Yoneda's Lemma for Internal Higher Categories

Home , Higher category theory, Simplex category, Yoneda lemma

YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES

LOUIS MARTINI

Abstract. We develop some basic concepts in the theory of higher categories internal to an arbitrary ∞- topos. We deﬁne internal left and right ﬁbrations and prove a version of the Grothendieck construction and of Yoneda’s lemma for internal categories.

Contents 1. Introduction 2 Motivation 2 Main results 3 Related work 4 Acknowledgment 4 2. Preliminaries 4 2.1. General conventions and notation4 2.2. Set theoretical foundations5 2.3. ∞-topoi 5 2.4. Universe enlargement 5 2.5. Factorisation systems 8 3. Categories in an ∞-topos 10 3.1. Simplicial objects in an ∞-topos 10 3.2. Categories in an ∞-topos 12 3.3. Functoriality and base change 16 3.4. The (∞, 2)-categorical structure of Cat(B) 18 3.5. Cat(S)-valued sheaves on an ∞-topos 19 3.6. Objects and morphisms 21 3.7. The universe for groupoids 23 3.8. Fully faithful and essentially surjective functors 26 arXiv:2103.17141v2 [math.CT] 2 May 2021 3.9. Subcategories 31 4. Groupoidal ﬁbrations and Yoneda’s lemma 36 4.1. Left ﬁbrations 36 4.2. Slice categories 38 4.3. Initial functors 42 4.4. Covariant equivalences 49 4.5. The Grothendieck construction 54 4.6. Yoneda’s lemma 61 References 71

Date: May 4, 2021. 1 2 LOUIS MARTINI

1. Introduction Motivation. In various areas of geometry, one of the principal strategies is to study geometric objects by means of algebraic invariants such as cohomology, K-theory and (stable or unstable) homotopy groups. Usually these invariants are constructed in a functorial way, in the sense that they define (possibly higher) presheaves on a suitable (higher) category C of the geometric objects of interest. Usually, C is equipped with a Grothendieck topology that encodes the topological behaviour of the geometric objects contained in C, and one typically expects reasonable algebraic invariants to respect this topology in that they should define (higher) sheaves on C. In that way, one can study the global nature of these objects by means of their local behaviour. For example, if A is an abelian group and if n ≥ 0 is an integer, there is a higher sheaf Hn(−,A) for the étaletopology on the category Sch of schemes that takes values in the ∞-category Grp(S) of group n objects in ∞-groupoids, such that for any scheme X the group π0H (X,A) ∈ Grp(Set) is the nth étale cohomology group of X with coefficients in A [Lur09a, Section 7]. In modern flavours of geometry, the study of categorified invariants has come more and more into focus. For example, one of the fundamental invariants that one can associate to a scheme X is its unbounded derived category D(X). In order to turn this construction into a sheaf on Sch, i.e. in order to be able to glue derived categories, one first passes to an ∞-categorical enhancement: there is a sheaf X 7→ QCoh(X) on Sch that sends X to the symmetric monoidal stable ∞-category of quasi-coherent sheaves on X such that the homotopy category of QCoh(X) recovers the derived category D(X) of X. The properties of QCoh and their relations to the geometry of the scheme X have been investigated by numerous authors, for example by Lurie in [Lur18] and by Ben-Zvi, Francis and Nadler in [BZFN10]. Furthermore, the sheaf QCoh plays a fundamental role in geometric representation theory; in [Gai14] Gaitsgory studies section-wise module ∞- categories over the sheaf QCoh, i.e. the sheaf of ∞-categories that are acted on by QCoh. Another categorical invariant on schemes comes from motivic homotopy theory: the assignment that sends a scheme X to its unstable motivic homotopy ∞-category H(X) defines a sheaf for the Nisnevich topology on Sch, and similarly the assignment X 7→ SH(X) in which SH(X) is the stable motivic homotopy ∞-category on X defines a sheaf for the Nisnevich topology on Sch [Hoy17]. The latter plays a prominent role in the formalisation of the six operations in motivic homotopy theory, see for example [CD19] for an overview. It has been long known that for many choices of an (a priori higher) category A, the datum of an A-valued

sheaf on Sch is equivalent to that of an A-object internal to the ∞-topos ShS(Sch), or the 1-topos ShSet(Sch) in the case that A is actually a 1-category. For example, a sheaf of abelian groups on Sch is simply an

abelian group internal to ShSet(Sch), and the collection of structure sheaves OX for X ∈ Sch are encoded by a single ring object in ShSet(Sch), the sheaf represented by the affine line. By making use of the internal logic of the ∞-topos of sheaves on Sch, one can therefore study such invariants in the same way as one studies their non-parametrized counterpart in the ∞-category of ∞-groupoids, see for example the PhD thesis of Blechschmidt [Ble17] for an application of these ideas to algebraic geometry in the 1-categorical case. The study of higher invariants, i.e. sheaves on Sch that take values in a reasonable ∞-category A, thus naturally leads to the emerging field of homotopy type theory [Uni13]. In fact, it is now known [Shu19] that much of homotopy type theory has a model in an arbitrary ∞-topos. By extending the same line of thought to the case of categorical invariants on schemes, the datum of a sheaf of ∞-categories on Sch is equivalent to that of a category internal to the ∞-topos ShS(Sch). One should therefore be able to formulate and study the properties of such sheaves by means of studying their categorical properties when viewed internally in the ∞-topos ShS(Sch). Such an internal perspective on category theory is not only useful for the study of categorical invariants on schemes, but in other situations as well. For example, work of Barkwick, Glasman and Haines [BGH20, BH19] shows that the proétale ∞- topos Sproét of any coherent scheme S can be naturally regarded as a pyknotic ∞-category, i.e. as a category YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 3

internal to the ∞-topos Pyk(S) of pyknotic ∞-groupoids. Moreover, Sebastian Wolf shows in [Wol20] that

internally in Pyk(S) the ∞-topos Spro´et turns out to simply be given by a presheaf category, which amounts

to a significant simplification of the structure of Sproét. Although bare homotopy type theory is insufficient to argue synthetically about such structures (owing to the presence of non-invertible arrows), Riehl and Shulman [RS17] have proposed an additional layer to homotopy type theory that rectifies this problem and that is powerful enough to support a synthetic formulation of basic higher categorical constructions. In the present paper, our goal is to study sheaves of ∞-categories from a semantic point of view: as categories internal to an ∞-topos B. On the one hand, this is meant to provide tools for studying categorical invariants in geometry, such as those that are mentioned above. On the other hand, the study of categories in an ∞-topos B should provide categorical semantics for the type theory developed by Riehl and Shulman. We do not aim to turn this into a formal statement, nevertheless we believe that developing the semantical side of the story might serve as a bridge to foster future development of the syntactic theory as well.

Main results. The present paper constitutes the first in a series of papers aimed at developing the theory of categories internal to an arbitrary ∞-topos B. Arguably, Yoneda’s lemma is at the very core of any flavour of category theory, so in this paper we will focus on a proof of this result in the context of internal higher categories. We define the ∞-category Cat(B) of categories in B as the full subcategory of the ∞-category of simplicial objects in B that satisfy the Segal condition and univalence, mimicking the definition of complete Segal spaces given by Rezk [Rez01] as previously done by Lurie [Lur09b] and Rasekh [Ras18]. Since complete Segal spaces provide a model for the ∞-category of ∞-categories, the latter can be identified with Cat(S), and one obtains an equivalence Cat(B) ' ShCat(S)(B) between the ∞-category of categories in B and the

∞-category of Cat(S)-valued sheaves on B. Consequently, the sheaf B/− that sends an object A ∈ B to the slice ∞-category B/A defines a (large) category Ω in B that we will refer to as the internal universe of B and which represents the reflection of B within itself. We furthermore show that Cat(B) is cartesian closed, hence that for any two categories C and D in B there is a category [C, D] ∈ Cat(B) of functors between C and D. The first of the two main results in this paper is an internal version of the Grothendieck construction: For

any category C in B, we deﬁne a large category LFibC in B whose objects are given by left ﬁbrations over C. We then show: Theorem 4.5.2. There is a canonical equivalence

[C, Ω] ' LFibC that is natural in C. By making use of the Grothendieck construction, we derive our second main result, an internal version op of Yoneda’s lemma. For any category C in B, we construct a mapping bifunctor mapC(−, −): C × C → Ω whose transpose gives rise to the Yoneda embedding h: C → [Cop, Ω]. Let ev: Cop × [Cop, Ω] → Ω be the evaluation functor, i.e. the counit of the adjunction Cop × − a [Cop, −]. We then show: Theorem 4.6.8. There is a commutative diagram

Cop × [Cop, Ω] h×id [Cop, Ω]op × [Cop, Ω]

map[Cop,Ω](−,−) ev Ω of (large) categories in B. In particular, theorem 4.6.8 implies that the Yoneda embedding h is a fully faithful functor. 4 LOUIS MARTINI

Related work. The idea of developing category theory internal to some topos is not new; in as early as 1963, Lawvere formulated axioms for a theory of categories [Law63], and in subsequent years Bénabou developed the theory of internal categories in a presheaf topos through the notion of fibred categories. For a detailed exposition of the theory of categories in a 1-topos, the reader may consult [Joh02]. In the world of higher category theory, Riehl and Shulman [RS17] have proposed a synthetic approach to the theory of ∞-categories. The theory of categories in an ∞-topos as presented herein is expected to provide categorical semantics to their type theory (cf. [RS17, Remark A.14]). We therefore believe that our version of Yoneda’s lemma can be formally deduced from their work. However, our approach is different in that it is derived from the internal Grothendieck construction, which is not featured in [RS17] and which is a useful result in its own right. On the semantic side of the story, Barwick, Dotto, Glasman, Nardin and Shah [BDG+16] have developed the theory of ∞-categories that are parametrised by a base ∞-category. Such parametrised ∞-categories are a special case of the more general notion of categories in an arbitrary ∞-topos; they precisely correspond to categories in presheaf ∞-topoi. Lastly, we note that Rasekh has previously worked out some aspects of the theory of internal higher categories in [Ras18].

Acknowledgment. I would like to thank my advisor Rune Haugseng for his support and help throughout the process of writing this paper. I furthermore thank Simon Pepin Lehalleur for introducing me to the world of higher category theory and for his invaluable guidance during the early stages of this project. I would also like to thank Mathieu Anel for helpful discussions.

2. Preliminaries 2.1. General conventions and notation. Throughout this paper we freely make use of the language of higher category theory. We will generally follow a model-independent approach to higher categories. This means that as a general rule, all statements and constructions that are considered herein will be invariant under equivalences in the ambient ∞-category, and we will always be working within such an ambient ∞-category. For example, this means that all constructions involving ∞-categories and functors between ∞- categories will be assumed to take place in the ∞-category of ∞-categories. In the same vein, a set will be a discrete ∞-groupoid, and a 1-category will be an ∞-category all of whose mapping ∞-groupoids are discrete. These conventions in particular imply that we will understand the adjective unique in the homotopical sense, i.e. as the condition that there is a contractible ∞-groupoid of choices. We denote by ∆ the simplex category, i.e. the category of non-empty totally ordered ﬁnite sets with order-preserving maps. Every natural number n ∈ N can be considered as an object in ∆ by identifying n with the totally ordered set [n] = {0, . . . n}. For i = 0, . . . , n we denote by δi :[n − 1] → [n] the unique injective map in ∆ whose image does not contain i. Dually, for i = 0, . . . n we denote by σi :[n + 1] → [n] the unique surjective map in ∆ such that the preimage of i contains two elements. Furthermore, if S ⊂ n is an arbitrary subset of k elements, we denote by δS :[k] → [n] the unique injective map in ∆ whose image is precisely S. In the case that S is an interval, we will denote by σS :[n] → [n − k] the unique surjective map op that sends S to a single object. If C is an ∞-category, we refer to a functor C• : ∆ → C as a simplicial

object in C. We write Cn for the image of n ∈ ∆ under this functor, and we write di, si, dS and sS for the image of the maps δi, σi, δS and σS under this functor. Dually, a functor C• : ∆ → C is referred to as a cosimplicial object in C. In this case we denote the image of δi, σi, δS and σS by di, si, dS and σS. The 1-category ∆ embeds fully faithfully into the ∞-category of ∞-categories by means of identifying posets as 0-categories and order-preserving maps between posets with functors between such 0-categories. We denote by ∆n the image of n ∈ ∆ under this embedding. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 5

2.2. Set theoretical foundations. Once and for all we will fix three Grothendieck universes U ∈ V ∈ W that contain the first infinite ordinal ω. A set is small if it is contained in U, large if it is contained in V and very large if it is contained in W. An analogous naming convention will be adopted for ∞-categories and ∞-groupoids. The large ∞-category of small ∞-groupoids is denoted by S, and the very large ∞-category of large ∞-groupoids by bS. The (even larger) ∞-category of very large ∞-groupoids will be denoted by bS. Similarly, we denote the large ∞-category of small ∞-categories by Cat(S), the very large ∞-category of large ∞-categories by Cat(bS), and the even larger ∞-category of very large ∞-categories by Cat(bS).

2.3. ∞-topoi. A large ∞-category B is defined to be an ∞-topos if there exists a small ∞-category C such op that B arises as a left exact and accessible localization of PShS(C) = Fun(C , S), see [Lur09a, Section 6] for alternative characterizations and the basic theory. An algebraic morphism between two ∞-topoi A and B is a functor f ∗ : A → B that commutes with small colimits and finite limits. We denote by Fun∗(A, B) the full subcategory of Fun(A, B) that is spanned by the algebraic morphisms between A and B. Dually, a ∗ geometric morphism between ∞-topoi is a functor f∗ : B → A that admits a left adjoint f which defines an

algebraic morphism (i.e. which commutes with ﬁnite limits). We denote by Fun∗(B, A) the full subcategory of Fun(B, A) spanned by the geometric morphisms. By the adjoint functor therorem, one ﬁnds Fun∗(A, B)op '

Fun∗(B, A). We let RTop be the subcategory of Cat(bS) that is spanned by the ∞-topoi and geometric morphisms, and we denote by LTop the subcategory of Cat(bS) that is spanned by the ∞-topoi and algebraic morphisms. There is an equivalence RTopop ' LTop that sends an ∞-topos to itself and a geometric morphism to its left adjoint. The ∞-category S of small ∞-groupoids is a ﬁnal object in RTop; for any ∞-topos B we denote by Γ: B → S the unique geometric morphism and refer to this functor as the global sections functor. Explicitly, this functor is given by mapB(1, −) where 1 ∈ B denotes a ﬁnal object. Dually, we denote the unique algebraic morphism from S to B by const: S → B and refer to this map as the constant sheaf functor.

