorem 5.3.4] in answer to a MathOverflow question. We thank Aaron Mazel-Gee and Omar Antolín Camarena for pointing out some inaccuracies in the first version of this paper. We also thank an anonymous referee for a very careful and helpful report.

2. Twisted Arrow ∞-Categories, (Co)ends, and Weighted (Co)limits

In this section we briefly recall the definitions of twisted arrow ∞-categories and (co)ends, and then use these to give a natural definition of weighted (co)limits in the ∞-categorical setting.

Definition 2.1. Let ϵ: Δ → Δ be the functor [n] ↦ [n] ⋆ [n]^{op}. The edgewise subdivision of a simplicial set S is the composite ϵ^{∗}S = S ∘ ϵ.

Definition 2.2. Let 𝒞 be an ∞-category. The twisted arrow ∞-category Tw(𝒞) of 𝒞 is the simplicial set ϵ^{∗}𝒞. Thus in particular

Hom(Δ^{n}, Tw(𝒞)) ≅ Hom(Δ^{n} ⋆ (Δ^{n})^{op}, 𝒞).

The natural transformations Δ^{•}, (Δ^{•})^{op} → Δ^{•} ⋆ (Δ^{•})^{op} induce a projection Tw(𝒞) → 𝒞 × 𝒞^{op}.

Remark 2.3. The twisted arrow ∞-category, which was originally introduced by Joyal, has previously been extensively used by Barwick [Bar13, Bar17] and collaborators [BGN14], and by Lurie [Lur14, §5.2.1]. By [Lur14, Proposition 5.2.1.3] the projection Tw(𝒞) → 𝒞 × 𝒞^{op} is a right fibration; in particular, the simplicial set Tw(𝒞) is an ∞-category if 𝒞 is. The functor 𝒞^{op} × 𝒞 → 𝒮 classified by this right fibration is the mapping space functor Map_{𝒞} (–, –) by [Lur14, Proposition 5.2.1.11].

Warning 2.4. There are two possible conventions for defining the edgewise subdivision (and therefore also the twisted arrow ∞-category); we follow that of Lurie in [Lur14, §5.2.1]. Alternatively, one can define the edgewise subdivision using the functor [n] ↦ [n]^{op} ⋆ [n], in which case Tw(𝒞) → 𝒞^{op} × 𝒞 is a left fibration — this is the convention used in the papers of Barwick cited above.

Example 2.5. The twisted arrow category Tw([n]) of the category [n] is the partially ordered set with objects (i, j) where 0 ≤ i ≤ j ≤ n and with (i, j) ≤ (i′, j′) if i ≤ i′ ≤ j′ ≤ j.

A natural definition of (co)ends in the ∞-categorical setting is then the following.

Definition 2.6. If F: 𝒞 × 𝒞^{op} → 𝒟 is a functor of ∞-categories, the coend of F is the colimit of the composite functor

Tw(𝒞) → 𝒞 × 𝒞^{op} → 𝒟.

Similarly, if G: 𝒞^{op} × 𝒞 → 𝒟 is a functor of ∞-categories, then the end of G is the limit of the composite functor

Tw(𝒞)^{op} → 𝒞^{op} × 𝒞 → 𝒟.

Remark 2.7. These ∞-categorical notions of ends and coends are also discussed in [Gla16, §2]. In the context of simplicial categories, a homotopically correct notion of coends was extensively used by Cordier and Porter [CP97]; see their paper for a discussion of the history of such definitions.

Now we can consider weighted (co)limits:

Definition 2.8. Let ℛ be a presentably symmetric monoidal ∞-category, i.e. a presentable ∞-category equipped with a symmetric monoidal structure such that the tensor product preserves colimits in each variable, and let ℳ be a right ℛ-module in Pr^{L}. Then ℳ is in particular tensored and cotensored over ℛ, i.e. there are functors

(– ⊗ –): ℳ × ℛ → ℳ,
(–)^{(–)} : ℛ^{op} × ℳ → ℳ,

such that for every x ∈ ℛ the functor –⊗x: ℳ → ℳ is left adjoint to (–)^{x}. Given functors F: 𝒞 → ℳ and W: 𝒞^{op} → ℛ, the W -weighted colimit colim_{𝒞}^{W} F of F is defined to be the coend colim_{Tw(𝒞)} F (–)⊗W (–). Similarly, given F: 𝒞 → ℳ and W: 𝒞 → ℛ, the W -weighted limit lim_{𝒞}^{W} F of F is the end lim_{Tw(𝒞)^{op}} F (–)^{W (–)}.

We are interested in the case where both ℛ and ℳ are the ∞-category Cat_{∞} of ∞-categories, with the tensoring given by Cartesian product and the cotensoring by Fun(–, –). In this case there are two special weights for every ∞-category 𝒞: we have functors 𝒞_{/–} : 𝒞 → Cat_{∞} and 𝒞_{–/}: 𝒞^{op} → Cat_{∞} sending x ∈ 𝒞 to 𝒞_{/x} and 𝒞_{x/}, respectively. Precisely, these functors are obtained by straightening the source and target projections 𝒞^{Δ^{1}} → 𝒞, which are respectively Cartesian and coCartesian. Using these functors, we can define lax and oplax (co)limits:

Definition 2.9. Suppose F: 𝒞 → Cat_{∞} is a functor. Then:

• The oplax colimit of F is the colimit of F weighted by 𝒞_{–/}, i.e.

colim_{Tw(𝒞)}F (–) × 𝒞_{–/}.

• The lax colimit of F is the colimit of F weighted by (𝒞^{op})_{/–}, i.e.

colim_{Tw(𝒞)}F (–) × (𝒞^{op})_{/–}.

• The lax limit of F is the limit of F weighted by 𝒞_{/–}, i.e.

lim_{Tw(𝒞)^{op}} Fun(𝒞_{/–}, F (–)).

• The oplax limit of F is the limit of F weighted by (𝒞^{op})_{–/}, i.e.

lim_{Tw(𝒞)^{op}} Fun((𝒞^{op})_{–/}, F (–)).

3. CoCartesian Fibrations over a Simplex

In this preliminary section we study coCartesian fibrations over the simplices Δ^{n}, and observe that in this case the description of a coCartesian fibration as an oplax colimit follows easily from results of Lurie in [Lur09a, §3.2]. More precisely, we will prove:

Proposition 3.1. There is an equivalence

colim_{Tw([n])} ϕ(–) × [n]_{–/} ⟶^{∼} Un_{[n]}^{co}(ϕ)

of functors Fun([n], Cat_{∞}) → Cat_{∞}, natural in Δ^{op}.

To see this we first recall fron [Lur09a, §3.2] the definition and some features of the mapping simplex of a functor ϕ: [n] → Set_{Δ}^{+} and show that its fibrant replacement is a coCartesian fibration classified the corresponding functor Δ^{n} → Cat_{∞}.

Definition 3.2. Let ϕ: [n] → Set_{Δ}^{+} be a functor. The mapping simplex M_{[n]}(ϕ) → Δ^{n} has k-simplices given by a map σ: [k] → [n] together with a k-simplex Δ^{k} → ϕ(σ(0)). In particular, an edge of M_{[n]}(ϕ) is given by a pair of integers 0 ≤ i ≤ j ≤ n and an edge f in ϕ(i); let S be the set of edges of M_{[n]}(ϕ) where the edge f is marked. Then M_{[n]}^{♮}(ϕ) is the marked simplicial set

(M_{[n]}(ϕ), S). This gives a functor M_{[n]}^{♮}: Fun([n], Set_{Δ}^{+}) → (Set_{Δ}^{+})_{/Δ^{n}}, pseudonatural in Δ^{op} (with respect to composition and pullback) — see Appendix A for a discussion of pseudonatural transformations.

Definition 3.3. Let ϕ: [n] → Set_{Δ} be a functor. The relative nerve N_{[n]}(ϕ) → Δ^{n} has k-simplices given by a map σ: [k] → [n] and for every ordered subset J ⊆ [k] with greatest element j, a map Δ^{J} → ϕ(σ(j)) such that for J′ ⊆ J the diagram

\begin{matrix}Δ^{J′} & ϕ(σ(j′)) \\ Δ^{J} & ϕ(σ(j)) \end{matrix}

commutes. Given a functor \overline{ϕ}: [n] → Set_{Δ}^{+} we define N_{[n]}^{+}(\overline{ϕ}) to be the marked simplicial set (N_{[n]}(ϕ), M) where ϕ is the underlying functor [n] → Set_{Δ} of \overline{ϕ}, and M is the set of edges Δ^{1} → N_{[n]}ϕ determined by

• a pair of integers 0 ≤ i ≤ j ≤ n,

• a vertex x ∈ ϕ(i),

• a vertex y ∈ ϕ(j) and an edge ϕ(i → j)(x) → y that is marked in \overline{ϕ}(j).

This determines a functor N_{[n]}^{+} : Fun([n], Set_{Δ}^{+}) → (Set_{Δ}^{+})_{/Δ^{n}}, pseudonatural in Δ^{op}.

Remark 3.4. By [Lur09a, Proposition 3.2.5.18], the functor N_{[n]}^{+} is a right Quillen equivalence from the projective model structure on Fun([n], Set_{Δ}^{+}) to the coCartesian model structure on (Set_{Δ}^{+})_{/Δ^{n}}. In particular, if ϕ: [n] → Set_{Δ}^{+} is a functor such that ϕ(i) is fibrant (i.e. is a quasicategory marked by its set of equivalences) for every i, then N_{[n]}^{+}(ϕ) is a coCartesian fibration.

Definition 3.5. There is a natural transformation ν_{[n]} : M_{[n]}^{♮}(–) → N_{[n]}^{+}(–)

that sends a k-simplex (σ: [k] → [n], Δ^{k} → ϕ(σ(0))) in M_{[n]}^{♮}(σ) to the k-simplex of N_{[n]}^{+}(ϕ) determined by the composites Δ^{J} → Δ^{k} → ϕ(σ(0)) → ϕ(σ(j)). This is clearly pseudonatural in maps in Δ^{op}, i.e. we have a pseudofunctor Δ^{op} → Fun([1], Cat) that to [n] assigns

ν_{[n]} : [1] × Fun([n], Set_{Δ}^{+}) → (Set_{Δ}^{+})_{/Δ^{n}}.

Proposition 3.6. Suppose ϕ: [n] → Set_{Δ}^{+} is fibrant. Then the natural map ν_{[n],ϕ}: M_{[n]}^{♮}(ϕ) → N_{[n]}^{+}(ϕ) is a coCartesian equivalence.

Proof. Since N_{[n]}^{+}(ϕ) → Δ^{n} is a coCartesian fibration by [Lur09a, Proposition 3.2.5.18], it follows from [Lur09a, Proposition 3.2.2.14] that it suffices to check that ν_{[n],ϕ} is a “quasi-equivalence” in the sense of [Lur09a, Definition 3.2.2.6]. Thus we need only show that the induced map on fibres M_{[n]}^{♮}(ϕ)_{i} → N_{[n]}^{+}(ϕ)_{i} is a categorical equivalence for all i = 0, …, n. But unwinding the definitions we see that this can be identified with the identity map ϕ(i) → ϕ(i) (marked by the equivalences). □

Let Un_{[n]}^{+,co} : Fun([n], Set_{Δ}^{+}) → (Set_{Δ}^{+})_{/Δ^{n}} be the coCartesian version of the marked unstraightening functor defined in [Lur09a, §3.2.1]. By [Lur09a, Remark 3.2.5.16] there is a natural transformation λ_{[n]} : N_{[n]}^{+} → Un_{[n]}^{+,co}, which is a weak equivalence on fibrant objects by [Lur09a, Corollary 3.2.5.20]. Since this is also pseudonatural in Δ^{op}, combining this with Proposition 3.6 we immediately get:

Corollary 3.7. For every [n] ∈ Δ^{op} there is a natural transformation λ_{[n]}ν_{[n]} : M_{[n]}^{♮}(–) → Un_{[n]}^{+,co}(–), and this is pseudonatural in [n] ∈ Δ^{op}. If

ϕ: [n] → Set_{Δ}^{+} is fibrant, then the map M_{[n]}^{♮}(ϕ) → Un_{[n]}^{+,co}(ϕ) is a coCartesian equivalence.

It is immediate from the definition that M_{[n]}^{♮}(ϕ) is the pushout

\begin{matrix}ϕ(0)^{♮} × (Δ^{\{1,...,n\}})^{♯} & ϕ(0)^{♮} × (Δ^{n})^{♯} \\ M_{[n−1]}^{♮}(ϕ|_{\{1,...,n\}}) & M_{[n]}^{♮}(ϕ). \end{matrix}

Moreover, since all objects are cofibrant in the model structure on marked simplicial sets and the top horizontal map is a cofibration, this is a homotopy pushout. Combining this with Corollary 3.7, we get the following:

Lemma 3.8. Suppose F: [n] → Cat_{∞} is a functor, and that ℰ → Δ^{n} is the corresponding coCartesian fibration. Let ℰ′ be the pullback of ℰ along the inclusion Δ^{{1,...,n}} ↪ Δ^{n}. Then there is a pushout square

\begin{matrix}F (0) × Δ^{\{1,...,n\}} & F (0) × Δ^{n} \\ ℰ′ & ℰ \end{matrix}

in Cat_{∞}.

Unwinding the definition, we see that M_{[n]}^{♮}(ϕ) is the colimit of the diagram 1234 David Gepner, Rune Haugseng, Thomas Nikolaus

ϕ(0) × (Δ^{n})^{♯}

ϕ(0) × (Δ^{{1,...,n}})^{♯}

ϕ(1) × (Δ^{{1,...,n}})^{♯}

ϕ(1) × (Δ^{{2,...,n}})^{♯}

ϕ(2) × (Δ^{{2,...,n}})^{♯}

⋮

ϕ(n − 1) × (Δ^{{n−1,n}})^{♯}

ϕ(n − 1) × (Δ^{{n}})^{♯}

ϕ(n) × (Δ^{{n}})^{♯}.

By Example 2.5 the category indexing this colimit is a cofinal subcategory of the twisted arrow category Tw([n]) of [n] — this is easy to check using [Lur09a, Corollary 4.1.3.3] since both categories are partially ordered sets. Hence we may identify M_{[n]}^{♮}(ϕ) with the coend

colim_{Tw([n])}(ϕ(–) × N[n]_{–/}).