2.4. Universe enlargement. For any two large ∞-categories C and A, an A-valued presheaf on C is a functor Cop → A and an A-valued sheaf an A-valued presheaf that preserves small limits (whenever they

exist). We denote the ∞-category of A-valued presheaves on C by PShA(C) and the full subcategory spanned

by the A-valued sheaves on C by ShA(C). For any ∞-topos B, we define its universe enlargement B = Sh (B) as the very large ∞-category of S- b Sb b valued sheaves on B. By [Lur09a, Remark 6.3.5.17] the ∞-category Bb is an ∞-topos relative to V. Moreover, one can make the assignment B 7→ Bb into a functor as follows: Consider the functor PSh (−): RTop → Cat(S) that acts by sending a map f : B → A in RTop to the Sb b ∗ map (−) ◦ f ∗ : PSh (B) → PSh (A). Since f ∗ commutes with small colimits, the functor (−) ◦ f ∗ restricts Sb Sb to a functor fˆ : B → A. As a consequence, if R PSh (−) → RTop is the cocartesian fibration that is ∗ b b Sb classified by the functor PSh (−), then the full subcategory of R PSh (−) that is spanned by pairs (B,F ) Sb Sb with B ∈ RTop and F ∈ B ⊂ PSh (B) is stable under cocartesian arrows and therefore defines a cocartesian b Sb subfibration of R PSh (−) over RTop. Moreover, by making use the adjunction f ∗ a f one sees that the Sb ∗ ˆ functor f∗ sits inside a commutative square

f B ∗ A

fˆ∗ Bb Ab,

hence the same argumentation implies that the full subcategory of R PSh (−) spanned by pairs (B,A) with Sb B ∈ RTop and A ∈ B deﬁnes a cocartesian subﬁbration of R PSh (−) over RTop too. Consequently, one Sb 6 LOUIS MARTINI

obtains a functor RTop → Cat(bS), B 7→ Bb together with a natural transformation

B7→B RTop Cat(bS)

B7→Bb

that is given by the inclusion B ,→ Bb. ˆ By [Lur09a, Remark 6.4.6.18] the functor f∗ deﬁnes a geometric morphism between ∞-topoi relative to the universe V, and the associated left adjoint fˆ∗ is obtained as the restriction of the functor of left Kan extension (f ∗) : PSh (A) → PSh (B) to A. Since the functor B 7→ B above therefore takes values in the ! Sb Sb b b ∞-category RTop\ of ∞-topoi relative to the universe V, taking opposite categories therefore results in a functor LTop → Cat(bS), B 7→ Bb that sends the geometric morphism f ∗ to fˆ∗. By construction, the functor fˆ∗ sits inside the commutative square f ∗ B A

fˆ∗ Bb Ab,

hence an analogous argument as above shows that the inclusion B ,→ Bb deﬁnes a natural transformation

B7→B LTop Cat(bS).

B7→Bb

∗ ˆ∗ Note that if f admits a further left adjoint f!, then the map f is given by precomposition with f!.

Remark 2.4.1. If B is an ∞-topos, there are a priori two ways to define the universe enlargement Bb relative to the universe W: either by applying the above construction to the pair U ∈ W, i.e. by defining Bb = Sh (B), or by applying this construction first to the pair U ∈ V and then to the pair V ∈ W, i.e. b Sb by setting Bb = Sh (B), where the right-hand side now denotes the ∞-categories of functors Bop → bS that b Sb b b b commute with V-small limits. It turns out that either approach results in the same object: in fact, upon

identifying U with a regular cardinal in V, we may identify B with the ∞-category InddU(B), hence [Lur09a, Proposition 5.3.5.10] implies that the inclusion B ,→ Bb induces an equivalence

FunU-ﬁlt.(Bb, bSop) ' Fun(B, bSop)

in which the left-hand side denotes the full subcategory of Fun(Bb, bSop) that is spanned by those functors that preserve U-ﬁltered colimits. Now [Lur09a, Proposition 5.5.1.9] implies that the above equivalence restricts to an equivalence Sh (B) ' Sh (B), noting that its proof does not require bSop to be presentable (relative to Sb b Sb b the universe V) but merely to admit V-small colimits.

Recall that the assignment A 7→ B/A deﬁnes a fully faithful functor B ,→ RTop/B. By [GHN17, Corol- lary 9.9] there is a functorial equivalence

PSh (B ) ' PSh (B) Sb /− Sb /− YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 7

of functors Bop → Cat(bS) that is given on each object A ∈ B by the left Kan extension of the functor B → PSh (B) that is induced by the Yoneda embedding B ,→ PSh (B) along the Yoneda embedding /A Sb /A Sb B ,→ PSh (B ). /A Sb /A Lemma 2.4.2. For every A ∈ B, the equivalence PSh (B ) ' PSh (B) restricts to an equivalence Sb /A Sb /A

Bd/A ' Bb/A.

Proof. In the commutative diagram

B PSh (B ) d/A Sb /A

ϕ '

B B PSh (B) /A b/A Sb /A the task is to ﬁnd the dashed arrow ϕ that completes the diagram and to show that this functor is an equivalence of ∞-categories. To that end, note that one has Bd/A ' InddU(B/A), which by [Lur09a, Proposi- tion 5.3.5.10] implies that composition with the Yoneda embedding gives rise to an equivalence

U Fun (Bd/A, D) ' Fun(B/A, D) for any ∞-category D which admits U-small filtered colimits. Here the left-hand side denotes the ∞- category of U-continuous functors, i.e. of those functors that commute with U-filtered colimits. This result implies that the functor ϕ in the diagram above is well-defined and makes the diagram indeed commute. By construction, ϕ must be fully faithful. On the other hand, combining the equivalence of ∞-categories

Bb ' InddU(B) with the fact that the projection Bb/A → Bb creates colimits shows that every object in Bb/A is

obtained as the colimit of a functor J → B/A → Bb/A where J is a U-ﬁltered ∞-category. Since ϕ commutes with U-ﬁltered colimits, this shows that this functor is essentially surjective. Proposition 2.4.3. There is a canonical equivalence

Bd/− ' Bb/− of functors Bop → Cat(bS).

Proof. By lemma 2.4.2, the functorial equivalence PSh (B ) ' PSh (B) Sb /− Sb /− restricts objectwise to an equivalence on the level of sheaves, hence the result follows. We ﬁnish this section by discussing the preservation of structure under universe enlargement:

Proposition 2.4.4. For any ∞-topos B the inclusion B ,→ Bb commutes with small limits and colimits, and the internal mapping bifunctor on Bb restricts to the internal mapping bifunctor on B.

Proof. Since the Yoneda embedding h: B ,→ PSh (B) commutes with small limits and since B is a local- Sb b isation of PSh (B), the embedding B ,→ B preserves small limits. The case of small colimits is proved Sb b in [Lur09a, Remark 6.3.5.17]. Lastly, since the product bifunctor on Bb restricts to the product bifunctor on B, it suﬃces to show that for A, B ∈ B ,→ Bb their internal mapping object [A, B] ∈ Bb is contained in B. Now B is a left exact localisation of PSh (B) and therefore an exponential ideal in PSh (B), hence it suﬃces b Sb Sb to show that the internal mapping object [h(A), h(B)] ∈ PSh (B) is representable. This follows from the Sb computation

[h(A), h(B)] ' mapPSh (B)(h(−) × h(A), h(B)) ' mapB(− × A, B) ' mapB(−, [A, B]) Sb 8 LOUIS MARTINI

in which we make repeated use of the Yoneda lemma and in which the object [A, B] ∈ B on the right-hand side denotes the internal mapping object in B. 2.5. Factorisation systems. Let C be an ∞-category. Given two maps f : a → b and g : c → d in C, we say that f and g are orthogonal if the commutative square

g∗ mapC(b, c) mapC(b, d)

f ∗ f ∗ g∗ mapC(a, c) mapC(a, d) is cartesian. We denote the orthogonality relation between f and g by f⊥g, and we will say that f is left orthogonal to g and g is right orthogonal to f. In particular, f and g being orthogonal implies that any lifting square a c f g b d has a unique solution. If C is cartesian closed, we furthermore say that f and g are internally orthogonal if the square g [b, c] ∗ [b, d]

f ∗ f ∗ g∗ [a, c] [a, d] is cartesian. By deﬁnition, this is equivalent to c × f⊥g for every c ∈ C. We will denote the internal

orthogonality relation between f and g by f g . If C has a terminal object 1 ∈ C, then an object c ∈ C is said to be local with respect to the map f : a → b

in C if the terminal map πc : c → 1 is right orthogonal to f, i.e. if f⊥πc holds. Similarly, c is internally local

with respect to f if f πc holds. If S is an arbitrary family of maps in C, we will denote by S⊥ the collection of maps that are right orthogonal to any map in S, and by ⊥S the collection of maps that are left orthogonal to any map in S.

Deﬁnition 2.5.1. Let C be an ∞-category. A factorisation system is a pair (L, R) of families of maps in C such that (1) Any map f in C admits a factorisation f ' rl with r ∈ R and l ∈ L. (2) L⊥ = R as well as ⊥R = L.

The following proposition summarizes some properties of factorisation systems whose proof is a straightforward consequence of the deﬁnition:

Proposition 2.5.2. Let C be an ∞-category and let (L, R) be a factorisation system in C. Then (1) The intersection L ∩ R is precisely the collection of equivalences in C; (2) if g ∈ L, then fg ∈ L if and only f ∈ L; dually, if f ∈ R then fg ∈ R if and only if g ∈ R; (3) R is stable under pullbacks and L is stable under pushouts; (4) both R and L are stable under taking retracts. (5) R is stable under all limits that exist in Fun(∆1, C), and dually L is stable under all colimits that 1 exist in Fun(∆ , C). If (L, R) is a factorisation system in an ∞-category C, then R deﬁnes a full subcategory of the arrow ∞-category Fun(∆1, C). The factorisation of maps in C then deﬁnes a left adjoint to this inclusion [Lur09a, Lemma 5.2.8.19]. More precisely, if f : d → c is a map in C and if rl : d → e → c is the factorisation of f into YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 9

maps l ∈ L and r ∈ R, then the assignment f 7→ r extends to a functor Fun(∆1, C) → R that is left adjoint to the inclusion. The unit of this adjunction is then given by the square d l e f r c id c. By dualization, this also shows that the inclusion L ,→ Fun(∆1, C) admits a right adjoint. Note that the fact that the inclusion R ,→ Fun(∆1, C) admits a left adjoint moreover proves that for any

c ∈ C the induced inclusion R/c ,→ C/c is reﬂective. If f : d → c is an object in C/c and if f ' rl is its factorisation, then the map r is the image of f under the localisation functor and the map l is the component of the counit of the adjunction at f. In particular, this shows that the map f : d → c is contained in L if

and only if it is sent to the terminal object by the localisation functor C/c → R/c, and the essential image of

the inclusion R/c ,→ C/c is spanned by those objects in C/c that are local with respect to the class of maps

in C/c that are sent to L via the projection functor (πc)! : C/c → C. Remark 2.5.3. Suppose that (L, R) is a factorization system in an ∞-category C that has a ﬁnal object

1 ∈ C, and let f : c → d be a map in C. Let L: C → R/1 be a left adjoint to the inclusion, and consider the commutative diagram f c d

L(f) L(c) L(d)

1 id 1

in which the two vertical compositions are determined by the factorisation of the two terminal maps πc : c → 1

and πd : d → 1 into maps in L and R. If f is contained in L, then item (2) of proposition 2.5.2 implies that L(f) must be contained in L as well. On the other hand, item (2) of proposition 2.5.2 also implies that L(f) is contained in R. Hence L(f) must be an equivalence. In other words, the functor L sends maps in

L to equivalences in R/1. The converse is however not true in general, i.e. not every map that is sent to an equivalence by L must necessarily be contained in L. A notable exception is the case where πd : d → 1 is already contained in L. In this case, the map d → L(d) is an equivalence, hence L(f) being an equivalence does imply that f is contained in L. Lastly, let us discuss how a factorisation system can be generated by a set of maps: Proposition 2.5.4 ([Lur09a, Proposition 5.5.5.7]). Let C be a presentable ∞-category and let S be a small ⊥ ⊥ set of maps in C. Then there is a factorisation system (L, R) in C with R = S and L = R. In the situation of proposition 2.5.4, the assignment S 7→ L can be viewed as a certain closure operation that is referred to as saturation. Recall the deﬁnition of a saturated class: Deﬁnition 2.5.5. Let C be a presentable ∞-category and let S be a class of maps in C. Then S is saturated if (1) S contains all equivalences in C and is closed under composition; (2) S is closed under small colimits in Fun(∆1, C); (3) S is closed under pushouts. By proposition 2.5.2, the left class in any factorisation system is saturated. Now if S is a small set of maps and if (L, R) is the induced factorisation system in C as provided by proposition 2.5.4, then L is the universal saturated class that contains S, in the following sense: 10 LOUIS MARTINI

Proposition 2.5.6. Let C be a presentable ∞-category and let S be a small set of maps in C. Let (L, R) be the associated factorisation system. Then L is the smallest saturated class of maps that contains S.