Moreover, since we can write this colimit as an iterated pushout along cofibrations, this is a homotopy colimit. From this we can prove Proposition 3.1 using the results of appendix A together with the following observation:

Lemma 3.9. Let G: C → D be a right Quillen functor between model categories. Suppose f: X → X̄ and g: Y → Ȳ are weak equivalences such that X̄ and Ȳ are fibrant, and G(f) and G(g) are weak equivalences in D. Then if h: X → Y is a weak equivalence, the morphism G(h) is also a weak equivalence in D.

Proof. Choose a factorization of the composite g ∘ h: X → Ȳ as a trivial cofibration i: X ↪ X′ followed by a fibration p: X′ → Ȳ. We then have a commutative diagram

X  X̄

X′

Y  Ȳ h f i p q g

where the dotted arrow q exists since X̄ is fibrant and i is a trivial cofibration, and all morphisms are weak equivalences by the 2-of-3 property. By assumption G takes f and g to weak equivalences, and as G is a right Quillen functor by Brown’s Lemma it also takes q and p to weak equivalences as these are weak equivalences between fibrant objects. By the 2-of-3 property we can then conclude first that G(i) is a weak equivalence and then that G(h) is a weak equivalence. □

Proof of Proposition 3.1. We will prove this by applying Proposition A.30 to a relative Grothendieck fibration constructed in the same way as in Proposition A.31. The only difference is that the mapping simplex of a functor ϕ: [n] → Set_{Δ}^{+} is not in general fibrant. We must therefore consider a larger relative subcategory of (Set_{Δ}^{+})_{/Δ^{n}} containing the mapping simplices of fibrant functors whose associated ∞-category is still Cat_{∞/Δ^{n}}^{cocart}. By [Lur09a, Proposition 3.2.2.7] every mapping simplex admits a weak equivalence to a fibrant object that is preserved under pullbacks along all morphisms in Δ. We therefore think of M_{[n]}^{♮} and Un_{[n]}^{+,co} as functors from fibrant objects in Fun([n], Set_{Δ}^{+}) to objects in (Set_{Δ}^{+})_{/Δ^{n}} that admit a weak equivalence to a fibrant object that is preserved by pullbacks — by Lemma 3.9 all weak equivalences between such objects are preserved by pullbacks, so we still get functors of relative categories. It remains to show that inverting the weak equivalences in this subcategory gives the same ∞-category as inverting the weak equivalences in the subcategory of fibrant objects. This follows from [BK12, 7.5], since any fibrant replacement functor gives a homotopy equivalence of relative categories. □

4. Free Fibrations

Our goal in this section is to prove that for any ∞-category 𝒞, the forgetful functor

Cat_{∞/𝒞}^{cart} → Cat_{∞/𝒞}

has a left adjoint, given by the following explicit formula:

Definition 4.1. Let 𝒞 be an ∞-category. For p: ℰ → 𝒞 any functor of ∞- categories, let F (p) denote the map ℰ ×_{𝒞^{{1}}} 𝒞^{Δ^{1}} → 𝒞^{{0}}, where the pullback is along the target fibration 𝒞^{Δ^{1}} → 𝒞 given by evaluation at 1 ∈ Δ^{1}, and the projection F (p) is induced by evaluation at 0. Then F defines a functor Cat_{∞/𝒞} → Cat_{∞/𝒞}.

We will call the projection F (p): ℰ ×_{𝒞} 𝒞^{Δ^{1}} → 𝒞 the free Cartesian fibration on p: ℰ → 𝒞 — the results of this section will justify this terminology.

Example 4.2. The free Cartesian fibration on the identity 𝒞 → 𝒞 is the source fibration F: 𝒞^{Δ^{1}} → 𝒞, given by evaluation at 0 ∈ Δ^{1}.

Lemma 4.3. The functor F factors through the subcategory Cat_{∞/𝒞}^{cart} → Cat_{∞/𝒞}.

Proof. By [Lur09a, Corollary 2.4.7.12] the projection F (p) → 𝒞 is a Cartesian fibration for any p: ℰ → 𝒞, and a morphism in F (p) is Cartesian if and only if its image in ℰ is an equivalence. It is thus clear that for any map ϕ: ℰ → ℱ in Cat_{∞/𝒞}, the induced map F (ϕ) preserves Cartesian morphisms, since the diagram

\begin{matrix}ℰ ×_{𝒞} 𝒞^{Δ^{1}} & ℱ ×_{𝒞} 𝒞^{Δ^{1}} \\ ℰ & ℱ \end{matrix}

commutes. □

Remark 4.4. If p: ℰ → 𝒞 is a functor, the objects of F (p) can be identified with pairs (e, ϕ: c → p(e)) where e ∈ ℰ and ϕ is a morphism in 𝒞. Similarly, a morphism in F (p) can be identified with the data of a morphism α: e′ → e in 𝒞 and a commutative diagram

\begin{matrix}c′ & p(e′) \\ c & p(e). \end{matrix}

If (e, ϕ) is an object in F (p) and ψ: c′ → c is a morphism in 𝒞, the Cartesian morphism over ψ with target (e, ϕ) is the obvious morphism from (e, ϕψ).

Theorem 4.5. Let 𝒞 be an ∞-category. The functor F: Cat_{∞/𝒞} → Cat_{∞/𝒞}^{cart} is left adjoint to the forgetful functor U: Cat_{∞/𝒞}^{cart} → Cat_{∞/𝒞}.

Remark 4.6. Analogues of this result in the setting of ordinary categories (as well as enriched and internal variants) can be found in [Str80] and [Web07].

Composition with the degeneracy s_{0} : Δ^{1} → Δ^{0} induces a functor 𝒞 → 𝒞^{Δ^{1}} (which sends an object of 𝒞 to the constant functor Δ^{1} → 𝒞 with that value). Since the composition of this with both of the evaluation maps 𝒞^{Δ^{1}} → 𝒞 is the identity, this induces a natural map ℰ → ℰ ×_{𝒞} 𝒞^{Δ^{1}} over 𝒞, i.e. a natural transformation

η: id → UF

of functors Cat_{∞/𝒞} → Cat_{∞/𝒞}. We will show that this is a unit transformation in the sense of [Lur09a, Definition 5.2.2.7], i.e. that it induces an equivalence

Map_{𝒞}^{Cart}(F (ℰ), ℱ) → Map_{𝒞} (UF (ℰ), U(ℱ)) → Map_{𝒞} (ℰ, U(ℱ))

for all ℰ → 𝒞 in Cat_{∞/𝒞} and ℱ → 𝒞 in Cat_{∞/𝒞}^{cart} . We first check this for the objects of Cat_{∞/𝒞} with source Δ^{0} and Δ^{1}, which (in a weak sense) generate Cat_{∞/𝒞} under colimits. If a map Δ^{0} → 𝒞 corresponds to the object x ∈ 𝒞, then its image under F is the projection 𝒞_{/x} → 𝒞. (Strictly speaking, the image is the “alternative overcategory” 𝒞^{/x} in the notation of [Lur09a, §4.2.1], but this is naturally weakly equivalent to 𝒞_{/x} by [Lur09a, Proposition 4.2.1.5].) Thus in this case we need to show the following:

Lemma 4.7.

(i) For every x ∈ 𝒞, the map Map_{𝒞}^{Cart}(𝒞_{/x}, ℰ) → Map_{𝒞} ({x}, ℰ) ≃ ιℰ_{x} is an equivalence.

(ii) More generally, for any 𝒳 ∈ Cat_{∞}, the map

Map_{𝒞}^{Cart}(𝒞_{/x} × 𝒳, ℰ) → Map_{𝒞} ({x} × 𝒳, ℰ) ≃ Map(𝒳, ℰ_{x})

is an equivalence.

Proof. The inclusion of the ∞-category of right fibrations over 𝒞 into Cat_{∞/𝒞}^{cart} has a right adjoint, which sends a Cartesian fibration p: ℰ → 𝒞 to its restriction to the subcategory ℰ_{cart} of ℰ where the morphisms are the p-Cartesian morphisms. The map Map_{𝒞}^{Cart}(𝒞_{/x}, ℰ) → ιℰ_{x} thus factors as

Map_{𝒞}^{Cart}(𝒞_{/x}, ℰ) ⟶^{∼} Map_{𝒞} (𝒞_{/x}, ℰ_{cart}) → ιℰ_{x},

where Map_{𝒞} (𝒞_{/x}, ℰ_{cart}) is the mapping space in the ∞-category of right fibrations over 𝒞, which is modelled by the contravariant model structure on (Set_{Δ})_{/𝒞} constructed in [Lur09a, §2.1.4]. By [Lur09a, Proposition 4.4.4.5], the inclusion {x} → 𝒞_{/x} is a trivial cofibration in this model category. Since this is a simplicial model category by [Lur09a, Proposition 2.1.4.8], it follows immediately that we have an equivalence

Map_{𝒞} (𝒞_{/x}, ℰ_{cart}) ⟶^{∼} Map_{𝒞} ({x}, ℰ_{cart}).

This proves (i). To prove (ii) we simply observe that since the model category is simplicial, the product {x} × K → 𝒞_{/x} × K is also a trivial cofibration for any simplicial set K. □

For the case of maps Δ^{1} → 𝒞, the key observation is:

Proposition 4.8. If Δ^{1} → 𝒞 corresponds to a map f: x → y in 𝒞, then the diagram

\begin{matrix}𝒞_{/x} & 𝒞_{/x} × Δ^{1} \\ 𝒞_{/y} & 𝒞^{Δ^{1}} ×_{𝒞} Δ^{1} \end{matrix}

is a pushout square in Cat_{∞/𝒞}^{cart} , where the top map is induced by the inclusion {0} ↪ Δ^{1}.

Proof. Since colimits in Cat_{∞/𝒞}^{cart} ≃ Fun(𝒞^{op}, Cat_{∞}) are detected fibrewise, it suffices to show that for every c ∈ 𝒞, the diagram on fibres is a pushout in Cat_{∞}. This diagram can be identified with

Map_{𝒞} (c, x)  Map_{𝒞} (c, x) × Δ^{1}

Map_{𝒞} (c, y)  𝒞_{c/} ×_{𝒞} Δ^{1}.

This is a pushout by Lemma 3.8, since 𝒞_{c/} ×_{𝒞} Δ^{1} → Δ^{1} is the left fibration corresponding to the map of spaces Map_{𝒞} (c, x) → Map_{𝒞} (c, y) induced by composition with f. □

Corollary 4.9. For every map σ: Δ^{1} → 𝒞 and every Cartesian fibration ℰ → 𝒞, the map

η_{σ}^{∗}: Map_{𝒞}^{Cart}(𝒞^{Δ^{1}} ×_{𝒞} Δ^{1}, ℰ) → Map_{𝒞} (Δ^{1}, ℰ)

is an equivalence.

Proof. By Proposition 4.8, if the map σ corresponds to a morphism f: x → y in 𝒞, we have a pullback square

Map_{𝒞}^{Cart}(𝒞^{Δ^{1}} ×_{𝒞} Δ^{1}, ℰ) Map_{𝒞}^{Cart}(𝒞_{/y}, ℰ)

Map_{𝒞}^{Cart}(𝒞_{/x} × Δ^{1}, ℰ)  Map_{𝒞}^{Cart}(𝒞_{/x}, ℰ).

The map η_{σ}^{∗} fits in an obvious map of commutative squares from this to the square

Map_{𝒞} (Δ^{1}, ℰ)  Map_{𝒞} ({y}, ℰ)

Map_{𝒞} ({x} × Δ^{1}, ℰ)  Map_{𝒞} ({x}, ℰ),

where the right vertical map is given by composition with Cartesian morphims over f. Since ℰ → 𝒞 is a Cartesian fibration, this is also a pullback square (this amounts to saying morphisms in ℰ over f are equivalent to composites of a morphism in ℰ_{x} with a Cartesian morphism over f). But now, by Lemma 4.7, we have a natural transformation of pullback squares that’s an equivalence everywhere except the top left corner, so the map in that corner is an equivalence too. □

To complete the proof, we now only need to observe that F preserves colimits:

Lemma 4.10. F preserves colimits.

Proof. Colimits in Cat_{∞/𝒞}^{cart} are detected fibrewise, so we need to show that for every x ∈ 𝒞, the functor 𝒞_{x/} ×_{𝒞} (–): Cat_{∞/𝒞} → Cat_{∞} preserves colimits. But 𝒞_{x/} → 𝒞 is a flat fibration by [Lur14, Example B.3.11], so pullback along it preserves colimits as a functor Cat_{∞/𝒞} → Cat_{∞/𝒞_{x/}} (since on the level of model categories the pullback functor is a left Quillen functor by [Lur14, Corollary B.3.15]), and the forgetful functor Cat_{∞/𝒞_{x/}} → Cat_{∞} also preserves colimits.

□

Proof of Theorem 4.5. By Lemma 4.10 the source and target of the natural map

Map_{𝒞}^{Cart}(F (ℰ), ℱ) → Map_{𝒞} (ℰ, U(ℱ))

both take colimits in ℰ to limits of spaces. Since Cat_{∞} is a localization of 𝒫(Δ), every object of Cat_{∞/𝒞} is canonically the colimit of a diagram consisting of objects of the form Δ^{n} → 𝒞, so it suffices to show that the map is an equivalence for such objects. But Δ^{n} can in turn be identified with the colimit Δ^{1} ⨿_{Δ^{0}} ⋯ ⨿_{Δ^{0}} Δ^{1}, so it suffices to check that the map is an equivalence when ℰ = Δ^{0} and Δ^{1}. Thus the result follows from Lemma 4.7 and Corollary 4.9. □

Proposition 4.11.

(i) Suppose X → S is a map of ∞-categories and K is an ∞-category. Then there is a natural equivalence F (K × X) ≃ K × F (X).

(ii) The unit map X → F (X) induces an equivalence of ∞-categories

Fun_{S}^{cart}(F (X), Y) ⟶^{∼} Fun_{S}(X, Y).

Proof. (i) is immediate from the definition of F. Then (ii) follows from the natural equivalence

Map(K, Fun_{S}(A, B)) ≃ Map_{S}(K × A, B) ≃ Map_{S}^{cart}(F (K × A), B)

≃ Map_{S}^{cart}(K × F (A), B) ≃ Map(K, Fun_{S}^{cart}(F (A), B)).

□

5. Natural Transformations as an End

It is a familiar result from ordinary category theory that for two functors F, G: C → D the set of natural transformations from F to G can be identified with the end of the functor C^{op} × C → Set that sends (C, C′) to Hom_{D}(F (C), G(C′)). Our goal in this section is to prove the analogous result for ∞-categories:

Proposition 5.1. Let F, G: 𝒞 → 𝒟 be two functors of ∞-categories. Then the space Map_{Fun(𝒞,𝒟)} (F, G) of natural transformations from F to G is naturally equivalent to the end of the functor

𝒞^{op} × 𝒞 ⟶^{(F^{op},G)} 𝒟^{op} × 𝒟 ⟶^{Map_{𝒟}} 𝒮.

A proof of this is also given in [Gla16, Proposition 2.3]; we include a slightly different proof for completeness.

Lemma 5.2. Suppose i: 𝒞_{0} ↪ 𝒞 is a fully faithful functor of ∞-categories. Then for any ∞-category 𝒳 the functor Fun(𝒳, 𝒞_{0}) → Fun(𝒳, 𝒞) is also fully faithful.

Proof. A functor G: 𝒜 → ℬ is fully faithful if and only if the commutative square of spaces

Map(Δ^{1}, 𝒜)  Map(Δ^{1}, ℬ)

ι𝒜^{×2}  ιℬ^{×2}

is Cartesian. Thus, we must show that for any 𝒳, the square

Map(Δ^{1} × 𝒳, 𝒞_{0})  Map(Δ^{1} × 𝒳, 𝒞)

Map(𝒳, 𝒞_{0})^{×2}  Map(𝒳, 𝒞)^{×2}

is Cartesian. But this is equivalent to the commutative square of ∞-categories

\begin{matrix}𝒞_{0}^{Δ^{1}} & 𝒞^{Δ^{1}} \\ 𝒞_{0}^{×2} & 𝒞^{×2} \end{matrix}

being Cartesian. By [Lur09a, Corollary 2.4.7.11] the vertical maps in this diagram are bifibrations in the sense of [Lur09a, Definition 2.4.7.2], so by [Lur09a, Propositions 2.4.7.6 and 2.4.7.7] to prove that this square is Cartesian it suffices to show that for all x, y ∈ 𝒞_{0} the induced map on fibres (𝒞_{0}^{Δ^{1}})_{(x,y)} → (𝒞^{Δ^{1}})_{(ix,iy)} is an equivalence. But this can be identified with the map Map_{𝒞_{0}} (x, y) → Map_{𝒞} (ix, iy), which is an equivalence as i is by assumption fully faithful. □

Proof of Proposition 5.1. By [Lur09a, Corollary 3.3.3.4], we can identify the limit of the functor

ϕ: Tw(𝒞)^{op} → 𝒞^{op} × 𝒞 ⟶^{(F^{op},G)} 𝒟^{op} × 𝒟 ⟶^{Map_{𝒟}} 𝒮

with the space of sections of the corresponding left fibration. By [Lur14, Proposition 5.2.1.11], the left fibration classified by Map_{𝒟} is the projection Tw(𝒟)^{op} → 𝒟^{op} × 𝒟, so the left fibration classified by ϕ is the pullback of this along Tw(𝒞)^{op} → 𝒞^{op} × 𝒞 → 𝒟^{op} × 𝒟. Thus the space of sections is equivalent to the space of commutative diagrams

Tw(𝒞)^{op}  Tw(𝒟)^{op}

𝒞^{op} × 𝒞  𝒟^{op} × 𝒟,

i.e. the space of maps from Tw(𝒞) to the pullback of Tw(𝒟) in the ∞-category of left fibrations over 𝒞^{op} × 𝒞. Using the “straightening” equivalence between this ∞-category and that of functors 𝒞^{op} ×𝒞 → 𝒮 we can identify our limit with the space of maps from y_{𝒞} to F^{∗} ∘ y_{𝒟} ∘ G in Fun(𝒞^{op} × 𝒞, 𝒮) ≃ Fun(𝒞, 𝒫(𝒞)). Since F^{∗} has a left adjoint F_{!}, we have an equivalence

Map_{Fun(𝒞,𝒫(𝒞))} (y_{𝒞} , F^{∗}y_{𝒟} ∘ G) ≃ Map_{Fun(𝒞,𝒫(𝒟))} (F_{!}y_{𝒞} , y_{𝒟} ∘ G).

But by [Lur09a, Proposition 5.2.6.3] the functor F_{!}y_{𝒞} is equivalent to y_{𝒟} ∘ F, and so the limit is equivalent to Map_{Fun(𝒞,𝒫(𝒟))} (y_{𝒟} ∘ F, y_{𝒟} ∘ G). The Yoneda embedding y_{𝒟} is fully faithful by [Lur09a, Proposition 5.1.3.1], so Lemma 5.2 implies that the functor Fun(𝒞, 𝒟) → Fun(𝒞, 𝒫(𝒟)) given by composition with y_{𝒟} is fully faithful, hence we have an equivalence

Map_{Fun(𝒞,𝒫(𝒟))} (y_{𝒞} ∘ F, y_{𝒟} ∘ G) ≃ Map_{Fun(𝒞,𝒟)} (F, G),

which completes the proof. □

6. Enhanced Mapping Functors

The Yoneda embedding for ∞-categories, constructed in [Lur09a, Proposition 5.1.3.1] or [Lur14, Proposition 5.2.1.11], gives for any ∞-category 𝒞 a mapping space functor Map_{𝒞} : 𝒞^{op} ×𝒞 → 𝒮. In some cases, this is the underlying functor to spaces of an interesting functor 𝒞^{op} × 𝒞 → Cat_{∞} — in particular, this is the case if 𝒞 is the underlying ∞-category of an (∞, 2)-category. For a definition and a comparison of different models of (∞, 2)-categories see [Lur09b].

Definition 6.1. A mapping ∞-category functor for an ∞-category 𝒞 is a functor

MAP_{𝒞} : 𝒞^{op} × 𝒞 → Cat_{∞}

together with an equivalence from the composite 𝒞^{op} × 𝒞 → Cat_{∞} ⟶^{ι} 𝒮 to the mapping space functor Map_{𝒞}.

Lemma 6.2. Suppose 𝒞 is an (∞, 2)-category with underlying ∞-category 𝒞′. Then 𝒞′ has a mapping ∞-category functor that sends (C, D) to the ∞-category of maps from C to D in 𝒞.

Proof. This follows from the same argument as in [Lur09a, §5.1.3], using the model of (∞, 2)-categories as categories enriched in marked simplicial sets, cf. [Lur09b]. □

Example 6.3. The ∞-category Cat_{∞} of ∞-categories has a mapping ∞- category functor

MAP_{Cat_{∞}} := Fun,

defined using the construction of Cat_{∞} as the coherent nerve of the simplicial category of quasicategories.

Lemma 6.4. Suppose 𝒞 is an ∞-category with a mapping ∞-category functor MAP_{𝒞}. Then for any ∞-category 𝒟 the functor ∞-category 𝒞^{𝒟} has a mapping ∞-category functor MAP_{𝒞^{𝒟}} given by the composite

(𝒞^{𝒟} )^{op} × 𝒞^{𝒟} → Fun(𝒟^{op} × 𝒟, 𝒞^{op} × 𝒞) → Fun(Tw(𝒟)^{op}, Cat_{∞}) ⟶^{lim} Cat_{∞},

where the second functor is given by composition with the projection Tw(𝒟)^{op} → 𝒟^{op} × 𝒟 and MAP_{𝒞}.

Proof. We must show that the underlying functor to spaces ι ∘ MAP_{𝒞^{𝒟}} is equivalent to Map_{𝒞^{𝒟}} . Since ι preserves limits (being a right adjoint), this follows immediately from Proposition 5.1. □

Definition 6.5. Suppose 𝒞 is an ∞-category with a mapping ∞-category functor MAP_{𝒞}. We say that 𝒞 is tensored over Cat_{∞} if for every C ∈ 𝒞 the functor MAP_{𝒞} (C, –) has a left adjoint – ⊗ C: Cat_{∞} → 𝒞; in this case these adjoints determine an essentially unique functor ⊗: Cat_{∞} × 𝒞 → 𝒞.

Example 6.6. The ∞-category Cat_{∞} is obviously tensored over Cat_{∞} via the Cartesian product ×: Cat_{∞} × Cat_{∞} → Cat_{∞}.

Lemma 6.7. Suppose 𝒞 is an ∞-category with a mapping ∞-category MAP_{𝒞} that is tensored over Cat_{∞}. Then for any ∞-category 𝒟, the mapping ∞- category functor for 𝒞^{𝒟} defined in Lemma 6.4 is also tensored over Cat_{∞}, via the composite

Cat_{∞} × 𝒞^{𝒟} → Cat_{∞}^{𝒟} × 𝒞^{𝒟} ≃ (Cat_{∞} × 𝒞)^{𝒟} → 𝒞^{𝒟}

where the first functor is given by composition with the functor 𝒟 → ∗ and the last by composition with the tensor functor for 𝒞.

Proof. We must show that for every functor F: 𝒟 → 𝒞 there is a natural equivalence

Map_{𝒞^{𝒟}} (𝒳 ⊗ F, G) ≃ Map_{Cat_{∞}}(𝒳, MAP_{𝒞^{𝒟}} (F, G)).

By Proposition 5.1 and the definition of ⊗ for 𝒞^{𝒟} , there is a natural equivalence

Map_{𝒞^{𝒟}} (𝒳 ⊗ F, G) ≃ lim_{Tw(𝒟)^{op}} Map_{𝒞} (𝒳 ⊗ F (–), G(–)).

Now using that 𝒞 is tensored over Cat_{∞}, this is naturally equivalent to

lim_{Tw(𝒟)^{op}} Map_{Cat_{∞}}(𝒳, MAP_{𝒞} (F (–), G(–))).

Moving the limit inside, this is

Map_{Cat_{∞}}(𝒳, lim_{Tw(𝒟)^{op}} MAP_{𝒞} (F (–), G(–))),

which is Map_{Cat_{∞}}(𝒳, MAP_{𝒞^{𝒟}} (F, G)) by definition. □

Example 6.8. For any ∞-category 𝒟, the ∞-category Cat_{∞}^{𝒟} is tensored over Cat_{∞}: 𝒳 ⊗ F is the functor D ↦ 𝒳 × F (D).

In the case where 𝒞 is the ∞-category Cat_{∞} of ∞-categories, Lemma 6.4 gives a mapping ∞-category functor

Nat_{𝒟^{op}} := MAP_{Fun(𝒟^{op},Cat_{∞})}

for Fun(𝒟^{op}, Cat_{∞}), for any ∞-category 𝒟. However, using the equivalence Fun(𝒟^{op}, Cat_{∞}) ≃ Cat_{∞/𝒟}^{cart} we can construct another such functor: the space of maps from ℰ to ℰ′ in Cat_{∞/𝒟}^{cart} is the underlying ∞-groupoid of the ∞-category Fun_{𝒟}^{cart}(ℰ, ℰ′), the full subcategory of Fun_{𝒟} (ℰ, ℰ′) spanned by the functors that preserve Cartesian morphisms. We will now prove that these two functors are equivalent:

Proposition 6.9. For every ∞-category 𝒞 there is a natural equivalence

Fun_{𝒞}^{cart}(ℰ, ℰ′) ≃ Nat_{𝒞^{op}} (St_{𝒞} ℰ, St_{𝒞} ℰ′).

Proof. By the Yoneda Lemma it suffices to show that there are natural equivalences

Map_{Cat_{∞}}(𝒳, Fun_{𝒞}^{cart}(ℰ, ℰ′)) ≃ Map_{Cat_{∞}}(𝒳, Nat_{𝒞^{op}} (St_{𝒞} ℰ, St_{𝒞} ℰ′)).

It is easy to see that Map_{Cat_{∞}}(𝒳, Fun_{𝒞}^{cart}(ℰ, ℰ′)) is naturally equivalent to Map_{Cat}{cart \atop ∞/𝒞} (𝒳 × ℰ, ℰ′) — these correspond to the same components of Map_{Cat_{∞}}(𝒳, Fun_{𝒞} (ℰ, ℰ′)). The equivalence St_{𝒞} preserves products, so this is equivalent to the mapping space

Map_{Fun(𝒞^{op},Cat_{∞})} (St_{𝒞} (𝒳 × 𝒞) × St_{𝒞} ℰ, St_{𝒞} ℰ′).

But the projection 𝒳 × 𝒞 → 𝒞 corresponds to the constant functor c^{∗}𝒳: 𝒞^{op} → ∗ → Cat_{∞} with value 𝒞 (since the Cartesian fibration classified by this composite is precisely the pullback of 𝒳 → ∗ along 𝒞 → ∗). Thus there is a natural equivalence

Map_{Cat_{∞}}(𝒳, Fun_{𝒞}^{cart}(ℰ, ℰ′)) ≃ Map_{Fun(𝒞^{op},Cat_{∞})} (c^{∗}𝒳 × St_{𝒞} ℰ, St_{𝒞} ℰ′).

But by Lemma 6.7, the ∞-category Fun(𝒞^{op}, Cat_{∞}) is tensored over Cat_{∞} and this is naturally equivalent to Map_{Cat_{∞}}(𝒳, Nat_{𝒞^{op}} (St_{𝒞} ℰ, St_{𝒞} ℰ′)), as required.

□

7. Cartesian and CoCartesian Fibrations as Weighted Colimits

In ordinary category theory it is a familiar fact that the Grothendieck fibration classified by a functor F: C^{op} → Cat can be identified with the lax colimit of F, and the Grothendieck opfibration classified by a functor F: C → Cat with the oplax colimit of F. In this section we will show that Cartesian and coCartesian fibrations admit analogous descriptions. It is immediate from our formula for the free Cartesian fibration that the sections of a Cartesian fibration are given by the oplax limit of the corresponding functor:

Proposition 7.1. The ∞-category of sections of the Cartesian fibration classified by F is given by the oplax limit of F. In other words, there is a natural equivalence

Fun_{𝒞} (𝒞, Un_{𝒞} (F)) ≃ lim_{Tw(𝒞)^{op}} Fun(𝒞_{–/}, F (–))

of functors Fun(𝒞^{op}, Cat_{∞}) → Cat_{∞}.

Proof. By Theorem 4.5 and Proposition 6.9 we have natural equivalences

Fun_{𝒞} (𝒞, Un_{𝒞} (F)) ≃ Fun_{𝒞}^{cart}(F (𝒞), Un_{𝒞} (F)) ≃ ≃ Nat_{𝒞^{op}} (𝒞_{–/}, F) ≃ lim_{Tw(𝒞)^{op}} Fun(𝒞_{–/}, F (–)).□

Definition 7.2. Let F: 𝒞 → Cat_{∞} be a functor, and let ℱ → 𝒞 be its associated coCartesian fibration. Given an ∞-category 𝒳, write Φ_{𝒳}^{F} for the simplicial set over 𝒞 with the universal property

Hom(K ×_{𝒞} ℱ, 𝒳) ≅ Hom_{𝒞} (K, Φ_{𝒳}^{F}).