Proof. To begin with, note that the property of a class being saturated is preserved under taking arbitrary intersections, hence the smallest saturated class containing S is well-deﬁned and is explicitly given by the intersection \ S = T S⊂T over all saturated classes of maps that contain S. We need to show that any saturated T ⊃ S contains L as well. But since L = ⊥R and as (L, R) is a factorisation system, this is equivalent to T ⊥ ⊂ R, which in turn ⊥ follows immediately from S ⊂ T and S = R. An analogous construction can be carried out when replacing orthogonality by internal orthogonality in the case where C is cartesian closed:

Proposition 2.5.7 ([ABFJ20, Proposition 3.2.9]). Let C be a presentable and cartesian closed ∞-category and let S be a small set of maps in C. Then there is a factorisation system (L, R) in C such that R = S ⊥ and L = R = R. Since a map r in a cartesian closed ∞-category C is internally right orthogonal to a map l if and only if r is right orthogonal to c × l for any c ∈ C, proposition 2.5.4 implies:

Proposition 2.5.8. Let C be a presentable and cartesian closed ∞-category and let S be a small set of maps in C. Let (L, R) be the factorisation system provided by proposition 2.5.7. Then L is the smallest saturated class of maps that contains the set {c × f | c ∈ C, f ∈ S}. Example 2.5.9. Let C be a presentable and cartesian closed ∞-category. We say that a map f : c → d is a monomorphism if it is internally right orthogonal to the codiagonal 1 t 1 → 1, where 1 is the ﬁnal object in C. By construction, a map in C is a monomorphism if and only if its diagonal is an equivalence. Dually, we say that a map is a strong epimorphism if it is internally left orthogonal to every monomorphism. By proposition 2.5.7, one obtains a factorisation system in which the left class are strong epimorphisms and the right class are monomorphisms. If C is an ∞-topos, then strong epimorphisms are precisely covers, i.e. eﬀective epimorphisms in the terminology of [Lur09a]. This follows from the fact that covers and monomorphisms form a factorization system in any ∞-topos [Lur09a, Example 5.2.8.16], combined with the fact that the left class of a factorization system is uniquely determined by the right class.

3. Categories in an ∞-topos In this chapter, we introduce the language and some basic concepts of the theory of categories in an ∞-topos B. We will confine ourselves to discussing only those aspects of the theory that will be needed for the discussion of the Grothendieck construction and Yoneda’s lemma in chapter4. Sections 3.1 to 3.5 set up the general framework and discuss basic properties of the ∞-category Cat(B) of categories in B. Section 3.6 features a brief discussion of the objects and morphisms of a category in B. In section 3.7 we define and investigate the universe Ω for groupoids in B, and finally sections 3.8 and 3.9 contain a discussion of fully faithful functors and full subcategories.

3.1. Simplicial objects in an ∞-topos. Let B be an arbitrary ∞-topos and let B∆ denote the ∞-topos of simplicial objects in B. By postcomposition with the adjunction (const a Γ): B → S one obtains an

induced adjunction (const a Γ): B∆ → S∆ on the level of simplicial objects. This deﬁnes a functor

(−)∆ : RTop → RTop/S∆ YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 11

from the ∞-category of ∞-topoi with geometric morphisms as maps to the slice ∞-category of ∞-topoi over

S∆.

We deﬁne the tensoring of B∆ over S∆ as the bifunctor

− ⊗ −: S∆ × B∆ → B∆

that is given by the composition (− × −) ◦ (const × idB∆ ). Dually, we deﬁne the powering of B∆ over S∆ as the bifunctor (−) op (−) : S∆ × B∆ → B∆

that is given by the composition [−, −] ◦ (const × idB∆ ), where [−, −] denotes the internal mapping object op in B∆. Let FunB(−, −): B∆ × B∆ → S∆ be the bifunctor given by Γ ◦ [−, −]. We then obtain equivalences (−) mapB∆ (−, (−) ) ' mapB∆ (− ⊗ −, −) ' mapS∆ (−, FunB(−, −)). Remark 3.1.1. For any object A in the ∞-topos B, we may regard A as a constant simplicial object via

the diagonal functor B ,→ B∆. Since products in B∆ are computed objectwise, the endofunctor A × − on

B∆ is equivalent to the functor that is given by postcomposing simplicial objects with the product functor A × −: B → B. The latter admits a right adjoint [A, −]: B → B, and since postcomposition with an adjunction induces an adjunction on the level of functor ∞-categories, the uniqueness of adjoints implies

that the internal mapping objects functor [A, −]: B∆ → B∆ is obtained by applying the internal mapping objects functor of B levelwise to simplicial objects in B. More precisely, the restriction of the internal op op mapping object bifunctor [−, −] on B∆ along the inclusion B × B∆ ,→ B∆ × B∆ is equivalent to the transpose of the composite functor

op op idBop × ev∆op op [−,−] B × B∆ × ∆ B × B B

op op in which ev∆op denotes the evaluation functor (i.e the counit of the adjunction − × ∆ a Fun(∆ , −)) and in which the second functor is the internal mapping object bifunctor on B.

In particular, this argument shows that the internal mapping objects bifunctor on B∆ restricts to the internal mapping objects bifunctor on B, which justiﬁes our choice of notation.

Observe that the diagonal functor ι: B ,→ B∆ admits a right adjoint that is given by the evaluation • functor (−)0 : B∆ → B. Let us denote by ∆ : ∆ ,→ S∆ the Yoneda embedding. Restricting the powering • ∆• bifunctor along ∆ then deﬁnes a functor (−) : B∆ → PShB∆ (∆). The computation

∆• ∆• mapB(−, ((−) )0) ' mapB∆ (ι(−), (−) ) • ' mapS∆ (∆ , FunB∆ (ι(−), −))

' Γ[ι(−), (−)•]

' mapB(−, (−)•) in which the penultimate equivalence follows from Yoneda’s lemma and remark 3.1.1 now shows :

Proposition 3.1.2. The composite functor

∆• (−) (−)0 B∆ PShB∆ (∆) B∆ in which the second arrow denotes postcomposition with the evaluation functor (−)0 : B∆ → B is equivalent to the identity functor on B∆. Lemma 3.1.3. Let S be a saturated class of maps in B that contains the maps s0 : ∆1 ⊗ C → C for every n C ∈ B∆. Then S contains the projection ∆ ⊗ A → A for every n ≥ 0 and every A ∈ B. 12 LOUIS MARTINI

Proof. We will use induction on n, the case n = 1 being true by assumption. Let us therefore assume that for an arbitrary integer n ≥ 1 the projection s0 : ∆n ⊗ A → A is contained in S. Then the composition (∆1 × ∆n) ⊗ A → ∆n ⊗ A → A in which the first map is induced by s0 : ∆1 → ∆0 is contained in S as well. Let α: ∆n+1 → ∆1 × ∆n be the map that is defined by α(0) = (0, 0) and α(k) = (1, k − 1) for all 1 ≤ k ≤ n + 1, and let β : ∆1 × ∆n → ∆n+1 be defined by β(0, k) = 0 and β(1, k) = k + 1 for all 0 ≤ k ≤ n. Then the composition βα is equivalent to the identity on ∆n+1, and we therefore obtain a retract diagram

β⊗id ∆n+1 ⊗ A α⊗id (∆1 × ∆n) ⊗ A ∆n+1 ⊗ A

A id A id A n+1 which shows that the map ∆ ⊗ A → A is contained in S as well.

Proposition 3.1.4. For any simplicial object C ∈ B∆, the following are equivalent:

(1) C is contained in the essential image of the diagonal functor ι: B ,→ B∆. (2) C is internally local with respect to the map s0 : ∆1 → ∆0.

Proof. If C is internally local with respect to s0 : ∆1 → ∆0, then lemma 3.1.3 and proposition 3.1.2 imply

that the simplicial maps C0 → Cn are equivalences in B for all n ≥ 0, which implies that C is a constant simplicial object and therefore in the essential image of the diagonal functor. Conversely, let A ∈ B be an arbitrary object. By making use of the adjunction colim∆op a ι and the fact that the functor colim∆op commutes with finite products as ∆op is a sifted category, the object ι(A) is internally local to s0 whenever 0 1 0 A is internally local to colim∆op (s ): colim∆op ∆ → colim∆op ∆ in B. Since the colimit of a representable presheaf on a small ∞-category is always the final object 1 ∈ S, the latter map must be an equivalence, hence the result follows. Definition 3.1.5. A simplicial object in B is said to be a groupoid in B if it satisfies the equivalent conditions

from proposition 3.1.4. We denote by Grpd(B) ,→ B∆ the full subcategory spanned by the groupoids in B.

3.2. Categories in an ∞-topos. We now proceed by deﬁning a category in an ∞-topos B to be a simplicial object in B that satisﬁes the Segal conditions and univalence. To that end, we will make use of the following combinatorial constructions:

n 1 1 n n Deﬁnition 3.2.1. For any n ≥ 1, let I = ∆ t∆0 · · ·t∆0 ∆ ⊂ ∆ denote the spine of ∆ , i.e the simplicial sub-∞-groupoid of ∆n that is spanned by the inclusions d{i−1,i} : ∆1 ⊂ ∆n for i = 1, . . . , n. Furthermore, let E1 be the simplicial ∞-groupoid that is deﬁned by the pushout square

0 0 ∆1 t ∆1 s ts ∆0 t ∆0

d{0,2}td{1,3} y ∆3 E1. We refer to E1 as the walking equivalence.

Remark 3.2.2. Many authors define the walking equivalence as the simplicial set that arises as the nerve of the category with two objects and a unique isomorphism between them. The simplicial set E1 from definition 3.2.1 is different in that it is comprised of a map together with separate left and right inverses.

Remark 3.2.3. In the situation of definition 3.2.1, the colimits that define In and E1 can be computed as the ordinary (i.e 1-categorical) pushouts in the 1-category of simplicial sets. Indeed, colimits in S∆ are computed levelwise and so are ordinary colimits in Set∆, hence it suffice to show that for any integer k ≥ 0 n 1 the colimits in S of the diagrams in Set ,→ S that define Ik and Ek can be computed by the ordinary colimits YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 13

of these diagrams in Set. As the Quillen model structure on the 1-category of simplicial sets is left proper, ordinary pushouts along monomorphisms of simplicial sets present homotopy pushouts in S. Now In is an n iterated pushout along monomorphisms in S∆, hence Ik is an iterated pushout along monomorphisms in Set ,→ S, hence the claim follows for In. Regarding the simplicial ∞-groupoid E1, note that E1 ﬁts into the commutative diagram

∆1 ∆0

d{1,3} {0,2} y 1 d 3 3 0 ∆ ∆ ∆ t∆1 ∆

y 0 0 3 y 1 ∆ ∆ t∆1 ∆ E

of simplicial ∞-groupoids. Since the pushouts in the lower left and in the upper right corner are computed by the ordinary pushouts of simplicial sets, the claim now follows from the straightforward observation that 1 3 3 0 the composition ∆ ,→ ∆ → ∆ t∆1 ∆ is a monomorphism.

Deﬁnition 3.2.4. An object C ∈ B∆ is a category in B if (Segal conditions) C is internally local with respect to the map I2 ,→ ∆2, and (univalence) C is internally local with respect to the map E1 → ∆0.

We denote the full subcategory of B∆ that is spanned by the categories in B by Cat(B).

2 2 Lemma 3.2.5. The saturation of the set of maps {I ⊗ K,→ ∆ ⊗ K | K ∈ B∆} is equal to the saturation of the set of maps {In ⊗ A,→ ∆n ⊗ A | n ≥ 0,A ∈ B}.

2 Proof. It suﬃces to show that a map f : C → D in B∆ is right orthogonal to all maps of the form I ⊗ K,→ 2 n n ∆ ⊗ K with K ∈ B∆ if and only if f is right orthogonal to all maps of the form I ⊗ A,→ ∆ ⊗ A with n ≥ 0 and A ∈ B. This means that we need to show that for every A ∈ B the map of simplicial ∞-groupoids n n f∗ : FunB(A, C) → FunB(A, D) that is induced by f is right orthogonal to the maps I ,→ ∆ for all n ≥ 0 2 2 if and only if the map f∗ : FunB(K,C) → FunB(K,D) is right orthogonal to the inclusion I ,→ ∆ for every simplicial object K in B. Let C be a small ∞-category such that B arises as a left exact localisation

of PShS(C). Then B∆ is a localisation of PShS(∆ × C), which means that any simplicial object K ∈ B∆ is a small colimit of objects in ∆ × C. We may therefore assume without loss of generality that K is of the k k form ∆ ⊗ A for some A ∈ B and some k ≥ 0. In this case, the map f∗ : FunB(∆ ⊗ A, C) → FunB(A, D) 2 2 is right orthogonal to the inclusion I ,→ ∆ if and only if the map f∗ : FunB(A, C) → FunB(A, D) is right orthogonal to the map I2 × ∆k ,→ ∆2 × ∆k. As a consequence, we may assume without loss of generality that B ' S, that A ' 1 and that K ' ∆k for an arbitrary k ≥ 0. Let us ﬁrst show that any saturated class S of maps containing the inclusions I2 × ∆k ,→ ∆2 × ∆k for arbitrary k ≥ 0 contains all spine inclusions In ,→ ∆n. Since the boundary ∂∆k can be obtained as a colimit of simplices, the set S contains all maps of the form I2 × ∂∆k ,→ ∆2 × ∆k and therefore all maps of the form

2 k 2 k 2 k (I × ∆ ) tI2∂∆k (∆ × ∆ ) ,→ ∆ × ∆

k k as well. By [Lur09a, Proposition 2.3.2.1], this implies that the set of inner horn inclusions Λi ,→ ∆ for all k ≥ 0 and all 0 < i < n are contained in S too. As by [Joy08, Proposition 2.13] the spine inclusions In ,→ ∆n are contained in the saturation of the set of inner horn inclusions, this shows that these must be elements of S. 14 LOUIS MARTINI

Conversely, let T be any saturated class of maps containing all spine inclusions In ,→ ∆n. We need to show that T contains the maps I2 × ∆k ,→ ∆2 × ∆k for k ≥ 0. Consider the commutative square

I2 × Ik ∆2 × Ik

I2 × ∆k ∆2 × ∆k. In order to show that the lower horizontal map is contained in T we only need to show that the remaining k 1 1 three maps are elements of T , and by making use of the decomposition I ' ∆ t∆0 · · · t∆0 ∆ this follows once we show that the maps In × ∆1 ,→ ∆n × ∆1 are contained in T , which was shown by Rezk in [Rez01, Lemma 10.3]. Lemma 3.2.6. Let S be a strongly saturated class of maps in B that contains the maps E1 ⊗ A → A and n n 1 I ⊗ A,→ ∆ ⊗ A for all A ∈ B and all n ≥ 0. Then S contains the maps E ⊗ C → C for all C ∈ B∆.