By [Lur09a, Corollary 3.2.2.13] the projection Φ_{𝒳}^{F} → 𝒞 is a Cartesian fibration.

Proposition 7.3. The Cartesian fibration Φ_{𝒳}^{F} → 𝒞 corresponds to the functor

Fun(F (–), 𝒳): 𝒞^{op} → Cat_{∞}.

Proof. We first consider the case where 𝒞 is a simplex Δ^{n}. By Proposition 3.1 there are natural equivalences

colim_{Tw([n])} ϕ(–) × [n]_{–/} ⟶^{∼} Un_{[n]}^{co}(ϕ)

for any ϕ: [n] → Cat_{∞}, natural in Δ^{op}. Thus by Proposition 7.1 there are natural equivalences

Fun_{Δ^{n}}(Δ^{n}, Φ_{𝒳}^{ϕ}) ≃ Fun(Un_{[n]}^{co}(ϕ), 𝒳) ≃ lim_{Tw([n])^{op}} Fun([n]_{–/}, Fun(ϕ(–), 𝒳))
≃ Fun_{Δ^{n}}(Δ^{n}, Un_{[n]}(Fun(ϕ(–), 𝒳))).

Since this equivalence is natural in Δ^{op} and Cat_{∞} is a localization of the presheaf ∞-category 𝒫(Δ), we get by the Yoneda lemma a natural equivalence

Φ_{𝒳}^{ϕ} ≃ Un_{[n]}^{cart}(Fun(ϕ(–), 𝒳))).

Since Cat_{∞} is an accessible localization of 𝒫(Δ), any ∞-category 𝒞 is naturally equivalent to the colimit of the diagram Δ_{/𝒞}^{op} → Cat_{∞/𝒞} → Cat_{∞}. Now given F: 𝒞 → Cat_{∞}, we have, since pullback along a Cartesian fibration preserves colimits,

Un_{𝒞}^{cart}(Fun(F (–), 𝒳))) ≃ colim_{σ∈Δ_{/𝒞}^{op}} Un_{[n]}(Fun(F σ(–), 𝒳)) ≃ colim_{σ∈Δ_{/𝒞}^{op}} Φ_{𝒳}^{F σ} ≃ Φ_{𝒳}^{F},

which completes the proof. □

Theorem 7.4. The coCartesian fibration classified by a functor F: 𝒞 → Cat_{∞} is given by the oplax colimit of F. In other words, there is a natural equivalence

Un_{𝒞}^{co}(F) ≃ colim_{Tw(𝒞)}F (–) × 𝒞_{–/}

of functors Fun(𝒞, Cat_{∞}) → Cat_{∞}.

Proof. Let F: 𝒞 → Cat_{∞} be a functor. Then by Proposition 7.3, we have a natural equivalence

Fun(Un_{𝒞} (F), 𝒳) ≃ Fun_{𝒞} (𝒞, Φ_{𝒳}^{F}).

By Proposition 7.1 we have a natural equivalence between the right-hand side and

lim_{Tw(𝒞)^{op}} Fun(𝒞_{–/}, Fun(F (–), 𝒳)) ≃ Fun( colim_{Tw(𝒞)}F (–) × 𝒞_{–/}, 𝒳).

By the Yoneda lemma, it follows that Un_{𝒞} (F) is naturally equivalent to colim_{Tw(𝒞)} F (–) × 𝒞_{–/}. □

Corollary 7.5. Any ∞-category 𝒞 is the oplax colimit of the constant functor 𝒞 → Cat_{∞} with value ∗.

Proof. The identity 𝒞 → 𝒞 is the coCartesian fibration classified by this functor. □

Corollary 7.6. The Cartesian fibration classified by a functor F: 𝒞^{op} → Cat_{∞} is given by the lax colimit of F. In other words, there is a natural equivalence

Un_{𝒞} (F) ≃ colim_{Tw(𝒞^{op} )} F (–) × 𝒞_{/–}

of functors Fun(𝒞^{op}, Cat_{∞}) → Cat_{∞}.

Proof. We have a natural equivalence Un_{𝒞} (F) ≃ Un_{𝒞^{op}}^{co} (F^{op})^{op}. Since (–)^{op} is an automorphism of Cat_{∞} it preserves colimits, so by Theorem 7.4 we have

Un_{𝒞} (F) ≃ ( colim_{Tw(𝒞^{op})} F (–)^{op} × (𝒞^{op})_{–/})^{op} ≃ colim_{Tw(𝒞^{op})} F (–) × 𝒞_{/–}.  □

Similarly, dualizing Proposition 7.1 gives:

Corollary 7.7. The ∞-category of sections of the coCartesian fibration classified by F is given by the lax limit of F. In other words, there is a natural equivalence

Fun_{𝒞} (𝒞, Un_{𝒞}^{co}(F)) ≃ lim_{Tw(𝒞)^{op}} Fun(𝒞_{/–}, F (–))

of functors Fun(𝒞, Cat_{∞}) → Cat_{∞}.

8. 2-Representable Functors

Suppose 𝒞 is an ∞-category equipped with a mapping ∞-category functor MAP_{𝒞} : 𝒞^{op} × 𝒞 → Cat_{∞}. We say a functor F: 𝒞^{op} → Cat_{∞} is 2-representable by C ∈ 𝒞 if F is equivalent to MAP_{𝒞} (–, C). Similarly, we say a Cartesian fibration p: ℰ → 𝒞 is 2-representable by C if p is classified by the functor MAP_{𝒞} (–, C). Our goal in this section is to prove that if a Cartesian fibration p: ℰ → 𝒞 is 2-representable then, under mild hypotheses, the same is true for the induced map ℰ^{𝒟} → 𝒞^{𝒟} for any ∞-category 𝒟. We begin by giving a somewhat unwieldy description of the functor classifying such fibrations for an arbitrary Cartesian fibration p:

Proposition 8.1. Suppose p: ℰ → 𝒞 is a Cartesian fibration corresponding to a functor F: 𝒞^{op} → Cat_{∞}. Then for any ∞-category 𝒟 the functor ℰ^{𝒟} → 𝒞^{𝒟} given by composition with p is a Cartesian fibration classified by a functor F_{𝒟} : (𝒞^{𝒟} )^{op} → Cat_{∞} that sends a functor ϕ: 𝒟 → 𝒞 to

lim_{Tw(𝒟)^{op}} Fun(𝒟_{–/}, F ∘ ϕ(–)).

Proof. The induced functor ℰ^{𝒟} → 𝒞^{𝒟} is a Cartesian fibration by [Lur09a, Proposition 3.1.2.1]. For f: K × 𝒟 → 𝒞 we have a natural equivalence

Fun_{𝒞^{𝒟}} (K, ℰ^{𝒟} ) ≃ Fun_{𝒞} (K × 𝒟, ℰ) ≃ Fun_{K×𝒟} (K × 𝒟, f^{∗}ℰ).

But then by Proposition 7.1 we have a natural equivalence

Fun_{K×𝒟} (K × 𝒟, f^{∗}ℰ) ≃ lim_{Tw(K×𝒟)^{op}} Fun((K × 𝒟)_{–/}, F ∘ f(–)),

and then as Tw preserves products (being a right adjoint) we can rewrite this as

lim_{Tw(K)^{op}} Fun(K_{–/}, lim_{Tw(𝒟)^{op}} Fun(𝒟_{–/}, F ∘ f(–))) ≃ lim_{Tw(K)^{op}} Fun(K_{–/}, F_{𝒟} ∘ f(–)),

which we can identify, using Proposition 7.1, with

Fun_{K}(K, f^{∗}Un_{𝒞^{𝒟}} (F_{𝒟} )) ≃ Fun_{𝒞^{𝒟}} (K, Un_{𝒞^{𝒟}} (F_{𝒟} )).

Now applying the Yoneda Lemma completes the proof. □

Definition 8.2. Suppose 𝒞 is an ∞-category equipped with a mapping ∞- category functor. We say that 𝒞 is cotensored over Cat_{∞} if for every C ∈ 𝒞 the functor MAP_{𝒞} (–, C): 𝒞 → Cat_{∞}^{op} has a right adjoint C^{(–)} : Cat_{∞}^{op} → 𝒞; in this case these adjoints determine an essentially unique functor (–)^{(–)} : Cat_{∞}^{op} ×𝒞 → 𝒞. If 𝒞 is also tensored over Cat_{∞}, being cotensored is equivalent to the functor – ⊗ 𝒳 having a right adjoint (–)^{𝒳} for all ∞-categories 𝒳.

Corollary 8.3. Let 𝒞 be an ∞-category equipped with a mapping ∞-category functor MAP_{𝒞} that is tensored and cotensored over Cat_{∞}, and suppose p: ℰ → 𝒞 is a Cartesian fibration that is 2-representable by C ∈ 𝒞. Then for any ∞-category 𝒟 the fibration ℰ^{𝒟} → 𝒞^{𝒟} is 2-representable by the functor C^{𝒟_{–/}} .

Proof. By Proposition 8.1 this fibration corresponds to the functor sending ϕ: 𝒟 → 𝒞 to

lim_{Tw(𝒟)^{op}} Fun(𝒟_{–/}, F ∘ ϕ(–)),

where F is the functor corresponding to p. If F is 2-representable by C, this is equivalent to

lim_{Tw(𝒟)^{op}} Fun(𝒟_{–/}, MAP_{𝒞} (ϕ(–), C) ≃ lim_{Tw(𝒟)^{op}} MAP_{𝒞} (ϕ(–), C^{𝒟_{–/}} ),

which is MAP_{𝒞^{𝒟}} (ϕ, C^{𝒟_{–/}} ) by definition of the mapping ∞-category of 𝒞^{𝒟} . □

9. Some Cartesian Fibrations Identified

In this section we will use our results so far to explicitly identify the Cartesian fibrations classified by certain classes of functors. This is the key input needed to prove our presentability result in the next section. We start with some notation:

Definition 9.1. If p: ℰ → ℬ is a functor of ∞-categories, we denote by ℰ_{ℬ}^{▷} the pushout

ℬ ⨿_{ℰ×{1}} ℰ × Δ^{1},

and by ℰ_{ℬ}^{◁} the pushout

ℬ ⨿_{ℰ×{0}} ℰ × Δ^{1}.

Warning 9.2. The notations ℰ_{ℬ}^{▷} and ℰ_{ℬ}^{◁} are somewhat abusive, as these simplicial sets depend on the functor p rather than just on ℰ and ℬ. Moreover, if ℬ = Δ^{0} then the definition does not reduce to ℰ^{▷} and ℰ^{◁} but rather the “alternative joins” ℰ ⋄ Δ^{0} and Δ^{0} ⋄ ℰ in the notation of [Lur09a, §4.2.1]. However, these are weakly equivalent to the usual joins by [Lur09a, Proposition 4.2.1.2].

Remark 9.3. Observe that (ℰ_{ℬ}^{▷} )^{op} ≃ (ℰ^{op})_{ℬ^{op}}^{◁} and (ℰ_{ℬ}^{◁} )^{op} ≃ (ℰ^{op})_{ℬ^{op}}^{▷} .

We then have the following simple observation:

Lemma 9.4. Given a functor p: ℰ → ℬ, write i: ℬ ↪ ℰ_{ℬ}^{◁} and j: ℬ ↪ ℰ_{ℬ}^{▷} for the inclusions in the pushout diagrams defining ℰ_{ℬ}^{◁} and ℰ_{ℬ}^{▷} , respectively. Then for any ∞-category 𝒟, we have:

(i) The functor

i^{∗}: Fun(ℰ_{ℬ}^{◁} , 𝒟) → Fun(ℬ, 𝒟)

given by composition with i is a Cartesian fibration classified by the functor Fun(ℬ, 𝒟)^{op} → Cat_{∞} that sends F to Fun(ℰ, 𝒟)_{F ∘p/}.

(ii) The functor

j^{∗}: Fun(ℰ_{ℬ}^{▷} , 𝒟) → Fun(ℬ, 𝒟)

given by composition with j is a coCartesian fibration classified by the functor Fun(ℬ, 𝒟) → Cat_{∞} that sends F to Fun(ℰ, 𝒟)_{/F ∘p}.

Proof. We will prove (i); the proof of (ii) is similar. By the definition of ℰ_{ℬ}^{◁} we have a pullback square

Fun(ℰ_{ℬ}^{◁} , 𝒟)  Fun(ℰ × Δ^{1}, 𝒟)

Fun(ℬ, 𝒟)  Fun(ℰ, 𝒟) i^{∗} ev_{0} p^{∗},

where the right vertical map can be identified with the evaluation-at-0 functor

Fun(ℰ, 𝒟)^{Δ^{1}} → Fun(ℰ, 𝒟)

This is the Cartesian fibration classified by the undercategory functor Fun(ℰ, 𝒟)_{(–)/}, hence the pullback i^{∗} is the Cartesian fibration classified by the composite functor Fun(ℰ, 𝒟)_{p^{∗}(–)/}. □

Remark 9.5. If 𝒟 has pushouts, then i^{∗} is also a coCartesian fibration, with coCartesian morphisms given by taking pushouts. Similarly, if 𝒟 has pullbacks then j^{∗} is also a Cartesian fibration, with Cartesian morphisms given by taking pullbacks.

In particular, given a map p: ℰ → ℬ we see that 𝒫(ℰ_{ℬ}^{◁} ) → 𝒫(ℬ) is a coCartesian and Cartesian fibration (recall that 𝒫(𝒞) = Fun(𝒞^{op}, 𝒮) denotes the presheaf ∞-category). The corresponding functors are given on objects by 𝒫(ℰ)_{/p^{∗}(–)}, with functoriality determined by composition and pullbacks, respectively. Our next goal is to give an alternative description of this functor:

Proposition 9.6. Let p: ℰ → ℬ be a functor of ∞-categories, and let j: ℬ → ℰ_{ℬ}^{◁} be the obvious inclusion. Then the functor j^{∗}: 𝒫(ℰ_{ℬ}^{◁} ) → 𝒫(ℬ) is a Cartesian fibration corresponding to the functor 𝒫(ℬ)^{op} ≃ RFib(ℬ)^{op} → Cat_{∞} that sends a right fibration 𝒴 → ℬ to 𝒫(𝒴 ×_{ℬ} ℰ).