Proof. Using a similar argumentation as in lemma 3.2.5, we may assume without loss of generality that B ' S, that A is the ﬁnal simplicial ∞-groupoid and furthermore that C ' ∆n. As furthermore by lemma 3.2.5 the map E1×In ,→ E1×∆n is contained in S for all n ≥ 0, we need only show that the induced map E1×In → In is contained in S. Using that In is a colimit of a diagram involving only ∆0 and ∆1, we may further restrict to the case n = 1, in which case the statement was proven by Rezk in [Rez01, Proposition 12.1]. Proposition 3.2.7. Let C be a simplicial object in B. The following conditions are equivalent: (1) C is a category in B; (2) for all n ≥ 2 the maps

Cn → C1 ×C0 · · · ×C0 C1 as well as the map

C0 → (C0 × C0) ×C1×C1 C3 are equivalences.

Proof. A simplicial object C in B satisﬁes the Segal condition if and only if it is local with respect to the 2 2 collection of maps I ⊗ E,→ ∆ ⊗ E for any simplicial object E ∈ B∆. On the other hand, the ﬁrst map of condition (2) is an equivalence if and only if C is local with respect to all maps In ⊗ A,→ ∆n ⊗ A for arbitrary A ∈ B. By lemma 3.2.5, these two conditions are equivalent. Similarly, lemma 3.2.6 implies that C is univalent if and only if the second map of condition (2) is an equivalence. Remark 3.2.8. Proposition 3.2.7 allows us to make sense of the notion of a category in C for any ∞-category

C with finite limits. That is, we may define a category in C to be a simplicial object C ∈ C∆ that satisfies the second condition of proposition 3.2.7.

Since Cat(B) is a localisation of B∆ at a small set of morphisms, one ﬁnds:

Proposition 3.2.9. The inclusion Cat(B) ,→ B∆ exhibits Cat(B) as an accessible localisation of B∆. In particular, the ∞-category Cat(B) is presentable.

Remark 3.2.10. As Cat(B) is an accessible localisation of B∆, the inclusion Cat(B) ,→ B∆ creates small limits. On the other hand, colimits of small diagrams are usually not created (or even preserved) by the inclusion, as these can be computed by applying the localisation functor B∆ → Cat(B) to the colimits of the underlying diagrams of simplicial objects in B. Filtered colimits, on the other hand, are created by the inclusion Cat(B) ,→ B∆. In fact, as such colimits commute with ﬁnite limits, it is immediate from proposition 3.2.7 that the colimit in B∆ of a small ﬁltered diagram of categories in B is contained in Cat(B) YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 15

and is therefore the colimit of this diagram in Cat(B). In particular, as colimits are universal in B∆, this observation implies that ﬁltered colimits are universal in Cat(B). As categories in B are internally local with respect to the maps I2 ,→ ∆2 and E1 → 1, the ∞-category Cat(B) is cartesian closed. More precisely, the construction of Cat(B) implies:

Proposition 3.2.11. The full subcategory Cat(B) ,→ B∆ is an exponential ideal and therefore in particular a cartesian closed ∞-category. In other words, if C is a category in B and D is a simplicial object in B, the internal mapping object [D, C] is a category in B.

Corollary 3.2.12. The localisation functor B∆ → Cat(B) preserves ﬁnite products.

Recall that the inclusion ι: B ,→ B∆ admits a right adjoint (−)0 and a left adjoint colim∆op (−). It is

moreover immediate from proposition 3.2.7 that ι factors through the inclusion Cat(B) ,→ B∆, i.e. that for any object A ∈ B the associated groupoid in B deﬁnes a category in B. Therefore, we obtain functors

(−)' > Grpd(B) Cat(B). > (−)gpd The functor (−)' is referred to as the core groupoid functor, and the functor (−)gpd is referred to as the groupoidification functor. If C is a category in B, the groupoid C' is to be thought of as the maximal groupoid that is contained in C (i.e. the subcategory spanned by all objects and equivalences in C), whereas the groupoid Cgpd should be regarded as the result of formally inverting all morphisms in C. Proposition 3.2.13. The groupoidification functor (−)gpd : Cat(B) → Grpd(B) commutes with small colimits and finite products. Proof. As the groupoidification functor is a left adjoint, it commutes with small colimits. Moreover, since the final object 1 ∈ Cat(B) is given by the constant simplicial object on 1 ∈ B and since groupoidification is a localisation functor, there is an equivalence 1gpd ' 1. It therefore suffices to consider the case of binary op products, which follows from ∆ being a sifted ∞-category. Proposition 3.2.14. The core groupoid functor (−)' : Cat(B) → Grpd(B) commutes with small filtered colimits and small limits. Proof. As the core groupoid functor is a right adjoint, it commutes with small limits. By 3.2.10, small filtered

colimits between categories in B are created by the inclusion Cat(B) ,→ B∆, hence it suﬃces to show that (−)0 : B∆ → B commutes with such colimits, which is immediate. Example 3.2.15. Proposition 3.2.7 shows that a category in S is precisely a complete Segal space as developed by Rezk [Rez01]. By a theorem of Joyal and Tierney [JT07] the ∞-category of complete Segal

spaces is a model for the ∞-category of ∞-categories Cat(S), which justiﬁes our notation. If Cat∞ denotes

any other model for the ∞-category of ∞-categories and if ∆ ,→ Cat∞ denotes the inclusion that is given by considering each n ∈ ∆ as a category and therefore a fortiori as an ∞-category, the equivalence Cat(S) '

Cat∞ is necessarily obtained by restricting the functor S∆ → Cat∞ that is induced by the embedding

∆ ,→ Cat∞ along the inclusion Cat(S) ,→ S∆. As the functor S∆ → Cat∞ preserves ﬁnal objects and small colimits, there is a commutative triangle ' Cat(S) Cat∞

ι S 16 LOUIS MARTINI

in which the left diagonal arrow is the functor that is induced by the diagonal functor S ,→ S∆ and the right diagonal arrow is the natural inclusion. Therefore the associated diagrams on right and left adjoints,

respectively, commute as well, i.e. the equivalence Cat(S) ' Cat∞ commutes with both the formation of core ∞-groupoids and groupoidiﬁcation.

3.3. Functoriality and base change. In this section we discuss how to change the base ∞-topos for internal higher category theory. What makes this possible is the following general lemma:

Lemma 3.3.1. Let F : C → D be a functor between ∞-categories that admit ﬁnite limits such that F preserves pullbacks and such that for any map f : c → c0 in C the commutative square

F (f×f) F (c × c) F (c0 × c0)

F (f)×F (f) F (c) × F (c) F (c0) × F (c0) is cartesian. Then the induced functor C∆ → D∆ that is given by postcomposition with F sends categories in C to categories in D and therefore restricts to a functor

F : Cat(C) → Cat(D).

In particular, any left exact functor F between ﬁnitely complete ∞-categories induces a functor on the level of categories.

Proof. Since F preserves pullbacks the functor C∆ → D∆ that is given by postcomposition with F preserves the Segal conditions, and by assumption on F the commutative square

F (C0) F (C3) p (d{0,2},d{1,3})

(s0,s0) F (C0) × F (C0) F (C1) × F (C1)

is cartesian for every category C in C, hence the claim follows.

Let Z denote the subcategory of Cat(bS) spanned by the ∞-categories that admit finite limits, together R with those functors that satisfy the conditions of lemma 3.3.1. Let (−)∆ → Z be the cocartesian fibration R that classifies the functor (−)∆ : Z → Cat(bS). Lemma 3.3.1 then implies that the full subcategory of (−)∆ that is spanned by the pairs (C, C) with C ∈ Z and C a category in C is stable under taking cocartesian R arrows and therefore defines a cocartesian subfibration of (−)∆ over Z. Hence one obtains a functor

Cat: Z → Cat(bS).

Since both the forgetful functor RTop → Cat(bS) and the universe enlargement functor B 7→ Bb for ∞-topoi factor through Z and since moreover the inclusion B ,→ Bb commutes with small limits, postcomposition with the functor Cat gives rise to functors B 7→ Cat(B) as well as B 7→ Cat(Bb) together with a natural transformation B7→Cat(B)

RTop Cat(bS)

B7→Cat(Bb ) YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 17

that is given by the inclusion Cat(B) ,→ Cat(Bb). Analogously, one obtains two functors LTop ⇒ Cat(bS) together with a natural transformation

B7→Cat(B)

LTop Cat(bS).

B7→Cat(Bb )

As the inclusion C ,→ C∆ and its right adjoint (−)0 : C∆ → C are clearly functorial in C ∈ Z, we may now conclude:

Proposition 3.3.2. There are two commutative squares

Grpd(B) Grpd(Bb)

' (−)' (−)

Cat(B) Cat(Bb) that are functorial in B both with respect to maps in RTop and maps in LTop.

Let W be the full subcategory of Z that is spanned by the ∞-categories that admit colimits indexed by op ∆ and the functors that preserve such colimits. Then the map colim∆op : C∆ → C is functorial in C ∈ W, and as universe enlargement preserves small colimits one ﬁnds:

Proposition 3.3.3. There is a commutative square

Grpd(B) Grpd(Bb)

gpd (−)gpd (−)

Cat(B) Cat(Bb) that is functorial in B with respect to maps in LTop.

Deﬁnition 3.3.4. A category in Bb is said to be small if it is contained in the essential image of the inclusion Cat(B) ,→ Cat(Bb) and large otherwise.

Base change along étale geometric morphisms is particularly well-behaved, as we will discuss now. Let op Cat(B)/− : B → Cat(bS) be the functor that classifies the cartesian fibration

1 Fun(∆ , Cat(B)) ×Cat(B) B → B.

One now ﬁnds:

Proposition 3.3.5. There is an equivalence

Cat(B/−) ' Cat(B)/− of functors Bop → Cat(bS) that ﬁts into a commutative square

Cat(B/−) Cat(Bd/−)

' '

Cat(B)/− Cat(Bb)/− 18 LOUIS MARTINI

1 1 Proof. On account of the natural equivalence Fun(∆ , B∆) ' Fun(∆ , B)∆ and by lemma 3.7.3 and remark 3.5.7 below, one obtains an equivalence

(B/−)∆ ' (B∆)/−

op of functors B → Cat(bS). In order to provide an equivalence Cat(B/−) ' Cat(B)/−, it therefore suﬃces

to show that this equivalence of functors sends categories in B/A to objects in (B∆)/A whose underlying simplicial object in B is a category. By construction, the component of the above equivalence at A ∈ B sits inside the commutative diagram ' (B/A)∆ (B∆)/A

(πA)!◦(−) (πA)! B∆,

hence the claim follows from the fact that the forgetful functor (πA)! : B/A → B satisﬁes the conditions of lemma 3.3.1. Lastly, the existence of a commutative square as in the statement of the proposition follows

from the construction of the equivalence Cat(B/−) ' Cat(B)/− and lemma 2.4.2.

3.4. The (∞, 2)-categorical structure of Cat(B). For any ∞-topos B, we denote by

[−, −]: Cat(B)op × Cat(B) → Cat(B)

the internal mapping bifunctor in Cat(B). By proposition 3.2.11 this bifunctor is obtained by restricting

the internal mapping bifunctor of B∆ to the full subcategory Cat(B) ,→ B∆. As the product bifunctor on Cat(B) is obtained in the same fashion, the three bifunctors that are deﬁned in the beginning of section 3.1 restrict to bifunctors on the level of categories in B and in S. Explicitly, one obtains a tensoring bifunctor

− ⊗ −: Cat(S) × Cat(B) → Cat(B) which is given by the composition (− × −) ◦ (const × idCat(B)), a powering bifunctor

(−)(−) : Cat(S)op × Cat(B) → Cat(B)

that is given by the composition [−, −] ◦ (const × idCat(B)), and a functor ∞-category bifunctor

op FunB(−, −): Cat(B) × Cat(B) → Cat(S) which is deﬁned as the composition Γ ◦ [−, −]. These functors are equipped with equivalences

(−) mapCat(B)(−, (−) ) ' mapCat(B)(− ⊗ −, −) ' mapCat(S)(−, FunB(−, −)).

In particular, the second equivalence implies that postcomposing FunB(−, −) with the core ∞-groupoid ' functor (−) : Cat(S) → S recovers the mapping ∞-groupoid bifunctor mapCat(B)(−, −). The above constructions are well-behaved with respect to universe enlargement:

Proposition 3.4.1. The internal mapping bifunctor on Cat(Bb) restricts to the internal mapping bifunctor on Cat(B).

Proof. As the product bifunctor on Cat(Bb) clearly restricts to the product bifunctor on Cat(B), it suffices to show that for any two small categories C, D ∈ Cat(B) ,→ Cat(Bb) their internal mapping object [C, D] in Cat(Bb) is small as well. It suffices to show this on the level of simplicial objects, i.e. we need to show that the simplicial object [C, D] ∈ Bb ∆ is contained in B∆. Using proposition 3.1.2 one finds

n [C, D]n ' [∆ ⊗ C, D]0, YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 19

hence it suﬃces to show that [C, D]0 is contained in B. As in the proof of proposition 3.2.7, using that the n functor [−, D]0 sends colimits in Bb ∆ to limits in Bb, we may assume without loss of generality C ' ∆ ⊗ A. In this case one computes n [∆ ⊗ A, D]0 ' [A, D]n ' [A, Dn] in which the last step follows from remark 3.1.1, therefore the claim is a consequence of proposition 2.4.4. Combining proposition 3.4.1 with proposition 3.3.2, one now easily deduces:

Corollary 3.4.2. The tensoring, powering and mapping ∞-category bifunctors on Cat(Bb) restrict to the tensoring, powering and mapping ∞-category bifunctors on Cat(B).

Remark 3.4.3. The bifunctor FunB(−, −) gives rise to an (∞, 2)-categorical enhancement of Cat(B). More precisely, on account of Cat(B) being cartesian closed, this ∞-category is canonically enriched over itself [GH15, Section 7]. The functor Γ: Cat(B) → Cat(S) can then be used to change enrichment from Cat(B) to Cat(S), so that Cat(B) becomes a Cat(S)-enriched ∞-category, which is one of the known models for (∞, 2)-categories [Hau15].