To prove this, we need to identify the functor 𝒫(ℰ)_{/p^{∗}(–)} with the functor 𝒫(–×_{ℬ} ℰ) under the equivalence 𝒫(ℬ) ≃ RFib(ℬ), for which we use the following observation:

Proposition 9.7. Suppose p: 𝒦 → 𝒞 is a right fibration of ∞-categories. Then the functor

p_{!} : RFib(𝒦) → RFib(𝒞)_{/p}

given by composition with p is an equivalence. Moreover, this equivalence is natural in p ∈ RFib(𝒞) (with respect to composition with maps f: 𝒦 → ℒ over 𝒞, and also with respect to pullbacks along such maps).

Proof. The functor p_{!} is described by the left Quillen functor p_{!} : Set_{Δ/𝒦} → (Set_{Δ/𝒞})_{/p} given by composition with p, where Set_{Δ/𝒦} is equipped with the contravariant model structure of [Lur09a, §2.1.4] and (Set_{Δ/𝒞})_{/p} with the model structure induced from the contravariant model structure on Set_{Δ/𝒞}. It therefore suffices to show that p_{!} is a left Quillen equivalence. The functor p_{!} is obviously an equivalence of underlying categories, and we claim it is actually an equivalence of model categories. The cofibrations clearly correspond under p_{!}, being the monomorphisms of underlying simplicial sets in both cases, so by [Joy08, Proposition E.1.10] it suffices to show the fibrant objects are the same. In Set_{Δ/𝒦} these are the right fibrations 𝒳 → 𝒦 by [Lur09a, Corollary 2.2.3.12], while in (Set_{Δ/𝒞})_{/p} they are the diagrams

\begin{aligned}𝒳 & & 𝒦 \\ & 𝒞 \end{aligned}_{\begin{matrix}f \\ p \end{matrix}}

where f is a fibration in Set_{Δ/𝒞}. But as p is a right fibration, this is equivalent to f being a right fibration by [Lur09a, Corollary 2.2.3.14]. □

Corollary 9.8. Suppose p: 𝒦 → 𝒞 is a right fibration, corresponding to a functor F: 𝒞^{op} → 𝒮. Then the functor p_{!} : 𝒫(𝒦) → 𝒫(𝒞)_{/F} given by left Kan extension along p^{op} is an equivalence, natural in p ∈ RFib(𝒞) (with respect to left Kan extensions along maps f: 𝒦 → ℒ over 𝒞 and composition with the associated natural transformation, as well as with respect to composition with f and pullback along the natural transformation).

Proof. This follows from combining Proposition 9.7 with the naturality of the straightening equivalence between right fibrations and functors, which can be proved by the same argument as in the proof of Corollary A.32. □

Proof of Proposition 9.6. This follows from combining Lemma 9.4 with Corollary 9.8, since under the equivalence between presheaves and right fibrations the functor p^{∗}: 𝒫(ℬ) → 𝒫(ℰ) corresponds to pullback along p.

□

Corollary 9.9.

(i) Let p: ℰ → ℬ be a Cartesian fibration classified by a functor F: ℬ^{op} → Cat_{∞}, and write j for the inclusion ℬ ↪ ℰ_{ℬ}^{◁} . Then the functor j^{∗}: 𝒫(ℰ_{ℬ}^{◁} ) → 𝒫(ℬ) is a Cartesian fibration classified by the functor RFib(ℬ)^{op} → Cat_{∞} that sends 𝒴 → ℬ to Fun_{ℬ^{op}} (𝒴^{op}, Φ_{𝒮}^{F}), where Φ_{𝒮}^{F} → ℬ^{op} is the Cartesian fibration classified by the functor 𝒫∘F: ℬ → Cat_{∞}.

(ii) Let p: ℰ → ℬ be a coCartesian fibration classified by a functor F: ℬ → Cat_{∞}, and write j for the inclusion ℬ ↪ ℰ_{ℬ}^{◁} . Then the functor j^{∗}: 𝒫(ℰ_{ℬ}^{◁} ) → 𝒫(ℬ) is a Cartesian fibration classified by the functor RFib(ℬ)^{op} → Cat_{∞} that sends 𝒴 → ℬ to Fun_{ℬ^{op}} (𝒴^{op}, Φ̃_{𝒮}^{F}), where Φ̃_{𝒮}^{F} → ℬ^{op} is the coCartesian fibration classified by the functor 𝒫 ∘ F: ℬ^{op} → Cat_{∞}.

Proof. Combine Proposition 9.6 with Proposition 7.3 and its dual. □

Our next observation lets us identify several interesting functors with full subfunctors of the functor Fun_{ℬ^{op}} ((–)^{op}, Φ̃_{𝒮}^{F}), which will allow us to identify the corresponding Cartesian fibrations with full subcategories of 𝒫(ℰ_{ℬ}^{◁} ).

Lemma 9.10. Suppose F: ℬ^{op} → Cat_{∞} is a functor of ∞-categories corresponding to the Cartesian fibration p: ℰ → ℬ and the coCartesian fibration q: ℱ → ℬ^{op}. Let F̂: 𝒫(ℬ)^{op} → Cat_{∞} be the unique limit-preserving functor extending F. Then:

(i) The functor Fun_{ℬ}^{cart}(ℬ_{/–}, ℰ): ℬ^{op} → Cat_{∞} is equivalent to F. (ii) The functor Fun_{ℬ}^{cart}(–, ℰ): RFib(ℬ) → Cat_{∞} corresponds to F̂ under the

equivalence RFib(ℬ) ≃ 𝒫(ℬ). (iii) The functor Fun_{ℬ^{op}}^{cocart}((ℬ^{op})_{–/}, ℱ): ℬ^{op} → Cat_{∞} is equivalent to F. (iv) The functor Fun_{ℬ^{op}}^{cocart}((–)^{op}, ℱ): RFib(ℬ) → Cat_{∞} corresponds to F̂ un-

der the equivalence RFib(ℬ) ≃ 𝒫(ℬ).

Proof. We will prove (i) and (ii); the proofs of (iii) and (iv) are essentially the same. To prove (i), observe that the straightening equivalence between Cartesian fibrations and functors to Cat_{∞} gives us a natural equivalence

Map_{Cat_{∞}}(𝒞, Fun_{ℬ}^{cart}(ℬ_{/–}, ℰ)) ≃ Map_{ℬ}^{cart}(𝒞 × ℬ_{/–}, ℰ)
≃ Map_{Fun(ℬ^{op},Cat_{∞})} (𝒞 × y_{ℬ} (–), F)
≃ Map_{Fun(ℬ^{op},Cat_{∞})} (y_{ℬ} (–), F^{𝒞}).

Now the Yoneda Lemma implies that this is naturally equivalent to ιF^{𝒞} (–) ≃ Map_{Cat_{∞}}(𝒞, F (–)), and so we must have Fun_{ℬ}^{cart}(ℬ_{/–}, ℰ) ≃ F. This proves (i). To prove (ii), we first observe that the functor Fun_{ℬ}^{cart}(–, ℰ) preserves limits, since for any ∞-category 𝒞 we have

Map_{Cat_{∞}}(𝒞, Fun_{ℬ}^{cart}(–, ℰ)) ≃ Map_{ℬ}^{cart}((𝒞 × ℬ) ×_{ℬ} (–), ℰ)

and the Cartesian product in Cat_{∞/ℬ}^{cart} preserves colimits in each variable. Moreover, it follows from (i) that this functor extends F, since the right fibration ℬ_{/b} → ℬ corresponds to the presheaf y_{ℬ} (b) under the equivalence between RFib(ℬ) and 𝒫(ℬ). □

Definition 9.11. Suppose F: ℬ → Cat_{∞} is a functor. Then we write 𝒫F: ℬ^{op} → Cat_{∞} for the composite of F^{op} with 𝒫: Cat_{∞}^{op} → Ĉat_{∞}, and let 𝒫̂F: 𝒫(ℬ)^{op} → Ĉat_{∞} be the unique limit-preserving functor extending 𝒫F.

Proposition 9.12. Let F: ℬ → Cat_{∞} be a functor, with p: ℰ → ℬ an associated coCartesian fibration. We define 𝒫^{cocart}(ℰ_{ℬ}^{◁} ) to be the full subcategory of 𝒫(ℰ_{ℬ}^{◁} ) spanned by those presheaves ϕ: (ℰ_{ℬ}^{◁} )^{op} → 𝒮 such that for every co-Cartesian morphism ᾱ: e → α_{!}e in ℰ over α: b → b′ in ℬ, the commutative square

\begin{matrix}ϕ(α_{!}e, 1) & ϕ(e, 1) \\ ϕ(b′, 0) & ϕ(b, 0) \end{matrix}_{\begin{matrix}ϕ(ᾱ, 1) \\ ϕ(α, 0) \end{matrix}}

is a pullback square. Then the restricted projection 𝒫^{cocart}(ℰ_{ℬ}^{◁} ) → 𝒫(ℬ) is a Cartesian fibration classified by the functor 𝒫̂F.

Proof. Combining Lemma 9.10 with (the dual of) Proposition 7.3, we may identify 𝒫̂F with the functor Fun_{ℬ^{op}}^{cocart}((–)^{op}, Φ̃_{𝒮}^{F}). This is a natural full subcategory of Fun_{ℬ^{op}} ((–)^{op}, Φ̃_{𝒮}^{F}), the functor classified by the Cartesian fibration 𝒫(ℰ_{ℬ}^{◁} ) → 𝒫(ℬ) by Proposition 9.6. It follows that 𝒫̂F is classified by the projection to 𝒫(ℬ) of the full subcategory of 𝒫(ℰ_{ℬ}^{◁} ) spanned by those presheaves that correspond to objects of Fun_{ℬ^{op}}^{cocart}((–)^{op}, Φ̃_{𝒮}^{F}) under the identification of the fibres with Fun_{ℬ^{op}} ((–)^{op}, Φ̃_{𝒮}^{F}). By [Lur09a, Corollary 3.2.2.13], the coCartesian edges of Φ̃_{𝒮}^{F} are those that correspond to functors Δ^{1} ×_{ℬ^{op}} ℰ^{op} → 𝒮 that take Cartesian edges of ℰ^{op} to equivalences in 𝒮. Thus if ℱ → ℬ^{op} is a coCartesian fibration, a functor ℱ → Φ̃_{𝒮}^{F} over ℬ^{op} preserves coCartesian morphisms precisely if the classifying functor ℱ ×_{ℬ^{op}} ℰ^{op} → 𝒮 takes morphisms of the form (ϕ, ϵ) with ϕ coCartesian in ℱ and ϵ Cartesian in ℰ^{op} to equivalences in 𝒮. If 𝒴 → ℬ is a right fibration, this means that Fun_{ℬ^{op}}^{cocart}(𝒴^{op}, Φ̃_{𝒮}^{F}) corresponds to the full subcategory of 𝒫(𝒴 ×_{ℬ} ℰ) spanned by the presheaves (𝒴 ×_{ℬ} ℰ)^{op} → 𝒮 that take morphisms of the form (η, ϵ) with ϵ coCartesian in ℰ to equivalences in 𝒮. Suppose ϕ: ℬ^{op} → 𝒮 is the presheaf classified by 𝒴 → ℬ. Then unwinding the equivalence

𝒫(ℰ_{ℬ}^{◁} )_{ϕ} ≃ 𝒫(ℰ)_{/p^{∗}ϕ} ≃ RFib(ℰ)_{/p^{∗}𝒴} ≃ RFib(ℰ ×_{ℬ} 𝒴) ≃ 𝒫(ℰ ×_{ℬ} 𝒴),

we see that the presheaf Ψ̃ on ℰ ×_{ℬ} 𝒴 classified by Ψ: ℰ^{op} → 𝒮 over p^{∗}ϕ assigns to (e, y) ∈ ℰ×_{ℬ} 𝒴 the fibre Ψ(e)_{y} of the map Ψ(e) → ϕ(pe) at y. Thus Ψ̃(ᾱ, η) is an equivalence for every coCartesian morphism ᾱ: e → α_{!}e in ℰ if and only if for every y ∈ ϕ(p(α_{!}e)) the map on fibres Ψ(α_{!}e)_{y} → Ψ(e)_{ϕ(α)(e)} is an equivalence. This is equivalent to the commutative square

\begin{matrix}Ψ(α_{!}e) & Ψ(e) \\ ϕ(pα_{!}e) & ϕ(pe) \end{matrix}

being Cartesian. Thus Fun_{ℬ^{op}}^{cocart}(𝒴^{op}, Φ̃_{𝒮}^{F}) corresponds to the full subcategory 𝒫^{cocart}(ℰ_{ℬ}^{◁} )_{ϕ} of 𝒫(ℰ_{ℬ}^{◁} )_{ϕ}, which completes the proof. □

Definition 9.13. Let 𝒦 be a collection of small simplicial sets. We write Cat_{∞}^{𝒦} for the subcategory of Cat_{∞} with objects the small ∞-categories that admit Kindexed colimits for all K ∈ 𝒦 and morphisms the functors that preserve these. Given 𝒞 ∈ Cat_{∞}^{𝒦} we let 𝒫_{𝒦} (𝒞) denote the full subcategory of 𝒫(𝒞) spanned by those presheaves 𝒞^{op} → 𝒮 that take K-indexed colimits in 𝒞 to limits for all K ∈ 𝒦. This defines a functor 𝒫_{𝒦} : (Cat_{∞}^{𝒦})^{op} → Ĉat_{∞}.

Example 9.14. Let 𝒦(κ) be the collection of all κ-small simplicial sets. In this case we write Cat_{∞}^{κ} for Cat_{∞}^{𝒦(κ)} and 𝒫_{κ} for 𝒫_{𝒦(κ)} . For 𝒞 ∈ Cat_{∞}^{κ} the ∞-category 𝒫_{κ}(𝒞) is equivalent to Ind_{κ} 𝒞 by [Lur09a, Corollary 5.3.5.4].

Definition 9.15. Suppose 𝒦 is a collection of small simplicial sets and F: ℬ → Cat_{∞}^{𝒦} is a functor of ∞-categories. Then we write 𝒫_{𝒦} F: ℬ^{op} → Ĉat_{∞} for the composite of F^{op} with 𝒫_{𝒦} : (Cat_{∞}^{𝒦})^{op} → Ĉat_{∞}, and let { \atop 𝒫_{𝒦} F}: 𝒫(ℬ)^{op} → Ĉat_{∞} be the unique limit-preserving functor extending 𝒫_{𝒦} F.