3.5. Cat(S)-valued sheaves on an ∞-topos. Categories in an ∞-topos B can be alternatively regarded as sheaves of ∞-categories on B. To see this, ﬁrst recall from the discussion in section 2.4 that the Yoneda embedding induces a commutative square

B Bb

PSh (B) PSh (B) S Sb that is natural in B both with respect to maps in RTop and in LTop. Postcomposition with the functor

(−)∆ therefore gives rise to a natural commutative square

B∆ Bb ∆

PShS (B) PSh (B) ∆ Sb∆

in which the essential image of the two vertical maps is spanned by the collection of S∆-valued and bS∆-valued sheaves, respectively. Using proposition 3.2.7 it is immediate that this square further restricts to a natural commutative square Cat(B) Cat(Bb)

PSh (B) PSh (B). Cat(S) Cat(Sb) As limits in Cat(S) and in Cat(bS) are computed on the level of the underlying simplicial ∞-groupoids, the essential image of the two vertical maps is spanned by the collection of Cat(S)-valued and Cat(bS)-valued sheaves, respectively. One therefore obtains:

Proposition 3.5.1. The inclusions Cat(B) ,→ PSh (B) and Cat(B) ,→ PSh (B) induce a commu- Cat(S) b Cat(Sb) tative square Cat(B) Cat(Bb)

' ' Sh (B) Sh (B) Cat(S) Cat(Sb) 20 LOUIS MARTINI that is natural in B both with respect to maps in RTop and in LTop.

Remark 3.5.2. In what follows, we will often implicitly identify a category C in B with the associated Cat(S)-valued sheaf on B. In particular, if A ∈ B is an arbitrary object we will write C(A) for the ∞- category of A-sections of C.

Remark 3.5.3. On account of the natural equivalence PShCat(S)(C) ' Cart(C) between the ∞-category of Cat(S)-valued presheaves on a small ∞-category C and the ∞-category of cartesian ﬁbrations over C estab- lished by the Grothendieck construction, the inclusion Cat(B) ,→ PSh (B) gives rise to an embedding b Cat(Sb)

Cat(Bb) ,→ Cart(B) that is natural in B. For any large category C in B we will denote the image of C under this functor by R C → B.

The equivalence in proposition 3.5.1 can also be formulated using the mapping ∞-category bifunctor

FunB(−, −): If ι: B ,→ Cat(B) denotes the natural inclusion, the computation

• ∆• mapCat(S)(∆ , FunB(ι(−), −)) ' mapCat(B)(ι(−), (−) ) ' mapB(−, (−)•) in which the last equivalence follows from proposition 3.1.2 shows that the transpose of the bifunctor

op FunB(ι(−), −): B × Cat(B) → Cat(S)

recovers the natural inclusion Cat(B) ,→ PShCat(S)(B) and in particular the equivalence Cat(B) ' ShCat(S)(B) from proposition 3.5.1. It is therefore reasonable to deﬁne:

Deﬁnition 3.5.4. For any ∞-topos B and any object A ∈ B, the local sections functor over A is deﬁned as

the functor FunB(A, −): Cat(B) → Cat(S).

Remark 3.5.5. In the context of deﬁnition 3.5.4, the local sections functor over an object A ∈ B is equivalently given by the composite

∗ ΓB πA /A Cat(B) Cat(B/A) Cat(S).

∗ In fact, the equivalence of functors − × (πA)!(−) ' (πA)!(πA(−) ×A −) gives rise to the following chain of equivalences

mapCat(B)(− ⊗ (πA)!(−), −) ' mapCat(B)((πA)!(− ⊗ −), −) ' map (− ⊗ −, π∗ (−)) Cat(B/A) A that induces an equivalence of functors

∗ FunB((πA)!(−), −) ' FunB/A (−, πA(−)).

Remark 3.5.6. By construction, the equivalence Cat(B) ' ShCat(S)(B) ﬁts into two commutative squares

' Grpd(B) ShS(B)

(−)' (−)' ' Cat(B) ShCat(S)(B)

that are functorial in B with respect to maps both in RTop and in LTop. Here the two vertical maps on the ' right-hand side are given by postcomposition with the adjunction (ι a (−) ): S Cat(S). One moreover YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 21

has a commutative square ' Grpd(B) ShS(B)

(−)gpd (−)gpd ' Cat(B) ShCat(S)(B) that is functorial in B with respect to maps in LTop, where the vertical map on the right is given by postcomposition with the groupoidiﬁcation functor (−)gpd : Cat(S) → S.

Remark 3.5.7. The fact that the inclusion Cat(B) ,→ PShCat(S)(B) is obtained by the functor FunB(ι(−), −) implies that there is a commutative square

(−)(−) Cat(S)op × Cat(B) Cat(B)

op FunS(−,−)◦(−) Cat(S) × PShCat(S)(B) PShCat(S)(B)

in which the lower horizontal arrow is to be understood as the functor that sends a pair (X,F ) to the presheaf X FunS(X, −) ◦ F . In other words, the Cat(S)-valued presheaf that underlies the powering C of a category

C ∈ Cat(B) by an ∞-category X ∈ Cat(S) is given by the functor FunS(X, C(−)). By contrast, the Cat(S)-valued presheaf that underlies the tensoring X ⊗ C of the category C by the ∞-category X is not given by the functor X × C(−). In fact, the latter is in general not a sheaf. However, one can show that the presheaf that is associated with X ⊗ C is given by the sheaﬁﬁcation of the presheaf

X × C(−), i.e. by the image of X × C(−) under the left adjoint of the inclusion ShCat(S)(B) ,→ PShCat(S)(B).

3.6. Objects and morphisms. Let C be a category in an ∞-topos B, and let A ∈ B be an arbitrary object. An object c of C in context A is deﬁned to be a local section c: A → C, which is equivalently determined by

a map c: A → C0 since A is a groupoid in B.A morphism in C with context A is defined as an object f of ∆1 1 C , i.e. as a map f : ∆ ⊗ A → C, or equivalently as a map f : A → C1. Similarly one defines the notion of an n-morphism for any n ≥ 1 as a map ∆n ⊗ A → C. Any morphism f has a source and a target which are obtained by precomposing f : ∆1 ⊗ A → C with d1 : A → ∆1 ⊗ A and with d0 : A → ∆1 ⊗ A, respectively. If c and d are the source and target of such a morphism f, we also use the familiar notation f : c → d. For any 0 1 object c in C in context A, there is a morphism idc that is defined by the composite cs : ∆ ⊗ A → A → C.

Given two objects c, d in C in context A, we deﬁne the mapping groupoid mapC(c, d) ∈ B/A as the pullback

mapC(c, d) C1

(d1,d0) (c,d) A C0 × C0.

Equivalently, this object can be deﬁned by the pullback

∆1 mapC(c, d) C

(d1,d0) (c,d) A C × C,

see section 4.1 below. By construction, sections A → mapC(c, d) over A correspond to morphisms f : c → d in C in context A. Two maps f, g : c ⇒ d are said to be equivalent if they are equivalent as sections A ⇒ mapC(c, d) over A, in which case we write f ' g. Similarly, f and g are locally equivalent if there is a cover p: B A in B such that the precomposition of f and g with p yields equivalent maps in context B. 22 LOUIS MARTINI

Similarly to the case of two objects, if c0, . . . , cn are objects in context A in C, one writes mapC(c0, . . . , cn) n+1 n+1 for the pullback of (dn, . . . , d0): Cn → C0 along the map (c0, . . . , cn): A → C0 . Using the Segal conditions, one obtains an equivalence

mapC(c0, . . . , cn) ' mapC(c0, c1) ×A · · · ×A mapC(cn−1, cn).

By combining this identiﬁcation with the map mapC(c0, . . . , cn) → mapC(c0, cn) that is induced by the map d{0,n} : Cn → C1, one obtains a composition map

mapC(c0, c1) ×A · · · ×A mapC(cn−1, cn) → mapC(c0, cn).

Given maps fi : ci−1 → ci in C for i = 1, . . . , n, we write f1 ··· fn for their composition. By making use of the simplicial identities, it is straightforward to verify that composition is associative and unital, i.e. that the relations f(gh) ' (fg)h and f id ' f ' id f as well as their higher analogues hold whenever they make sense, see [Rez01, Proposition 5.4] for a proof. A morphism f : c → d in C is an equivalence if there are maps g : c → d and h: c → d (all in context A) 1 1 {1,2} 1 3 such that gf ' idc and fh ' idd. Let ∆ → E be the map that is induced by the inclusion d : ∆ ,→ ∆ . One then obtains the following characterization of equivalences in C:

Proposition 3.6.1. A map f : c → d in context A is an equivalence if and only if the map ∆1 ⊗ A → C that is determined by f factors through the map ∆1 ⊗ A → E1 ⊗ A.

Proof. Suppose that there are g, h: d ⇒ c together with equivalences gf ' idc and fh ' idd that witness f as an equivalence in C. The triple (h, f, g) then determines a map I3 ⊗ A → C which can be uniquely extended to a map ∆3 ⊗ A → C since C is a category in B. By construction, the restriction of this map {0,2} 1 3 {1,3} 1 3 along the inclusions d : ∆ ⊗ A → ∆ ⊗ A and d : ∆ ⊗ A → ∆ ⊗ A are equivalent to idd and idc, respectively. By deﬁnition, this means that ∆3 ⊗ A → C extends along the map ∆3 ⊗ A → E1 ⊗ A. Conversely, if the map ∆1 ⊗ A → C that is determined by f factors through the map ∆1 ⊗ A → E1 ⊗ A, it in particular determines a map ∆3 ⊗ A → C whose restriction along d{0,1} : ∆1 ⊗ A → ∆3 ⊗ A and {2,3} 1 3 1 d : ∆ ⊗A → ∆ ⊗A gives rise to two maps h, g : c ⇒ d in C. By construction of E and the deﬁnition of composition, the composites fh and gf factor through d: A → C and c: A → C, respectively, which means that these composites are equivalent to idc and idd.

Corollary 3.6.2. A map f : A → C1 deﬁnes an equivalence in C if and only if it factors through the map

s0 : C0 → C1.

Proof. Since C is a category in B, any map E1 ⊗ A → C extends uniquely along the projection E1 ⊗ A → A, hence the result follows from proposition 3.6.1. As a consequence of corollary 3.6.2, given two objects c, d in C in context A ∈ B, we may deﬁne the groupoid of equivalences eqC(c, d) ∈ B/A via the pullback square

eqC(c, d) C0

p (d1,d0) (c,d) A C0 × C0.

By construction, sections A → eqC(c, d) over A correspond to equivalences f : c → d in C in context A. We will say that two objects c, d: A ⇒ C are equivalent if there is an equivalence c ' d, i.e. a section A → eqC(c, d) over A. This is equivalent to the condition that the two maps c and d are equivalent as objects in mapB(A, C0). Finally, we say that c and d are locally equivalent if there is a cover p: B A in B such that precomposing c and d with p yields equivalent objects in context B. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 23

3.7. The universe for groupoids. Recall that for any large ∞-category C that admits pullbacks one can 1 define a Cat(bS)-valued presheaf C/− as the functor that classifies the codomain fibration d0 : Fun(∆ , C) → C. If C is an ∞-topos, then this presheaf is a sheaf [Lur09a, Proposition 6.1.3.10]. One may therefore define:

Deﬁnition 3.7.1. Let B be an ∞-topos. The universe for groupoids in B is deﬁned to be the category Ω

in Bb that corresponds to the Cat(bS)-valued sheaf B/− on B. The universe for groupoids Ω in an ∞-topos B plays a role akin to the ∞-category S of ∞-groupoids in higher category theory. It should therefore be regarded as a reﬂection of the ∞-topos B within itself. This is supported by the observation that there is an equivalence

FunB(A, Ω) ' B/A for every A ∈ B, which in particular shows that objects A → Ω correspond to objects in the slice ∞-topos

B/A. Our goal is to obtain a similar characterization for the maps in Ω. More precisely, we will show the following result:

Proposition 3.7.2. For any two objects g, h in Ω in context A ∈ B that correspond to objects P,Q ∈ B/A, there is an equivalence

mapΩ(g, h) ' [P,Q]/A in B/A, where the right-hand side denotes the internal mapping object in B/A. Before giving a proof of proposition 3.7.2, we need a few preparatory steps. We begin by characterizing the powering functor on the ∞-category of categories in B. To that end, if C is a category in B and if R (−) op p: C → B is the associated cartesian ﬁbration, postcomposing the powering functor C : Cat(bS) → R Cat(Bb) with the Grothendieck construction : Cat(Bb) ,→ Cart(B) ,→ Cat(bS)/B gives rise to a functor R (−) op op C ∈ Fun(Cat(bS) , Cat(bS)/B). By identifying B with the constant functor Cat(bS) → Cat(bS) with value B, one obtains an equivalence

op op Fun(Cat(bS) , Cat(bS)/B) ' Fun(Cat(bS) , Cat(bS))/B, R (−) op R (−) hence one may equivalently regard C as a functor Cat(bS) → Cat(bS) together with a map C → B. Lemma 3.7.3. Let C be a (large) category in B, and let p: R C → B be the cartesian fibration that is classified by C. Then the map R C(−) → B fits into a cartesian square

R C(−) Fun(−, R C)

p p∗ diag B Fun(−, B) in Fun(Cat(bS)op, Cat(bS)), where diag denotes the diagonal functor.

op Proof. Let F : Cat(bS) → Cat(bS) be the pullback of p∗ along diag. By postcomposition with the Yoneda embedding one obtains a cartesian square map (−,F (−)) map (− × −, R C) Cat(Sb) Cat(Sb) p (−)◦pr map (−, B) 0 map (− × −, B) Cat(Sb) Cat(Sb)

∗ in which pr0 denotes the projection onto the ﬁrst factor. By making use of the adjunction (πB)! a πB, this shows that there is an equivalence

∗ R map/B(−,F (−)) ' map/B(− ×B πB(−), C) 24 LOUIS MARTINI

op of presheaves on Cat(bS)/B ×Cat(bS). As by [Lur09a, Corollary 3.2.2.12] the functor F : Cat(bS) → Cat(bS)/B ∗ factors through the subcategory Cart(B) and so does πB, we conclude the proof with the chain of equivalences ∗ R R (−) mapCart(B)(−,F (−)) ' mapCart(B)(− ×B πB(−), C) ' mapCart(B)(−, C )

R (−) that gives rise to the desired equivalence of functors F ' C . Let − ∆0 : Cat(S) → Cat(S) denote the functor that sends an ∞-category C to the pushout

C id ×d0 C × ∆1

y ∆0 C ∆0,

0 and note that the map in the above square allows us to consider − ∆ as a functor Cat(S) → Cat(S)∆0/. Postcomposition with Fun(−, B) then gives rise to a functor

0 op Fun(− ∆ , B): Cat(bS) → Cat(bS)/B.