Proposition 9.16. Let 𝒦 be a collection of small simplicial sets and let F: ℬ → Cat_{∞}^{𝒦} be a functor, with p: ℰ → ℬ an associated coCartesian fibration. We define 𝒫_{𝒦}^{cocart}(ℰ_{ℬ}^{◁} ) to be the full subcategory of 𝒫^{cocart}(ℰ_{ℬ}^{◁} ) spanned by those presheaves ϕ: (ℰ_{ℬ}^{◁} )^{op} → 𝒮 such that for every colimit diagram q̄: K^{▷} → ℰ_{b} (for any b ∈ ℬ), the composite

(K^{◁▷})^{op} → (ℰ_{b}^{◁})^{op} → (ℰ_{ℬ}^{◁} )^{op} → 𝒮

is a limit diagram. Then the restricted projection 𝒫_{𝒦}^{cocart}(ℰ_{ℬ}^{◁} ) → 𝒫(ℬ) is a Cartesian fibration associated to the functor { \atop 𝒫_{𝒦} F}.

Proof. Since 𝒫_{𝒦} F (b) is a natural full subcategory of 𝒫F (b), we may identify the coCartesian fibration classified by 𝒫_{𝒦} F with the projection to ℬ^{op} of a full subcategory Φ̃_{𝒮}^{F}(𝒦) of Φ̃_{𝒮}^{F}. By Lemma 9.10 we may then identify the functor { \atop 𝒫_{𝒦} F} with Fun_{ℬ^{op}}^{cocart}((–)^{op}, Φ̃_{𝒮}^{F}(𝒦)), which is a natural full subcategory of Fun_{ℬ^{op}}^{cocart}((–)^{op}, Φ̃_{𝒮}^{F}). By Proposition 9.12 we may therefore identify the Cartesian fibration classified by { \atop 𝒫_{𝒦} F} with the projection to 𝒫(ℬ) of a full subcategory of 𝒫^{cocart}(ℰ_{ℬ}^{◁} ). It thus remains to identify those presheaves on ℰ_{ℬ}^{◁} that correspond to objects of the ∞-category Fun_{ℬ^{op}}^{cocart}((–)^{op}, Φ̃_{𝒮}^{F}(𝒦)) under the identification of the fibres with Fun_{ℬ^{op}} ((–)^{op}, Φ̃_{𝒮}^{F}). If 𝒴 → ℬ is a right fibration, it is clear that Fun_{ℬ^{op}} (𝒴^{op}, Φ̃_{𝒮}^{F}(𝒦)) corresponds to the full subcategory of 𝒫(𝒴 ×_{ℬ} ℰ) spanned by the presheaves (𝒴 ×_{ℬ} ℰ)^{op} → 𝒮 such that for every y ∈ 𝒴 over b ∈ ℬ, the restriction ({y} ×_{ℬ} ℰ)^{op} ≃ ℰ_{b}^{op} → 𝒮 preserves K-indexed limits for all K ∈ 𝒦. Suppose ϕ: ℬ^{op} → 𝒮 is the presheaf classified by 𝒴 → ℬ. Then unwinding the equivalence

𝒫(ℰ_{ℬ}^{◁} )_{ϕ} ≃ 𝒫(ℰ)_{/p^{∗}ϕ} ≃ RFib(ℰ)_{/p^{∗}𝒴} ≃ RFib(ℰ ×_{ℬ} 𝒴) ≃ 𝒫(ℰ ×_{ℬ} 𝒴),

we see that these presheaves on ℰ ×_{ℬ} 𝒴 correspond to presheaves Ψ: ℰ^{op} → 𝒮 over p^{∗}ϕ such that for every b ∈ ℬ, the restriction

ℰ_{b}^{op} → 𝒮_{/ϕ(b)}

has the property that for every y ∈ ϕ(b), the composite with the map 𝒮_{/ϕ(b)} → 𝒮 given by pullback along {y} → ϕ(b) takes K-indexed colimits to limits for all K ∈ 𝒦. Now recall that the ∞-category 𝒮_{/ϕ(b)} is equivalent to Fun(ϕ(b), 𝒮), with pullback to {y} corresponding to evaluation at y, and that limits in functor categories are computed pointwise. Thus we may identify our full subcategory with that of presheaves Ψ such that for every b ∈ ℬ, the restriction

ℰ_{b}^{op} → 𝒮_{/ϕ(b)}

takes K-indexed colimits to limits in 𝒮_{/ϕ(b)}. For any ∞-category 𝒞 and x ∈ 𝒞, a diagram K^{◁} → 𝒞_{/x} is a limit diagram if and only if the associated diagram K^{◁▷} → 𝒞 is a limit diagram. Therefore, the full subcategory of 𝒫^{cocart}(ℰ_{ℬ}^{◁} ) we have identified is precisely 𝒫_{𝒦}^{cocart}(ℰ_{ℬ}^{◁} ), which completes the proof. □

Corollary 9.17. Suppose F: ℬ → Cat_{∞}^{κ} is a functor, with p: ℰ → ℬ an associated coCartesian fibration. Let Înd_{κ}F: 𝒫(ℬ)^{op} → Ĉat_{∞} be the unique limitpreserving functor extending Ind_{κ} ∘F^{op} : ℬ^{op} → Ĉat_{∞}. Then the restricted projection 𝒫_{κ}^{cocart}(ℰ_{ℬ}^{◁} ) := 𝒫_{𝒦(κ)}^{cocart}(ℰ_{ℬ}^{◁} ) → 𝒫(ℬ) is a Cartesian fibration classi-

fied by the functor Înd_{κ}F.

10. Presentable Fibrations are Presentable

In ordinary category theory, an accessible fibration is a Grothendieck fibration p: E → C such that C is an accessible category, the corresponding functor F: C^{op} → Ĉat factors through the category of accessible categories and accessible functors, and F preserves κ-filtered limits for κ sufficiently large. In [MP89], Makkai and Paré prove that if p is an accessible fibration, then its source E is also an accessible category, and p is an accessible functor. The goal of this section is to prove an ∞-categorical variant of this result. As it makes the proof much clearer we will, however, restrict ourselves to considering only presentable fibrations of ∞-categories, defined as follows:

Definition 10.1. A presentable fibration is a Cartesian fibration p: ℰ → ℬ such that ℬ is a presentable ∞-category, the corresponding functor F: ℬ^{op} → Ĉat_{∞} factors through the ∞-category Pr^{R} of presentable ∞-categories and right adjoints, and F preserves κ-filtered limits for κ sufficiently large.

Remark 10.2. Suppose p: ℰ → ℬ is a presentable fibration. Since the morphisms of ℬ are all mapped to right adjoints under the associated functor, it follows that p is also a coCartesian fibration.

The goal of this section is then to prove the following:

Theorem 10.3. Let p: ℰ → ℬ be a presentable fibration. Then ℰ is a presentable ∞-category.

As in Makkai and Paré’s proof of [MP89, Theorem 5.3.4], we will prove this by explicitly describing the total space of the presentable fibration classified by a special class of functors as an accessible localization of a presheaf ∞-category. To state this result we first recall some notation from [Lur09a, §5.5.7]:

Definition 10.4. Suppose κ is a regular cardinal. As before, let Cat_{∞}^{κ} be the category of small ∞-categories that have all κ-small colimits, and functors that preserve these. Then Ind_{κ} gives a functor from Cat_{∞}^{κ,op} to the ∞-category Pr_{κ}^{R} of κ-presentable ∞-categories and limit-preserving functors that preserve κ-filtered colimits. Using the equivalence Pr^{L} ≃ (Pr^{R})^{op} we may equivalently regard this as a functor Ind_{κ}^{∨}: Cat_{∞}^{κ} → Pr_{κ}^{L}, where Pr_{κ}^{L} is the ∞-category of κ-presentable ∞-categories and colimit-preserving functors that preserve κcompact objects.

The key step in the proof of Theorem 10.3 can then be stated as follows:

Proposition 10.5. Suppose F: ℬ → Cat_{∞}^{κ} is a functor of ∞-categories with associated coCartesian fibration p: ℰ → ℬ. Let q: ℰ̂ → 𝒫(ℬ) be a Cartesian fibration classified by the unique limit-preserving functor Înd_{κ}F: 𝒫(ℬ)^{op} → Pr_{κ}^{R} extending Ind_{κ} ∘F^{op} : ℬ^{op} → Pr_{κ}^{R}. Then the ∞-category ℰ̂ is an accessible ∞- category, and q is an accessible functor.

We will prove Proposition 10.5 using Corollary 9.17 together with the following simple observation:

Lemma 10.6. Suppose 𝒞 is a small ∞-category, and let S = \{\overline{p}_{α}: K_{α}^{▷} → 𝒞\} be a small set of diagrams in 𝒞. Then the full subcategory of 𝒫(𝒞) spanned by presheaves that take the diagrams in S to limit diagrams in 𝒮 is accessible, and the inclusion of this into 𝒫(𝒞) is an accessible functor.

Proof. Let y_{𝒞} : 𝒞 → 𝒫(𝒞) denote the Yoneda embedding. A presheaf F: 𝒞^{op} → 𝒮 takes \overline{p}_{α}^{op} to a limit diagram if and only if it is local with respect to the map of presheaves

colim(y_{𝒞} ∘ \overline{p}|_{K_{α}}) → y_{𝒞} (∞),

where ∞ denotes the cone point. Thus if S′ is the set of these morphisms for \overline{p}_{α} ∈ S, the subcategory in question is precisely the full subcategory of S′- local objects. (This observation can also be found e.g. in the proof of [Lur09a, Proposition 5.3.6.2].) Since S, and hence S′, is by assumption a small set, it follows that this subcategory is an accessible localization of 𝒫(𝒞). In particular, it is itself accessible and the inclusion into 𝒫(𝒞) is an accessible functor. □

Proof of Proposition 10.5. By Proposition 9.17 the Cartesian fibration ℰ̂ → 𝒫(ℬ) can be identified with the restriction to the full subcategory 𝒫_{κ}^{cocart}(ℰ_{ℬ}^{◁} ) of the functor j^{∗}: 𝒫(ℰ_{ℬ}^{◁} ) → 𝒫(ℬ) induced by composition with the inclusion j: ℬ ↪ ℰ_{ℬ}^{◁} .

The ∞-category 𝒫_{κ}^{cocart}(ℰ_{ℬ}^{◁} ) is by definition the full subcategory of 𝒫(ℰ_{ℬ}^{◁} ) spanned by presheaves that take two classes of diagrams to limit diagrams in 𝒮 — one indexed by coCartesian morphisms in ℰ, which form a set, and one indexed by κ-small colimit diagrams in the fibres of p; these do not form a set, but we can equivalently consider only pushout squares and coproducts indexed by κ-small sets, which do form a set. It then follows from Lemma 10.6 that 𝒫_{κ}^{cocart}(ℰ_{ℬ}^{◁} ) is accessible and the inclusion 𝒫_{κ}^{cocart}(ℰ_{ℬ}^{◁} ) ↪ 𝒫(ℰ_{ℬ}^{◁} ) is an accessible functor. The functor j^{∗}: 𝒫(ℰ_{ℬ}^{◁} ) → 𝒫(ℬ) preserves colimits, since these are computed pointwise, and so the composite 𝒫_{κ}^{cocart}(ℰ_{ℬ}^{◁} ) → 𝒫(ℬ) is also an accessible functor. □

To complete the proof of Theorem 10.3 we now just need an easy Lemma:

Lemma 10.7. Suppose π: ℰ → ℬ is a coCartesian fibration such that both ℬ and the fibres ℰ_{b} for all b ∈ ℬ admit small colimits, and the functors f_{!} : ℰ_{b} → ℰ_{b′} preserve colimits for all morphisms f: b → b′ in ℬ. Then ℰ admits small colimits.

Proof. The coCartesian fibration π satisfies the conditions of [Lur09a, Corollary 4.3.1.11] for all small simplicial sets K, and so in every diagram

\begin{matrix}K & ℰ \\ K^{▷} & ℬ \end{matrix} p π q̄ p̄

there exists a lift p̄ that is a π-colimit of p. Given a diagram p: K → ℰ we can apply this with q̄ a colimit of π ∘ p to get a colimit p̄: K^{▷} → ℰ of p. □

Proof of Theorem 10.3. It follows from Lemma 10.7 that ℰ has small colimits. It thus remains to prove that ℰ is accessible and p is an accessible functor. Let F: ℬ^{op} → Ĉat_{∞} be a functor corresponding to p. Choose a regular cardinal κ so that ℬ is κ-presentable and F preserves κ-filtered limits. Since ℬ is κ-presentable, ℬ ≃ Ind_{κ} ℬ^{κ} is the full subcategory of 𝒫(ℬ^{κ}) spanned by the presheaves that preserve κ-small limits. Let F̂: 𝒫(ℬ^{κ})^{op} → Ĉat_{∞} be the unique limit-preserving functor extending F |_{ℬ^{κ,op}} ; then F is equivalent to the restriction of F̂ to Ind_{κ} ℬ^{κ}. If p̂: ℰ̂ → 𝒫(ℬ^{κ}) is a Cartesian fibration classified by F̂ we therefore have a pullback square

\begin{matrix}ℰ & ℰ̂ \\ ℬ & 𝒫(ℬ^{κ}), \end{matrix}

where the bottom map preserves κ-filtered colimits, so by [Lur09a, Proposition 5.4.6.6] it suffices to show that ℰ̂ is accessible and p̂ is an accessible functor. Since ℬ^{κ} is a small ∞-category, we can choose a cardinal λ such that F |_{ℬ^{κ,op}} factors through the ∞-category Pr^{R,λ} of λ-presentable ∞-categories and right adjoints that preserve λ-filtered colimits. By [Lur09a, Proposition 5.5.7.2] we can equivalently think of this, via the equivalence Pr^{R} ≃ (Pr^{L})^{op}, as a functor from ℬ^{κ} to the ∞-category Pr^{L,λ} of λ-presentable ∞-categories and functors that preserve colimits and λ-compact objects. Taking λ-compact objects defines a functor (–)^{λ}: Pr^{L,λ} → Cat_{∞}^{λ}. Then, defining F_{0} : (ℬ^{κ}) → Cat_{∞} to be (F^{op}|_{ℬ^{κ}} )^{λ}, we see that F ≃ Ind_{λ} F_{0}, and so ℰ̂ is accessible and p̂ is an accessible functor by Proposition 10.5. □

Appendix A. Pseudofunctors and the Naturality of Unstraightening

At several points in this paper we will need to know that the unstraightening functors Fun(𝒞^{op}, Cat_{∞}) → Cat_{∞/𝒞}^{cart} (and a number of similar constructions) are natural as we vary the ∞-category 𝒞. The obvious way to prove this is to consider the naturality of the unstraightening Un_{S}^{+} : Fun(ℭ(S), Set_{Δ}^{+}) → (Set_{Δ}^{+})_{/S} as we vary the simplicial set S. However, since pullbacks are only determined up to canonical isomorphism, these functors are not natural “on the nose”, but only up to natural isomorphism — i.e. they are only pseudonatural. In the body of the paper we have swept such issues under the rug, but in this appendix we indulge ourselves in a bit of 2-category theory to prove that pseudo-naturality on the level of model categories does indeed give naturality on the level of ∞-categories. We begin by reviewing Duskin’s nerve of bicategories [Dus02] and its basic properties. However, we will only need to consider the case of strict 2- and (2,1)-categories:

Definition A.1. A strict 2-category is a category enriched in Cat, and a strict (2,1)-category is a category enriched in Gpd. We write Cat_{2} for the category of strict 2-categories and Cat_{(2,1)} for the category of strict (2,1)-categories.