Proposition 3.7.4. The functor

0 op Fun(− ∆ , B): Cat(bS) → Cat(bS)/B

factors through the inclusion Cart(B) ,→ Cat(bS)/B such that there is a commutative square

Fun(−∆0,B) Cat(bS)op Cart(B)

Ω(−) ' Cat(B) PSh (B). b Cat(Sb)

Proof. By lemma 3.7.3, the functor R Ω(−) ﬁts into the pullback square

R Ω(−) Fun(− × ∆1, B)

p ∗ (id ×d0) diag B Fun(−, B)

in Fun(Cat(bS)op, Cat(bS)), which is precisely the square obtained by applying the functor Fun(−, B) to the 0 pushout square that deﬁnes − ∆ . For any n ≥ 0 there is a canonical equivalence ∆n ∆0 ' ∆n+1 in Cat(S). Hence postcomposing the • 0 •+1 cosimplicial ∞-category ∆ with the functor − ∆ gives rise to the cosimplicial object ∆ in Cat(S)∆0/. Further postcomposing this functor with Fun(−, B) for any ∞-topos B then results in the simplicial object

op n+1 ∆ → Cat(bS)/B, [n] 7→ Fun(∆ , B). By proposition 3.7.4, this simplicial object takes values in Cart(B). Let RFib(B) ⊂ Cart(B) be the full subcategory spanned by the right ﬁbrations into B, and recall that the Grothendieck construction restricts to an equivalence RFib(B) ' PSh (B) with respect to which the core Sb functor (−)' : PSh (B) → PSh (B) corresponds to the functor that is given by restricting a cartesian Cat(Sb) Sb ﬁbration P → B to the the subcategory of P spanned by the cartesian edges. On account of proposition 3.1.2, proposition 3.7.4 now implies:

Corollary 3.7.5. The core functor Cart(B) → RFib(B) ' PSh (B) sends the simplicial object Fun(∆•+1, B) Sb in Cart(B) to the universe Ω ∈ Cat(B) ,→ PSh (B) . b Sb ∆ YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 25

∗ Lemma 3.7.6. For any object A ∈ B, there is an equivalence πˆAΩB ' ΩB/A of categories in Bd/A.

∗ Proof. Since the functor πA admits a left adjoint (πA)!, the cartesian ﬁbration that corresponds to the ∗ categoryπ ˆAΩB is obtained as the pullback

R ∗ 1 πˆAΩB Fun(∆ , B) p

(πA)! B/A B.

1 It therefore suﬃces to observe that the above pullback can be identiﬁed with Fun(∆ , B/A).

Proof of proposition 3.7.2. We have to show that there is a cartesian square

[P,Q] Ω1 p (d1,d0) (g,h) A Ω0 × Ω0 in B or equivalently in PSh (B). As lemma 3.7.6 implies that the pullback functor b Sb ∗ πˆA : Bb → Bd/A carries the universe of B to the universe of B/A, we may assume without loss of generality that A ' 1 holds.

By, corollary 3.7.5, the map (d1, d0): Ω1 → Ω0 × Ω0 corresponds to the map of right ﬁbrations obtained by restricting the functor 2 2 Fun(∆ , B) → Fun(Λ2, B) to the subcategories spanned by the cartesian edges. Since the core functor (−)' : PSh (B) → PSh (B) Cat(S) Sb commutes with limits, it therefore suﬃces to show that there is a cartesian square

R [P,Q] Fun(∆2, B) p

2 B Fun(Λ2, B) in Cart(B), where the lower horizontal map sends an object A ∈ B to the diagram

A × Q

pr0 pr A × P 0 A.

Note that since the forgetful functor Cart(B) → Cat(bS) creates pullbacks, it suffices to consider the above square as a diagram of ∞-categories. Moreover, note that essentially by definition the right fibration R [G, H] → B arises as the pullback

R [P,Q] Fun(∆1, B)

p (d1,d0) (−×G,H) B B × B. By factoring the functor − × G into the composite

1 B Fun(∆1, B) d B 26 LOUIS MARTINI

in which the ﬁrst arrow acts by sending A ∈ B to the projection pr0 : A × G → A, we conclude that there is a pullback square R 2 [P,Q] Fun(Λ0, B)

p (d{0,2},d{1}) B Fun(∆1 t ∆0, B). Let K be the simplicial set that is obtained by the pushout

1 ∆0 d ∆1

d{1} 2 y Λ2 K

1 0 {0,2} 1 2 and let ι: ∆ t ∆ ,→ K be the map that is determined by the two inclusions d : ∆ → Λ2 and 0 0 1 2 d : ∆ → ∆ . Then the functor B → Fun(Λ2, B) that is determined by the global section (g, h): 1 → Ω0 ×Ω0 can be decomposed into the composition

1 0 ι∗ 2 B Fun(∆ t ∆ , B) Fun(K, B) Fun(Λ2, B)

in which ι∗ denotes the functor of right Kan extension along ι. Let L be the simplicial set that is deﬁned by the pushout 1 ∆0 d ∆1

d{1} y ∆2 L. One then obtains a commutative diagram

2 2 Fun(Λ0, B) Fun(L, B) Fun(∆ , B)

(d{0,2},d{1}) p p

1 0 ι∗ 2 Fun(∆ t ∆ , B) Fun(K, B) Fun(Λ2, B).

in Cat(bS) in which both squares are cartesian, hence the result follows. 3.8. Fully faithful and essentially surjective functors. A functor between two categories C and D in an ∞-topos B is simply deﬁned to be a map f : C → D in Cat(B). Using proposition 3.2.11, there is a category [C, D] in B whose objects in context A ∈ B are given by the functors A × C → D. A map in [C, D] is referred to as a morphism of functors or alternatively as a natural transformation, the datum of such a morphism (in context A) is given by a map ∆1 ⊗ (A × C) → D in Cat(B).

Deﬁnition 3.8.1. A functor C → D between categories in B is said to be fully faithful if it is internally right orthogonal to the map (d1, d0): ∆0 t ∆0 → ∆1. Dually, a functor is essentially surjective if is (internally) left orthogonal to the class of fully faithful functors.

Remark 3.8.2. A functor f : C → D between ∞-categories is essentially surjective in the sense of deﬁni- tion 3.8.1 if and only if every object in D is equivalent to an object in the image of f, i.e. if and only if it is essentially surjective in the usual sense of the term. We will show this in 3.9.5 below.

By proposition 2.5.7, the two classes of essentially surjective and fully faithful functors form an orthogonal factorisation system in Cat(B). In particular, one obtains:

Proposition 3.8.3. Let B be an ∞-topos and let f : C → D be a functor between categories in B. Then f is an equivalence in Cat(B) if and only if f is fully faithful and essentially surjective. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 27

Since fully faithful functors are by deﬁnition internally right orthogonal to the map (d1, d0): ∆0t∆0 → ∆1, the class of essentially surjective functors is stable under products with arbitrary categories in Cat(B). Dually, this means that the class of fully faithful functors must be stable under exponentiation. We therefore conclude:

Proposition 3.8.4. If f : C → D is a fully faithful functor in Cat(B) and if E is an arbitrary simplicial object B, the induced functor f∗ :[E, C] → [E, D] is fully faithful as well.

Let f : C → D be a functor between categories in B. By deﬁnition, f being fully faithful precisely means that the square

∆1 1 f 1 C∆ D∆

f×f C × C D × D is cartesian. Applying the core functor, this implies that the square

f1 C1 D1

f0×f0 C0 × C0 D0 × D0

is cartesian as well. In fact, the latter square being cartesian is even a suﬃcient criterion for f to be fully faithful. The proof of this statement requires the following combinatorial lemma:

Lemma 3.8.5. Lett A ∈ B be an arbitrary object and let S be a saturated class of morphisms in B∆ that contains the maps In ⊗ A,→ ∆n ⊗ A for all n ≥ 0. If S contains the map (d1, d0): A t A → ∆1 ⊗ A, then it also contains the map (d1, d0): (∆n ⊗ A) t (∆n ⊗ A) → (∆1 × ∆n) ⊗ A for any integer n ≥ 0.

Proof. By replacing B with B/A, we may assume without loss of generality A ' 1. As all maps in the statement of the lemma are contained in the essential image of const: S → B, we may further assume B ' S. Furthermore, as the inclusion In t In ,→ ∆n t ∆n is contained in S, it suﬃces to show that the map In t In ,→ ∆1 × ∆n is contained in S as well. By lemm 3.2.5 the map ∆1 × In ,→ ∆1 × ∆n is contained in S, hence we need only show that also the map In t In ,→ ∆1 × In is an element of S. Now In being deﬁned 1 1 as the colimit ∆ t∆0 · · · t∆0 ∆ , we can assume without loss of generality n = 1. Using the decomposition 1 1 2 2 ∆ × ∆ ' ∆ t∆1 ∆ , one easily sees that this map is obtained by glueing the two maps

(d{0,1}, d{2}): ∆1 t ∆0 ,→ ∆2

and (d{0}, d{1,2}): ∆0 t ∆1 ,→ ∆2

1 0 0 0 1 1 along the map (d , d ): ∆ t∆ ,→ ∆ in Fun(∆ , S∆). The proof is therefore finished once we show that the two maps above are contained in S. We show this for the first one, the case of the second one is completely analogous. Making use once more of the assumption that the spine inclusion I2 ,→ ∆2 is contained in S, it suffices to show that the map (d{0,1}, d{2}): ∆1 t ∆0 ,→ I2

is contained in S. This follows from the observation that this map is obtained by glueing ∆0 t ∆0 ,→ ∆1 1 0 1 and the identity on ∆ along the identity on ∆ in Fun(∆ , S∆). 28 LOUIS MARTINI

Proposition 3.8.6. A functor f : C → D between categories in B is fully faithful if and only if the square

f1 C1 D1

f0×f0 C0 × C0 D0 × D0 is cartesian.

Proof. We already observed above that f being fully faithful implies that the square is cartesian. Conversely, the square being cartesian is equivalent to f being right orthogonal to the set of maps

S = {(∆0 t ∆0) ⊗ A → ∆1 ⊗ A | A ∈ B}

in Cat(B). By lemma 3.8.5, the saturation of S in Cat(B) contains the maps (∆n t∆n)⊗A → (∆1 ×∆n)⊗A for A ∈ B and n ≥ 0, which translates into the statement that the induced square

∆1 ∆1 fn ∆1 (C )n (D )n

fn×fn Cn × Cn Dn × Dn

is a pullback square for all n ≥ 0. As limits in Cat(B) can be computed on the underlying simplicial objects, this shows that f is fully faithful.

Proposition 3.8.7. Let f : C → D be a functor between large categories in B. Then the following are equivalent: (1) The functor f is fully faithful;

(2) for any A ∈ B and any two objects c0, c1 : A → C in context A, the morphism

mapC(c0, c1) → mapD(f(c0), f(c1))

that is induced by f is an equivalence in Bb/A; (3) for every A ∈ B the functor f(A): C(A) → D(A) of ∞-categories is fully faithful; (4) the map of cartesian ﬁbrations over B that is determined by f is fully faithful.

Proof. The equivalence of the last two conditions follows from lemma 3.7.3 and remark 3.5.7. As moreover limits in Sh (B) are computed objectwise, it is clear that the ﬁrst and the third condition are equivalent. Cat(Sb) Suppose now that f is fully faithful. For any object A ∈ B and any two objects c0, c1 : A → C in context A, the map

mapC(c0, c1) → mapD(f(c0), f(c1)) is deﬁned by the commutative diagram

mapC(c0, c1) mapD(f(c0), f(c1))

f1 C1 D1 A id A

f0×f0 C0 × C0 D0 × D0.

in which the two vertical squares on the left and on the right are cartesian. As f is fully faithful, the square in the front is cartesian, hence the square in the back must be cartesian as well, which implies that the YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 29

second condition holds. Conversely, suppose that f induces an equivalence on mapping groupoids. As the

object C0 × C0 in Bb is obtained as the colimit of the diagram

B/C0×C0 → B ,→ Bb, we obtain a cover G A C0 × C0 A→C0×C0 in Bb. By assumption, pasting the front square in the above diagram with the pullback square F map (c , c ) C (c0,c1) C 0 1 1

F A C × C (c0,c1) 0 0 results in a pullback square, hence the claim follows from the fact that base change along a cover in an ∞-topos constitutes a conservative algebraic morphism. Lemma 3.8.8. The map ∆1 → ∆0 is essentially surjective as a functor of categories in B.

Proof. Algebraic morphisms preserve essential surjectivity since dually geometric morphisms preserve full faithfulness. We may therefore assume without loss of generality B ' S. Let S be the saturated class of 0 0 1 n n maps in S∆ that is generated by ∆ t ∆ ,→ ∆ , the spine inclusions I ,→ ∆ for n ≥ 0 as well as the map E1 → 1, where E1 denotes the walking equivalence. It suﬃces to show that ∆1 → ∆0 is contained in S. Let K,→ ∆3 be the unique map of simplicial ∞-groupoids that ﬁts into the commutative diagram

{0,1} ∆1 d ∆2

{1,2} d {1,2,3} y d ∆2 K

∆3. d{0,1,2} The inclusion K,→ ∆3 is contained in S: in fact, as the inclusion I3 ,→ ∆3 is an element of S and factors through K,→ ∆3, it suﬃces to observe that the map I3 ,→ K can be obtained by glueing two copies of the 2 2 1 1 inclusion I ,→ ∆ along the identity on ∆ in Fun(∆ , S∆). We now obtain a commutative diagram ∆1 t ∆1 K ∆3

y y ∆0 t ∆0 L E1

in which the upper left horizontal map is induced by postcomposing the inclusion d{0,2} : ∆1 ,→ ∆2 with the 2 3 two maps ∆ ⇒ K that are defined by the pushout square. As K,→ ∆ is contained in S, we conclude that the induced map L,→ E1 must be in S as well. As a consequence, the terminal map L → ∆0 is an element of S too. Let ∆1 → K be the composite map in the pushout square that defines K. Postcomposing with the map K → L from the previous diagram gives rise to a map ∆1 → L. We finish the proof by showing that this map is contained in S. Let H be defined by the pushout square ∆1 ∆0

d{0,2} y ∆2 H. 30 LOUIS MARTINI

Then the map ∆1 → L is recovered as the composite map in the cocartesian square

β ∆1 H α y H L,

in which the two maps α and β are given by postcomposing d{1,2} : ∆1 ,→ ∆2 and d{0,1} : ∆1 ,→ ∆2, respectively, with the map ∆2 → H. As a consequence, we only need to verify that α, β ∈ S. We will show this for α, the case of the map β is analogous. Consider the commutative diagram

0 ∆0 d ∆1 ∆0

d0 d0 y y 1 2 1 ∆ Λ0 ∆ α {1,2} d y ∆2 H

in which the composite of the two vertical maps in the middle is given by the inclusion d{0,2} : ∆1 ,→ ∆2. 2 2 As maps in S are stable under pushouts, we only need to show that the inclusion Λ0 ,→ ∆ is contained in S. Consider the commutative square

0 1 2 ∆ t ∆ Λ0

I2 ∆2

that is uniquely determined by the condition that the composite map is induced by d{0} : ∆0 ,→ ∆2 and d0 : ∆1 ,→ ∆2. As the lower horizontal map is contained in S by assumption on S, it suﬃces to show that the two maps from ∆0 t ∆1 are in S as well. This follows immediately from the observation that both of 0 0 1 these maps can be obtained as a pushout of the map ∆ t ∆ ,→ ∆ . Proposition 3.8.9. The groupoidiﬁcation functor preserves essential surjectivity when viewed as a functor (−)gpd : Cat(B) → Cat(B) . Dually, the core groupoid functor (−)' : Cat(B) → Cat(B) preserves full faithfulness.

Proof. Let f : C → D be an essentially surjective functor, and consider the commutative diagram

f C D

f gpd Cgpd Dgpd in which the two vertical maps are obtained from the adjunction unit. In order to show that f gpd is essentially surjective, it suffices to show that the adjunction unit C → Cgpd is essentially surjective for every category C in B. As this map is contained in the saturated class internally generated by ∆1 → ∆0, it suffices to show that the map ∆1 → ∆0 is essentially surjective, which is the content of lemma 3.8.8. Suppose now that f : C → D is fully faithful. To show that f ' is fully faithful as well, we need to show that it is contained in the right complement of the class of essentially surjective maps. But by the adjunction (−)gpd a (−)' (viewing both as endofunctors on Cat(B)) a map g is left orthogonal to f ' if and only if ggpd is left orthogonal to f, hence the claim follows from the first part of the proof. Proposition 3.8.10. Every fully faithful functor of categories in B is a monomorphism in Cat(B). YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 31

Proof. By lemma 3.8.8, the map ∆1 → ∆0 is essentially surjective, hence the map ∆0 t∆0 → ∆0 is essentially surjective as well. Since monomorphisms are precisely those maps that are internally right orthogonal to this map, the claim follows. For fully faithful functors between groupoids in an ∞-topos, the converse of proposition 3.8.10 is true as well:

Corollary 3.8.11. For a map f : G → H between groupoids in B, the following are equivalent: (1) f is fully faithful; (2) f is a monomorphism.

Proof. By proposition 3.8.10, if f is fully faithful, then f is a monomorphism in Cat(B). Conversely, f being a monomorphism precisely means that f is internally right orthogonal to the map ∆0 t ∆0 → ∆0. As the latter is the image of the map ∆0 t ∆0 → ∆1 under the groupoidiﬁcation functor (−)gpd : Cat(B) → Cat(B), the claim follows by adjunction from the assumption that f is internally right 0 0 0 orthogonal to ∆ t ∆ → ∆ . Dually, one ﬁnds for essentially surjective functors between groupoids:

Corollary 3.8.12. For a map f : G → H between groupoids in B, the following are equivalent: (1) f is essentially surjective;

(2) f0 is a cover in B.

Proof. Let ι: B ,→ Cat(B) be the inclusion. If f is essentially surjective and if g is a monomorphism in B, then ι(g) is a monomorphism between groupoids and therefore fully faithful by corollary 3.8.11. Since f0 is left orthogonal to g if and only if f is left orthogonal to ι(g), this shows that f0 is a cover.

Conversely, assuming f0 is a cover and that g is a fully faithful map between categories in B, then g0 is a monomorphism by proposition 3.8.10, hence f0 is left orthogonal to g0. Since f ' ι(f0), the adjunction between (−)0 and ι implies that this is equivalent to f being left orthogonal to g, i.e. to f being essentially surjective. 3.9. Subcategories. Let D be a category in an ∞-topos B. A map f : C → D is a monomorphism in Cat(B)

if and only if f is a (−1)-truncated object in Cat(B)/D. By [Lur09a, Proposition 6.2.1.3] the full subcategory

of (−1)-truncated objects in Cat(B)/D forms a small partially ordered set that we denote by Sub(D). By proposition 3.8.10, every fully faithful functor is a monomorphism. We may therefore deﬁne:

Deﬁnition 3.9.1. Lett D be a category in B.A full subcategory of D is a fully faithful functor C ,→ D. The collection of full subcategories of D spans a partially ordered subset of Sub(D) that we denote by Subfull(D).

As in ordinary category theory, a full subcategory of a category in an ∞-topos B should be uniquely speciﬁed by the collection of objects that are contained in it. Hereafter our goal is to turn this heuristic

into a precise statement. To that end, note that the functor (−)0 : B∆ → B admits a right adjoint Cˇ(−) that sends an object A ∈ B to its Cechˇ nerve Cˇ(A). Now if D is an arbitrary category in B, the functor

(−)0 :(B∆)/D → B/D0 admits a right adjoint h−iD that is given by the composition

ˇ η∗ B C (B ) (B ) /D0 ∆ /Cˇ(D0) ∆ /D

in which η : D → Cˇ(D0) denotes the adjunction unit. As Cˇ is fully faithful, so is the functor h−iD.

Lemma 3.9.2. For every category D in B and any monomorphism P,→ D0, the simplicial object hP iD is a category in B, and the functor hP iD → D is fully faithful. 32 LOUIS MARTINI

Proof. By construction, the map hP iD → D ﬁts into a cartesian square

hP iD D η

Cˇ(P ) Cˇ(D0).

To show that hP iD is a category in B, it therefore suﬃces to show that the map Cˇ(P ) → Cˇ(D0) is internally 1 2 2 right orthogonal to the two maps E → 1 and I ,→ ∆ . This is in turn equivalent to the map P,→ D0 1 2 2 being internally right orthogonal (in B) to the two maps (E )0 → 1 and (I )0 ,→ (∆ )0. As the ﬁrst one is a cover in B and the second one is an equivalence, this is immediate. By the same argumentation, the functor 0 0 1 hP iD → D is fully faithful precisely if the map P,→ D0 is internally right orthogonal to (∆ t ∆ )0 → (∆ )0, which follows from the observation that this map is an equivalence in B.

As a consequence of lema 3.9.2, the functor h−iD restricts to an embedding full h−iD : Sub(D0) ,→ Sub (D) of partially ordered sets.

full Proposition 3.9.3. For any category D in B, the map h−iD : Sub(D0) ,→ Sub (D) is an equivalence. Proof. It suﬃces to show that the map is essentially surjective. Let therefore C ,→ D be a full subcategory

of D. Using corollary 3.8.11, the induced map C0 → D0 is a monomorphism in B. We therefore obtain a factorization

C ,→ hC0iD ,→ D in which the ﬁrst map is fully faithful since the second map and the composite map are fully faithful. As

moreover the map C ,→ hC0iD induces an equivalence on level 0, it must be an equivalence on level 1 as well. Together with the Segal condition, this implies that this map is an equivalence, which completes the proof. Proposition 3.9.4. Let f : C → D be a functor between large categories in B and let E ,→ D be a full subcategory. Then the following are equivalent: (1) f factors through the inclusion E ,→ D; (2) f ' factors through E' ,→ D'; (3) for every object c in C in context A ∈ B its image f(c) is contained in E.

Proof. Clearly (1) implies (2). By making use of the adjunction (−)0 a h−iD and proposition 3.9.3, one ﬁnds that conversely (2) implies (1). A fortiori (2) implies (3). Conversely, suppose that for any c: A → C the

composite map A → C0 → D0 factors through E0 ,→ D0. As the map G A C0 A→C0 deﬁnes a cover in Bb, the lifting problem F A E A→C0 0

f C0 D0

admits a unique solution, which proves that (2) holds.

Corollary 3.9.5. A map f : C → D between categories in B is essentially surjective if and only if f0 : C0 →

D0 is a cover in B. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 33

Proof. Suppose ﬁrst that f0 is a cover, and let p C E i D be the factorisation of f into an essentially surjective and a fully faithful functor. We need to show that i is

an equivalence. Since i is fully faithful, proposition 3.9.3 implies that this is the case if and only if i0 is an equivalence. But since f0 is a cover, the map i0 is one as well and must therefore be an equivalence as it is already a monomorphism by proposition 3.8.10. Conversely, suppose that f is essentially surjective, and let

p i C0 P D0

be the factorisation of f0 into a cover and a monomorphism in B. We need to show that i is an equivalence.

By proposition 3.9.3, the object P determines a full subcategory hP iD of D and since f0 factors through P ,

proposition 3.9.4 implies that f factors through a map C → hP iD. It suﬃces to show that this functor is essentially surjective. Let C → E ,→ hP iD be the factorisation of this functor into an essentially surjective and a fully faithful functor. Then p: C0 → P factors through a monomorphism E0 ,→ P , but since p is a cover this map must be a cover as well and therefore an equivalence. Proposition 3.9.3 then implies that

the map E ,→ hP iD is an equivalence and therefore that the functor C → hP iD is essentially surjective, as desired.

Deﬁnition 3.9.6. Let f : C → D be a map in Cat(B) and let C E ,→ D be the factorisation of f into an essentially surjective and a fully faithful functor. Then the full subcategory E ,→ D is referred to as the essential image of f.

Deﬁnition 3.9.7. Let D be a category in B and let (di : Ai → D)i∈I be a small family of objects in D. The F essential image of the induced map i Ai → D is referred to as the full subcategory of D that is generated by the family (di)i∈I . F In the context of deﬁnition 3.9.7, let G,→ D0 be the image of the map (di)i∈I : i Ai → D0. Then corollary 3.9.5 implies that the full subcategory of D generated by the family (di)i∈I is given by hGiD.

Remark 3.9.8. Let (si : Ai → C)i∈I be a small family of maps in B and let r : B → C be another map such that for all i ∈ I there is a cover pi : B Ai such that r ' sipi. One then obtains a commutative diagram F p (id)i∈I F i i F B i B i Ai

(r)i∈I r C (si)i∈I which shows that the image of (si)i∈I in C coincides with the image of the map r. This shows that in the situation of deﬁnition 3.9.7, we may assume without loss of generality that the objects are pairwise locally non-equivalent in the sense that for any pair (i, j) ∈ I × I it is not possible to ﬁnd covers p: B Ai and q : B Aj such that dip ' djq.

Remark 3.9.9. Let D be a category in B and let (di : Ai → D)i∈I be a family of objects in D that are pairwise locally non-equivalent in the sense of remark 3.9.8. Then I is V-small since for any A ∈ B the set

of equivalence classes of maps A → D0 is small and the set of equivalence classes of objects in B is V-small. By viewing D as a category in Bb, the subcategory spanned by the family (di)i∈I is therefore well-deﬁned. Note that this is still a small category since any full subcategory of a small category must be small as well.

We conclude this section with a discussion of the poset of full subcategories of the universe Ω for groupoids in B. To that end, let us recall the notion of a local class in an ∞-topos [Lur09a, Section 6.1.3]: 34 LOUIS MARTINI

Definition 3.9.10. Let S be a collection of maps in B that is stable under pullbacks. Then the full subcategory of Fun(∆1, B) spanned by the maps in S forms a cartesian subfibration of the codomain fibration 1 d0 : Fun(∆ , B) → B that is classified by a Cat(bS)-valued presheaf S/− on B. The class S is said to be local if S/− is a sheaf and bounded if S/− takes values in Cat(S).

In the situation of deﬁnition 3.9.10,[Lur09a, Lemma 6.1.3.7] implies that the presheaf S/− is a sheaf if ' and only if (S/−) is an bS-valued sheaf, and since the latter takes values in S if and only if S/− takes values in Cat(S), one obtains:

Proposition 3.9.11. Let S be a collection of maps in B that is stable under pullbacks. Then the following are equivalent: (1) S is a (bounded) local class.

(2) S/− is a (Cat(S)-valued) sheaf. ' (3)( S/−) is an (S-valued) sheaf. The set of local classes in B can be identiﬁed with a subset of the partially ordered set Sub(Fun(∆1, B)) and therefore inherits a partial order. One now ﬁnds:

Proposition 3.9.12. There is an equivalence between the partially ordered set of local classes in B and Subfull(Ω) with respect to which bounded local classes in B correspond to small full subcategories of Ω.