Definition A.2. Suppose C and D are strict 2-categories. A normal oplax functor F: C → D consists of the following data:

(a) for each object x ∈ C, an object F (x) ∈ D,

(b) for each 1-morphism f: x → y in C, a 1-morphism F (f): F (x) → F (y),

(c) for each 2-morphism ϕ: f ⇒ g in C(x, y), a 2-morphism F (ϕ): F (f) ⇒ F (g) in D(F (x), F (y)),

(d) for each pair of composable 1-morphisms f: x → y, g: y → z in C, a 2-morphism η_{f,g}: F (g ∘ f) ⇒ F (g) ∘ F (f),

such that:

(i) for every object x ∈ C, the 1-morphism F (id_{x}) = id_{F (x)}, (ii) for every 1-morphism f: x → y in C, the 2-morphism F (id_{f}) = id_{F (f)}, (iii) for composable 2-morphisms ϕ: f ⇒ g, ψ: g ⇒ h in C(x, y), we have

F (ψ ∘ ϕ) = F (ψ) ∘ F (ϕ), (iv) for every morphism f: x → y, the morphisms η_{id_{x},f} and η_{f,id_{y}}: F (f) →

F (f) are both id_{F (f)},

(v) if ϕ: f ⇒ f′ is a 2-morphism in C(x, y) and ψ: g ⇒ g′ is a 2-morphism in C(y, z), then the diagram

\begin{matrix}F (g ∘ f) & F (g) ∘ F (f) \\ F (g′ ∘ f′) & F (g′) ∘ F (f′) \end{matrix}_{\begin{matrix}η_{f,g} \\ F (ψ ∘ ϕ) \\ F (ψ) ∘ F (ϕ) \\ η_{f′,g′} \end{matrix}}

commutes, (vi) for composable triples of 1-morphisms f: x → y, g: y → z, h: z → w,

the diagram

\begin{matrix}F (h ∘ g ∘ f) & F (h ∘ g) ∘ F (f) \\ F (h) ∘ F (g ∘ f) & F (h) ∘ F (g) ∘ F (f)) \end{matrix}_{\begin{matrix}η_{f,hg} \\ η_{gf,h} \\ η_{g,h} ∘ id \\ id ∘ η_{f,g} \end{matrix}}

commutes. We say a normal oplax functor F from C to D is a normal pseudofunctor if the 2-morphisms η_{f,g} are all isomorphisms. In particular, if the 2-category C is a (2,1)-category, all normal oplax functors C → D are normal pseudofunctors.

Remark A.3. In 2-category theory one typically considers the more general notions of (not necessarily normal) oplax functors and pseudofunctors, which do not satisfy F (id_{x}) = id_{F (x)} but instead include the data of natural maps F (id_{x}) → id_{F (x)} (which are isomorphisms for pseudofunctors). We only consider the normal versions because, as we will see below, these correspond to maps of simplicial sets between the nerves of strict 2- and (2,1)-categories.

Before we recall the definition of the nerve of a strict 2-category, we first review the definition of nerves for ordinary categories and simplicial categories:

Definition A.4. Let N: Cat → Set_{Δ} be the usual nerve of categories, i.e. if C is a category then NC_{k} is the set Hom([k], C) where [k] is the category corresponding to the partially ordered set {0, …, k}.

Remark A.5. Since Cat has colimits, the functor N has a left adjoint C: Set_{Δ} → Cat, which is the unique colimit-preserving functor such that C(Δ^{n}) = [n].

Lemma A.6. The functor C: Set_{Δ} → Cat takes inner anodyne morphisms to isomorphisms.

Proof. Let 𝔚 denote the class of monomorphisms of simplicial sets that are taken to isomorphisms by C. To see that 𝔚 contains the inner anodyne morphisms we apply [JT07, Lemma 3.5], which says that 𝔚 contains the inner anodyne maps if

(i) 𝔚 is weakly saturated, i.e. it contains the isomorphisms and is closed under composition, transfinite composition, cobase change, and codomain retracts,

(ii) 𝔚 has the right cancellation property, i.e. it if fg and g are in 𝔚 then f

is in 𝔚, (iii) 𝔚 contains the inclusions Sp^{n} ↪ Δ^{n}, where Sp^{n} denotes the n-spine,

i.e. the simplicial set Δ^{{0,1}} ⨿_{Δ^{{1}}} ⋯ ⨿_{Δ^{{n−1}}} Δ^{{n−1,n}}.

Here conditions (i) and (ii) follow immediately from the definition of 𝔚, as the functor C preserves colimits. It remains to prove (iii), i.e. to show that C(Sp^{n}) → C(Δ^{n}) is an isomorphism. Since C preserves colimits, this is the map of categories

[1] ⨿_{[0]} ⋯ ⨿_{[0]} [1] → [n].

But the category [n] is the free category on the graph with vertices 0, …, n and edges i → (i + 1), which obviously decomposes as a colimit in this way, and the free category functor on graphs preserves colimits. □

Proposition A.7. The functor C: Set_{Δ} → Cat preserves products.

Proof. Since C preserves colimits and the Cartesian products in Cat and Set_{Δ} both commute with colimits in each variable, it suffices to check that the natural map C(Δ^{n} × Δ^{m}) → C(Δ^{n}) × C(Δ^{m}) is an isomorphism for all n, m. Since products of inner anodyne maps are inner anodyne by [Lur09a, Corollary 2.3.2.4], the inclusion Sp^{n} × Sp^{m} → Δ^{n} × Δ^{m} is inner anodyne. Thus in the diagram

C(Sp^{n} × Sp^{m})  C(Sp^{n}) × C(Sp^{m})

C(Δ^{n} × Δ^{m})  C(Δ^{n}) × C(Δ^{m})

the vertical maps are isomorphisms by Lemma A.6. It hence suffices to prove that the upper horizontal map is an isomorphism. Since C preserves colimits and the Cartesian products preserve colimits in each variable, in the commutative diagram

C(Δ^{1} × Sp^{m}) ⨿_{C(Δ^{0}×Sp^{m})} C(Sp^{n} × Sp^{m})

C(Sp^{n+1} × Sp^{m})

C(Δ^{1}) × C(Sp^{m}) ⨿_{C(Δ^{0})×C(Sp^{m})} C(Sp^{n}) × C(Sp^{m})

C(Sp^{n+1}) × C(Sp^{m})

Lax Colimits and Free Fibrations in ∞-Categories 1259

C(Δ^{1} × Sp^{m}) ⨿_{C(Δ^{0}×Sp^{m})} C(Sp^{n} × Sp^{m}) C(Sp^{n+1} × Sp^{m})

C(Δ^{1}) × C(Sp^{m}) ⨿_{C(Δ^{0})×C(Sp^{m})} C(Sp^{n}) × C(Sp^{m})

C(Sp^{n+1}) × C(Sp^{m})

the vertical morphisms are isomorphisms. By inducting on n and m this implies that the map in question is an isomorphism for all n and m provided

C(Δ^{n} × Δ^{m}) → C(Δ^{n}) × C(Δ^{m})

is an isomorphism when n and m are both either 0 or 1. The cases where n or m is 0 are trivial, so it only remains to show that C(Δ^{1} × Δ^{1}) → [1] × [1] is an isomorphism. The simplicial set Δ^{1} × Δ^{1} is the pushout Δ^{2} ⨿_{Δ^{{0,2}}} Δ^{2}, so this amounts to showing that the analogous functor [2] ⨿_{[1]} [2] → [1] × [1] is an isomorphism, or equivalently that for any category C, the square

Hom([1] × [1], C)  Hom([2], C)

Hom([2], C)  Hom([1], C)

is Cartesian. But this claim is equivalent to the statement that a commutative square in C is the same as two compatible commutative triangles, which is obvious. □

Definition A.8. The functor N preserves products, being a right adjoint, and so induces a functor N_{∗}: Cat_{2} → Cat_{Δ}, given by applying N on the mapping spaces; this has a left adjoint C_{∗}: Cat_{Δ} → Cat_{2} given by composition with C, since C preserves products by Proposition A.7.

We now briefly recall the definition of the coherent nerve functor from simplicial categories to simplicial sets, following [Lur09a, §1.1.5]:

Definition A.9. Let P_{i,j} be the partially ordered set of subsets of {i, i + 1, …, j} containing i and j. Then ℭ(Δ^{n}) denotes the simplicial category with objects 0, …, n and

ℭ(Δ^{n})(i, j) = \begin{cases}∅, & i > j \\ NP_{i,j}, & otherwise \end{cases}

Composition of morphisms is induced by union of the subsets in the P_{i,j}’s.

Remark A.10. The simplicial set NP_{i,j} is isomorphic to (Δ^{1})^{×(j−i−1)} for j > i.

Definition A.11. The coherent nerve is the functor 𝔑: Cat_{Δ} → Set_{Δ} defined by

𝔑C_{k} = Hom(ℭ(Δ^{k}), C).

This has a left adjoint ℭ: Set_{Δ} → Cat_{Δ}, which is the unique colimit-preserving functor extending the cosimplicial simplicial category ℭ(Δ^{•}).

Definition A.12. Let N_{2} : Cat_{2} → Set_{Δ} denote the composite

Cat_{2} ⟶^{N_{∗}} Cat_{Δ} ⟶^{𝔑} Set_{Δ}.

This functor has a left adjoint C_{2}, which is the composite

Set_{Δ} ⟶^{ℭ} Cat_{Δ} ⟶^{C_{∗}} Cat_{2}.

Remark A.13. It is clear from the definitions given in [Dus02, §§6.1–6.7] that the functor N_{2} as we have defined it is simply the restriction of Duskin’s nerve for bicategories to strict 2-categories. (This nerve also implicitly appeared earlier in [Str87].)

Remark A.14. We can describe the strict 2-category C_{2}(Δ^{n}) as follows: its objects are 0, …, n. For i > j, the category C_{2}(Δ^{n})(i, j) is empty, and for j > i it is the partially ordered set P_{i,j} (which is isomorphic to [1]^{×(j−i−1)} if j > i). We can thus describe the low-dimensional simplices of the nerve N_{2}C of a strict 2-category C as follows:

• The 0-simplices are the objects of C.

• The 1-simplices are the 1-morphisms of C.

• A 2-simplex in N_{2}C is given by objects x_{0}, x_{1}, x_{2}, 1-morphisms f_{01} : x_{0} → x_{1}, f_{12} : x_{1} → x_{2}, f_{02} : x_{0} → x_{2}, and a 2-morphism ϕ_{012} : f_{02} ⇒ f_{12} ∘ f_{01}.

• A 3-simplex is given by – objects x_{0}, x_{1}, x_{2}, x_{3},

– 1-morphisms f_{ij}: x_{i} → x_{j} for 0 ≤ i < j ≤ 3,

– 2-morphisms ϕ_{012} : f_{02} ⇒ f_{12} ∘ f_{01}, ϕ_{123} : f_{13} ⇒ f_{23} ∘ f_{12}, ϕ_{023} : f_{03} ⇒ f_{23} ∘ f_{02} and ϕ_{013} : f_{03} ⇒ f_{13} ∘ f_{01}, such that the square

\begin{matrix}f_{03} & f_{13} ∘ f_{01} \\ f_{23} ∘ f_{02} & f_{23} ∘ f_{12} ∘ f_{01} \end{matrix}_{\begin{matrix}ϕ_{013} \\ ϕ_{023} \\ ϕ_{123} ∘ id \\ id ∘ ϕ_{012} \end{matrix}}

commutes.

Definition A.15. Let Δ_{≤k} denote the full subcategory of Δ spanned by the objects [n] for n ≤ k. The restriction sk_{k}: Set_{Δ} → Fun(Δ_{≤k}^{op}, Set) has a right adjoint

cosk_{k}: Fun(Δ_{≤k}^{op}, Set) → Set_{Δ}.

We say a simplicial set X is k-coskeletal if it is in the image of the functor cosk_{k}. Equivalently, X is k-coskeletal if every map ∂Δ^{n} → X extends to a unique n-simplex Δ^{n} → X when n > k.

Proposition A.16. For every strict 2-category C, the simplicial set N_{2}C is 3-coskeletal.

Remark A.17. A more general version of this result in the setting of bicategories appears in [Dus02].

Proof. We must show that every map ∂Δ^{k} → N_{2}C extends to a unique map from Δ^{k} if k > 3. Equivalently, we must show that given a map ℭ(∂Δ^{k}) → N_{∗}C it has a unique extension to ℭ(Δ^{k}) for k > 3. We can describe the simplicial category ℭ(∂Δ^{k}) and its map to ℭ(Δ^{k}) as follows:

• the objects of ℭ(∂Δ^{k}) are 0, … k,

• the maps ℭ(∂Δ^{k})(i, j) → ℭ(Δ^{k})(i, j) are isomorphisms except when i = 0 and j = k,

• the simplicial set ℭ(∂Δ^{k})(0, k) is the boundary of the (k − 1)-cube

ℭ(Δ^{k})(0, k) ≅ (Δ^{1})^{×(k−1)}. Thus extending a map F: ℭ(∂Δ^{k}) → N_{∗}C to ℭ(Δ^{k}) amounts to extending the map

ℭ(∂Δ^{k})(0, k) → NC(F (0), F (k))

to ℭ(Δ^{k})(0, k). But the inclusion ℭ(∂Δ^{k})(0, k) → ℭ(Δ^{k})(0, k) is a composition of pushouts of inner horn inclusions and the inclusion ∂Δ^{k−1} → Δ^{k−1}, and if k − 1 > 2 the nerve of a category has unique extensions along these. □

Theorem A.18 (Duskin, Bullejos–Faro–Blanco). Suppose C and D are strict 2-categories. Then the maps of simplicial sets N_{2}C → N_{2}D can be identified with the normal oplax functors C → D.

Remark A.19. The more general version of this result for bicategories appears to be an unpublished result of Duskin; for 2-categories it is proved by Bullejos, Faro, and Blanco as [BFB05, Proposition 4.3]. We do not include a complete proof here, but we will now briefly indicate how a map of nerves gives rise to a normal oplax functor. By Proposition A.16, a map N_{2}C → N_{2}D can be identified with a map F: sk_{3}N_{2}C → sk_{3}N_{2}D. Using Remark A.14 we can identify this with the data of a normal oplax functor as given in Definition A.2:

• The 0-simplices of N_{2}C are the objects of C, so F assigns an object F (c) ∈ D to every c ∈ C, which gives (a)

• The 1-simplices of N_{2}C are the 1-morphisms in C, with sources and targets given by the face maps [0] → [1], so F assigns a 1-morphism F (f): F (x) → F (y) to every 1-morphism f: x → y in C, which gives (b).

• Moreover, identity 1-morphisms correspond to degenerate edges in N_{2}C, so since these are preserved by any map of simplicial sets we get F (id_{x}) = id_{F (x)}, i.e. (i).

• The 2-simplices of N_{2}C are given by three 1-morphisms f: x → y, g: y → z, h: z → w (corresponding to the three face maps), and a 2-morphism ϕ: h ⇒ g ∘ f. In particular:

– Considering 2-simplices where the second edge is degenerate, which correspond to 2-morphisms in C, we see that F assigns a 2-morphism F (ϕ): F (h) ⇒ F (g) to every ϕ: h ⇒ g in C, which gives (c).

– Considering 2-simplices where the 2-morphism ϕ is the identity, we see (as this condition is not preserved by F) that F assigns a 2-morphism F (g ∘f) ⇒ F (g)∘F (f) to all composable pairs of 1-morphisms, which gives (d).

• Since F preserves degenerate 2-simplices, which correspond to identity 2-morphisms of the form f ∘ id ⇒ f and id ∘ f ⇒ f, we get (ii) and (iv).

• The 3-simplices of N_{2}C are given by – objects x_{0}, x_{1}, x_{2}, x_{3},

– 1-morphisms f_{ij}: x_{i} → x_{j} for 0 ≤ i < j ≤ 3,

– 2-morphisms ϕ_{012} : f_{02} ⇒ f_{12} ∘ f_{01}, ϕ_{123} : f_{13} ⇒ f_{23} ∘ f_{12}, ϕ_{023} : f_{03} ⇒ f_{23} ∘ f_{02} and ϕ_{013} : f_{03} ⇒ f_{13} ∘ f_{01}, such that the square

\begin{matrix}f_{03} & f_{13} ∘ f_{01} \\ f_{23} ∘ f_{02} & f_{23} ∘ f_{12} ∘ f_{01} \end{matrix}_{\begin{matrix}ϕ_{013} \\ ϕ_{023} \\ ϕ_{123} ∘ id \\ id ∘ ϕ_{012} \end{matrix}}

commutes. In particular, we have:

– If x_{1} = x_{2} = x_{3}, f_{12} = f_{13} = f_{23} = id_{x_{1}}, and ϕ_{123} = id_{id_{x_{1}}} , then this says ϕ_{013} = ϕ_{012} ∘ϕ_{023}, and since F preserves identities this gives (iii).

– In the case where the 2-morphisms are all identities, we get (vi).

– To get (v), we consider the 3-simplices where f_{12} = id, ϕ_{023} = id, and ϕ_{013} is the composite of ϕ_{012} and ϕ_{123}.

Definition A.20. The inclusion Gpd ↪ Cat of the category of groupoids preserves products, and so induces a functor Cat_{(2,1)} → Cat_{2}; we write N_{(2,1)} for the composite

Cat_{(2,1)} → Cat_{2} ⟶^{N_{2}} Set_{Δ}.

Corollary A.21. If C and D are strict (2,1)-categories, then a morphism of simplicial sets N_{(2,1)}C → N_{(2,1)}D can be identified with a normal pseudofunctor C → D.

Definition A.22. Recall that a relative category is a category C equipped with a subcategory W containing all isomorphisms; see [BK12] for a more extensive discussion. A functor of relative categories f: (C, W) → (C′, W′) is a functor f: C → C′ that takes W into W′. We write RelCat_{(2,1)} for the strict (2,1)- category of relative categories, functors of relative categories, and all natural isomorphisms between these.

We now want to prove that a normal pseudofunctor to RelCat_{(2,1)} determines a map of ∞-categories to Cat_{∞} via the following construction:

Definition A.23. If (C, W) is a relative category, let L(C, W) ∈ Set_{Δ}^{+} be the marked simplicial set (NC, NW_{1}). This defines a simplicial functor N_{∗}RelCat_{(2,1)} → Set_{Δ}^{+}.

Definition A.24. If (C, W) is a relative category, we write C[W^{−1}] for the ∞- category obtained by taking a fibrant replacement of the marked simplicial set L(C, W). More generally, if C is a strict (2,1)-category and W is a collection of 1-morphisms in C, we write C[W^{−1}] for the ∞-category obtained by fibrantly replacing the marked simplicial set (N_{(2,1)}C, W).

Lemma A.25. Let C be a strict (2,1)-category, and let F be a normal pseudofunctor F: C → RelCat_{(2,1)}. If W is a collection of 1-morphisms in C such that F takes the morphisms in W to weak equivalences of relative categories, then F determines a functor of ∞-categories C[W^{−1}] → Cat_{∞}, which sends x ∈ C to E_{x}[W_{x}^{−1}] where F (x) = (E_{x}, W_{x}).

Proof. By Proposition A.21 the normal pseudofunctor F corresponds to a map of simplicial sets N_{(2,1)}C → N_{(2,1)}RelCat_{(2,1)}. Composing this with the map 𝔑(L): N_{(2,1)}RelCat_{(2,1)} → 𝔑Set_{Δ}^{+} we get a map N_{(2,1)}C → 𝔑Set_{Δ}^{+}. We may regard this as a map of marked (large) simplicial sets

(N_{(2,1)}C, W) → (𝔑Set_{Δ}^{+}, W′),

where W′ is the collection of marked equivalences in Set_{Δ}^{+}. Now invoking [Lur14, Theorem 1.3.4.20] we conclude that Cat_{∞} is a fibrant replacement for the marked simplicial set (𝔑Set_{Δ}^{+}, W′), so this map corresponds to a map C[W^{−1}] → Cat_{∞} in the ∞-category Ĉat_{∞} underlying the model category of (large) marked simplicial sets. □

We will now make use of Grothendieck’s description of pseudofunctors to the (2,1)-category of categories to get a way of constructing pseudofunctors to RelCat_{(2,1)}:

Theorem A.26 (Grothendieck [Gro63]). Let C be a category. Then pseudofunctors from C^{op} to the strict 2-category CAT correspond to Grothendieck fibrations over C.

Remark A.27. Let us briefly recall how a pseudofunctor is constructed from a Grothendieck fibration, as this is the part of Grothendieck’s theorem we will actually use. A cleavage of a Grothendieck fibration p: E → B is the choice, for each (e ∈ E, f: b → p(e)), of a single Cartesian morphism over f with target e; cleavages always exist, by the axiom of choice. Given a choice of cleavage of p, we define the pseudofunctor C^{op} → CAT by assigning the fibre E_{b} to each b ∈ B, and for each f: b → b′ the functor f^{∗} assigns to e ∈ E_{b} the source of the Cartesian morphism over f with target e in the cleavage. Clearly, this pseudofunctor will be normal precisely when the cleavage is normal in the sense that the Cartesian morphisms over the identities in B are all chosen to be identities in E. Every Grothendieck fibration obviously has a normal cleavage, so from any Grothendieck fibration we can construct a normal pseudofunctor.

Definition A.28. A relative Grothendieck fibration is a Grothendieck fibration p: E → C together with a subcategory W of E containing all the p-Cartesian morphisms. In particular, the restricted projection W → C is also a Cartesian fibration. Moreover, for every x ∈ C the fibres (E_{x}, W_{x}) are relative categories, and the functor f^{∗} induced by each f in C is a functor of relative categories.

If (C, U) is a relative category, we say that the relative Grothendieck fibration is compatible with U if this functor f^{∗}: (E_{q}, W_{q}) → (E_{p}, W_{p}) is a weak equivalence of relative categories for every f: p → q in U.

The following is then an obvious consequence of Theorem A.26:

Lemma A.29. Relative Grothendieck fibrations over a category C correspond to normal pseudofunctors C^{op} → RelCat_{(2,1)}.

Proposition A.30. Let (E, W) be a relative Grothendieck fibration over C compatible with a collection U of morphisms in C. Then this induces a functor of ∞-categories

C[U^{−1}]^{op} → Cat_{∞}

that sends p ∈ C to E_{p}[W_{p}^{−1}].

Proof. Combine Lemmas A.29 and A.25. □

All the maps whose naturality we are interested in can easily be constructed as relative Grothendieck fibrations. We will explicitly describe this in the case of the unstraightening equivalence, and leave the other cases to the reader.

Proposition A.31. The unstraightening functors

Un_{S}^{+} : Fun_{Δ}(ℭ(S)^{op}, Set_{Δ}^{+})^{fib} → (Set_{Δ}^{+})_{/S}^{fib}

define a relative Grothendieck fibration over Set_{Δ} × Δ^{1} compatible with the categorical equivalences in Set_{Δ}.

Proof. Let E be the category whose objects are triples (i, S, X) where i = 0 or 1, S ∈ Set_{Δ}, and X is a fibrant map Y → S^{♯} in Set_{Δ}^{+} if i = 0 and a fibrant simplicial functor ℭ(S)^{op} → Set_{Δ}^{+} if i = 1; the morphisms (i, S, X) → (j, T, Y) consist of a morphism i → j in [1], a morphism f: S → T in Set_{Δ}, and the following data:

• if i = j = 1, X: ℭ(S) → Set_{Δ}^{+} and Y : ℭ(T) → Set_{Δ}^{+}, a simplicial natural transformation X → ℭ(f) ∘ Y,

• if i = j = 0, X is E → S and Y is F → T, a commutative square

\begin{matrix}E & F \\ S^{♯} & T^{♯} \end{matrix}

in Set_{Δ}^{+},

• if i = 1 and j = 0, Y is a functor ℭ(S)^{op} → Set_{Δ}^{+} and X is E → T, a commutative square

\begin{matrix}Un_{S}^{+}(X) & E \\ S & T. \end{matrix}

Composition is defined in the obvious way, using the natural maps of [Lur09a, Proposition 3.2.1.4]. We claim that the projection E → Δ^{1} × Set_{Δ} is a Grothendieck fibration. It suffices to check that Cartesian morphisms exist for morphisms of the form (id_{i}, f) and (0 → 1, id_{S}), which is clear. □

Corollary A.32. There is a functor of ∞-categories Cat_{∞}^{op} → Fun(Δ^{1}, Ĉat_{∞}) that sends 𝒞 to the unstraightening equivalence

Fun(𝒞^{op}, Cat_{∞}) ⟶^{∼} Cat_{∞/𝒞}^{cart} .

References

[BK12] C. Barwick and D. M. Kan, Relative categories: another model for the homotopy theory of homotopy theories, Indag. Math. (N.S.) 23 (2012), no. 1-2, 42–68.

[Bar13] Clark Barwick, On the Q-construction for exact ∞-categories (2013), available at

arXiv:1301.4725.

[Bar17] , Spectral Mackey functors and equivariant algebraic K-theory (I), Adv. Math. 304 (2017), 646–727, available at arXiv:1404.0108.

[BGN14] Clark Barwick, Saul Glasman, and Denis Nardin, Dualizing cartesian and cocartesian fibrations (2014), available at arXiv:1409.2165.

[BFB05] M. Bullejos, E. Faro, and V. Blanco, A full and faithful nerve for 2-categories, Appl. Categ. Structures 13 (2005), no. 3, 223–233.

[BN14] U. Bunke and T. Nikolaus, Twisted differential cohomology (2014), available at

arXiv:1406.3231.

[CP97] Jean-Marc Cordier and Timothy Porter, Homotopy coherent category theory, Trans. Amer. Math. Soc. 349 (1997), no. 1, 1–54.

[Dus02] John W. Duskin, Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories, Theory Appl. Categ. 9 (2002), No. 10, 198–308 (electronic).

[Gla16] Saul Glasman, A spectrum-level Hodge filtration on topological Hochschild homology, Selecta Math. (N.S.) 22 (2016), no. 3, 1583–1612, available at

arXiv:1408.3065.

[Gro63] Alexander Grothendieck, Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique, vol. 1960/61, Institut des Hautes Études Scientifiques, Paris, 1963.

[JT07] André Joyal and Myles Tierney, Quasi-categories vs Segal spaces, Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI, 2007, pp. 277–326, available at arXiv:math/0607820.

[Joy08] André Joyal, The theory of quasi-categories and its applications, Advanced course on simplicial methods in higher

categories,  CRM  Quaderns,  vol.  45-2,  2008,  available  at

http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf.

1266 David Gepner, Rune Haugseng, Thomas Nikolaus

[Lur09a] Jacob Lurie, Higher Topos Theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. Available at

http://math.harvard.edu/~lurie/papers/highertopoi.pdf.

[Lur09b] , (∞,2)-Categories and the Goodwillie Calculus I (2009), available at

http://math.harvard.edu/~lurie/papers/GoodwillieI.pdf.

[Lur14] , Higher Algebra, 2014. Available at

http://math.harvard.edu/~lurie/papers/higheralgebra.pdf.

[MP89] Michael Makkai and Robert Paré, Accessible categories: the foundations of categorical model theory, Contemporary Mathematics, vol. 104, American Mathematical Society, Providence, RI, 1989.

[Str80] Ross Street, Fibrations in bicategories, Cahiers Topologie Géom. Différentielle 21 (1980), no. 2, 111–160.

[Str87] , The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987), no. 3, 283–335.

[Web07] Mark Weber, Yoneda structures from 2-toposes, Appl. Categ. Structures 15 (2007), no. 3, 259–323.

David Gepner Department of Mathematics Purdue University West Lafayette Indiana USA dgepner@purdue.edu Rune Haugseng University of Copenhagen Copenhagen Denmark haugseng@math.ku.dk

Thomas Nikolaus Max-Planck-Institut für Mathematik Bonn Germany