Proof. If S is a local class, proposition 3.9.11 shows that S/− is a Cat(bS)-valued sheaf and therefore corresponds to a large category ΩS in B. By proposition 3.8.7, this is a full subcategory of the universe of B. Conversely, if C ,→ Ω exhibits C as a full subcategory of Ω, proposition 3.9.11 implies that the set of objects contained in the essential image of the associated inclusion R C ,→ Fun(∆1, B) of cartesian fibrations over B defines a local class. Clearly these operations are inverse to each other and order-preserving. Applying proposition 3.9.11 once more, one moreover sees that this equivalence restricts to an equivalence between the poset of bounded local classes and the poset of small full subcategories of Ω. Definition 3.9.13. A full subcategory C ,→ Ω of the universe is said to be a subuniverse in B.

In the situation of proposition 3.9.12, the case where S is a bounded local class deserves a more careful ' discussion. In this case, the sheaf S/− is represented by the small category ΩS in B, hence (S/−) is ' 1 representable by ΩS which by Yoneda’s lemma implies that the full subcategory S,→ Fun(∆ , B) admits ' ' a final object ϕS : ΩbS → ΩS that is referred to as the universal morphism in S. Hereafter, our goal is to reverse this discussion: Suppose that p: P → A is an arbitrary morphism in B, and denote by hpi the class of morphisms in B that arise as a pullback of p. Since hpi is stable under pullbacks, the full subcategory of Fun(∆1, B) spanned by the maps in hpi defines a cartesian fibration over B and is therefore classified by a Cat(bS)-valued presheaf hpi/− on B. We would like to investigate the conditions that ensure hpi to be a bounded local class in B, with p as a universal morphism.

Deﬁnition 3.9.14. A map p: P → A in B is univalent if hpi is a bounded local class.

' ' Remark 3.9.15. Let S be a bounded local class of morphisms in B and let ϕS : ΩbS → ΩS denote the associated universal morphism in S. Then a map in B arises as a pullback of ϕS if and only if it is contained in S, hence the map ϕS is univalent.

By proposition 3.9.11 the notion of univalence admits the following equivalent characterization:

Proposition 3.9.16. For a map p: P → A in B, the following conditions are equivalent: (1) p is univalent. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 35

(2) hpi/− is representable by a category in B ' (3)( hpi/−) is representable by a groupoid in B. Lemma 3.9.17. Let B be an ∞-topos and let C be a small ∞-category. A map f : Y → X in the ∞-topos Fun(C, B) is a cover if and only if f(c): Y (c) → X(c) is a cover for every c ∈ C.

Proof. For every c ∈ C, evaluation at c deﬁnes an algebraic morphism Fun(C, B) → B, and since equivalences Q in Fun(C, B) are determined objectwise, the induced algebraic morphism Fun(C, B) → c∈C B is conservative. Now f is a cover if and only if the inclusion Im(f) ,→ X is an equivalence. Since algebraic morphisms preserve the image factorisation of a map and since conservative functors reﬂect equivalences, the claim follows.

Suppose that p: P → A is a map in B and let g : A → Ω0 be the associated object in Ω. By deﬁnition of hpi and the fact that a map in PSh (B) is a cover if and only if it is objectwise given by a cover in S Sb b (cf. lemma 3.9.17), the image factorisation of g in PSh (B) is given by A (hpi )' ,→ Ω . Therefore p is Sb /− 0 ' univalent if and only if the cover A (hpi/−) is a monomorphism, which is the case if and only if g itself is a monomorphism. We therefore conclude:

Proposition 3.9.18 ([GK17, Proposition 3.8]). Let p: P → A be a map in B and let g : A → Ω0 be the associated object in Ω. Then p is univalent if and only if g is a monomorphism.

Corollary 3.9.19. Let p: P → A be a map in B and let g : A → Ω0 be the associated object in Ω. Let

pri : A × A → A be the projection onto the ith factor for i ∈ {0, 1}. Then p is univalent if and only if the canonical map ϕ: A → eqΩ(g pr0, g pr1) in B/A×A is an equivalence.

Proof. By proposition 3.9.18, the morphism p is univalent precisely if g : A → Ω0 is a monomorphism in Bb, which is equivalent to the commutative square g A Ω0

(id,id) (id,id) g×g A × A Ω0 × Ω0 being cartesian. On account of the cartesian square

eqΩ(g pr0, g pr1) Ω0 p (id,id) g×g A × A Ω0 × Ω0, we see that this is the case if and only if the map ϕ is an equivalence.

The object of morphisms of the category Ωhpi that is associated with a univalent map p: G → A in B admits an explicit description as well:

Proposition 3.9.20. Let p: P → A be a univalent morphism in B and let Ωhpi be the associated category ∗ ∗ in B. Then (Ωhpi)1 is equivalent to the internal mapping object [pr0 G, pr1 G] in B/A×A.

Proof. By construction there is a fully faithful functor Ωhpi ,→ Ω in Cat(Bb), which means that the square

(Ωhpi)1 Ω1

(Ωhpi)0 × (Ωhpi)0 Ω0 × Ω0 is cartesian. On the other hand, proposition 3.7.2 identiﬁes the pullback of the above diagram with ∗ ∗ [pr0 G, pr1 G], which ﬁnishes the proof. 36 LOUIS MARTINI

Remark 3.9.21. The theory of univalent maps in an ∞-topos has been previously worked out by Gepner and Kock in [GK17] and by Rasekh in [Ras18], using slightly diﬀerent methods.

4. Groupoidal fibrations and Yoneda’s lemma The main goal of this chapter is to formulate and prove Yoneda’s lemma for categories in an ∞-topos B. The proof will rely on the interplay between Ω-valued functors on a category C in B and left ﬁbrations p: P → C, a result that is commonly referred to as the Grothendieck construction. The collection of left ﬁbrations forms the right class of a factorisation system in Cat(B) whose left complement is comprised of initial functors. Sections 4.1 to 4.4 are devoted to a discussion of this factorisation system, and in section 4.5 we discuss the Grothendieck construction. Finally, we prove Yoneda’s lemma in section 4.6.

Remark 4.0.1. Our strategy for the proof of Yoneda’s lemma is inspired by Cisinski’s proof of Yoneda’s lemma for ∞-categories in [Cis19].

4.1. Left ﬁbrations. In this section we discuss left ﬁbrations between categories in an ∞-topos B and discuss some of their basic properties.

Definition 4.1.1. A functor P → C between categories in B is a left fibration if it is internally right orthogonal to the map d1 : ∆0 ,→ ∆1. Dually, p is a right fibration if it is internally right orthogonal to the map d0 : ∆0 ,→ ∆1. We denote by LFib and RFib, respectively, the full subcategories of Fun(∆1, Cat(Bb)) spanned by the left and right fibrations.

In what follows, we will mostly restrict the discussion to left fibrations. By dualizing, however, all statements carry over unchanged to right fibrations. In more precise terms, this dualization is obtained by taking opposite categories: Recall that the autoequivalence (−)op : Cat(S) → Cat(S) restricts to an op autoequivalence on ∆ ,→ Cat(S). By precomposition, one obtains an autoequivalence (−) : B∆ → B∆ that restricts to an autoequivalence of Cat(B). For any category C in B, the category Cop is referred to as the opposite category of C. Since the functor (−)op sends the inclusion d1 : ∆0 ,→ ∆1 to the map d0 : ∆0 ,→ ∆1, the autoequivalence (−)op sends right fibrations to left fibrations and vice versa.

1 1 Lemma 4.1.2. The saturated class of maps in B∆ that is generated by the maps d : E,→ ∆ ⊗ E for any simplicial object E in B coincides with the saturation of the set

{d{0} : A,→ ∆n ⊗ A | A ∈ B, n ≥ 0}.

1 1 Proof. Let S be the saturation of the set of maps d : E,→ ∆ ⊗ E for E ∈ B∆. Then for any A ∈ B and any n ≥ 0 the map d0 : (∆0 × ∆n) ⊗ A,→ (∆1 ⊗ ∆n) ⊗ A is contained in S as well. Let α: ∆n+1 ,→ ∆1 × ∆n be deﬁned by α(0) = (0, 0) and α(k) = (1, k − 1) for 1 ≤ k ≤ n, and let β : ∆1 × ∆n → ∆n+1 be deﬁned by β(0, k) = 0 and β(1, k) = k + 1 for any 0 ≤ k ≤ n. One then obtains a commutative diagram

∆0 ∆n ∆0

d{0} d1×id d{0} β ∆n+1 α ∆1 × ∆n ∆n+1,

{0} 0 n+1 1 n 1 n and since βα ' id∆n+1 the map d : ∆ ,→ ∆ is a retract of d × id: ∆ → ∆ × ∆ . By tensoring with A, this shows that the map d{0} : A,→ ∆n ⊗ A is contained in S for all n ≥ 1. Conversely, let S be the saturation of the set of maps d{0} : A,→ ∆n ⊗ A for n ≥ 0 and A ∈ B, and let E be a simplicial object in B. We need to show that the map d1 : E,→ ∆1 ⊗ E is contained in S. By the same argument as in lemma 3.2.5, we may assume without loss of generality E ' ∆n ⊗ A for some n ≥ 1 YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 37

1 n n+1 n+1 and some A ∈ B. Now with respect to the usual decomposition ∆ × ∆ ' ∆ t∆n · · · t∆n ∆ of the product ∆1 × ∆n into n + 1 copies of ∆n+1, the map d{0} : ∆n ,→ ∆1 × ∆n is given by the iterated pushout

{0} {0,...,n} 0 n n+1 n+1 d td{0} · · · td{0,...,n−1} d : ∆ t∆0 · · · t∆n−1 ∆ → ∆ t∆n · · · t∆n ∆ 1 {0,...,i} i in Fun(∆ , S∆). It therefore suﬃces to show that for every n ≥ 1 and every 0 ≤ i ≤ n the map d : ∆ ⊗ A,→ ∆n ⊗ A is contained in S, which follows immediately from the assumption by using item (2) of proposition 2.5.2. Proposition 4.1.3. A functor P → C between categories in B is a left ﬁbration if and only if for every n ≥ 1 the commutative diagram

Pn Cn

d{0} d{0}

P0 C0 is cartesian.

Proof. This follows immediately from lemma 4.1.2.

Proposition 4.1.4. For any simplicial object K in an ∞-topos B, the endofunctor [K, −]: B∆ → B∆ preserves left ﬁbrations.

Proof. Since left fibrations are by definition internally right orthogonal to d1 : ∆0 ,→ ∆1, they are also 1 0 1 internally right orthogonal to d : ∆ ⊗ K,→ ∆ ⊗ K, hence the result follows. Left fibrations are functors that are fibred in groupoids. To see this, note that since d0 : ∆0 ,→ ∆1 is a section of the unique map ∆1 → ∆0, one finds:

Lemma 4.1.5. Let S be a saturated class of maps in Cat(B) that contains the maps d0 : E ,→ ∆1 ⊗ E for 1 0 all E ∈ Cat(B). Then S contains the class of maps in Cat(B) that is internally generated by ∆ → ∆ . Deﬁnition 4.1.6. A functor C → D between categories in B is said to be conservative if it is internally right orthogonal to the map ∆1 → ∆0.

Lemma 4.1.7. Let S be a saturated class of maps in Cat(B) that contains the projections ∆1 ⊗ A → A for all A ∈ B. Then S contains the projection ∆1 ⊗ C → C for any category C in B.

Proof. As every category C is a colimit of categories of the form ∆n ⊗ A for some n ≥ 0 and some A ∈ B, we n n 1 1 may assume without loss of generality C ' ∆ . Since ∆ ' ∆ t∆0 · · ·t∆0 ∆ in Cat(B), we may furthermore 1 1 2 2 1 1 1 assume n = 1. In light of the decomposition ∆ ×∆ ' ∆ t∆1 ∆ , the projection ∆ ×∆ → ∆ is equivalent to the composition s1t id 0 2 2 ∆1 2 s 1 ∆ t∆1 ∆ ∆ ∆ . 0 1 2 1 It will therefore be enough to show that the two maps s , s : ∆ ⇒ ∆ are contained in S, which follows 0 0 immediately from the observation that these two maps are given by s t∆0 id and id t∆0 s in light of the 2 1 1 decomposition ∆ ' ∆ t∆0 ∆ . Lemma 4.1.7 immediately implies:

Proposition 4.1.8. A functor f : C → D between categories in B is conservative if and only if the square

f0 C0 D0

s0 s0

f1 C1 D1 38 LOUIS MARTINI is cartesian. Corollary 4.1.9. A functor f : C → D between categories in B is conservative if and only if the commutative square f ' C' D'

f C D is cartesian.

Proof. On account of the Segal conditions, a cartesian square between categories in B is cartesian if and only if it is cartesian on level 0 and level 1, hence the claim follows from the observation that the square in the statement of the corollary is trivially cartesian on level 0 and recovers the square from proposition 4.1.8 on level 1. Corollary 4.1.10. A functor C → D between categories in B is conservative if and only if for any object d: A → D the ﬁbre C|d = C ×A D is a groupoid in B.

Proof. If f is conservative, then for any object d: A → D the map C|d → A is conservative as well, hence C|d 1 0 is internally local with respect to ∆ → ∆ and therefore a groupoid. Conversely, if C|d is a groupoid for any ' ' object d: A → D, by choosing d to be the core inclusion i: D → D one obtains an equivalence C|i ' C , hence f is conservative on account of corollary 4.1.9. By using lemma 4.1.5 and corollary 4.1.10, one now concludes:

Proposition 4.1.11. Both left and right fibrations between categories in B are conservative. In particular, the fibre of a left or right fibration over any object in the codomain in context A ∈ B is a groupoid in B/A. Remark 4.1.12. Proposition 4.1.11 implies in particular that if C → G is a left fibration in Cat(B) in which G is a groupoid, then C is a groupoid as well. Conversely, one easily deduces from proposition 4.1.3 that any map between groupoids in B is a left (and also a right) fibration. In particular, this argument shows that both localisation functors Fun(∆1, Cat(B)) → RFib and Fun(∆1, Cat(B)) → LFib recover the groupoidification functor Cat(B) → Grpd(B) upon taking the fibre over the final object 1 ∈ Cat(B).

Equivalences between left ﬁbrations can be detected ﬁbrewise:

Proposition 4.1.13. A map f : P → Q between left ﬁbrations over a category C in B is an equivalence if and only if the induced map f|c : P|c → Q|c is an equivalence in Grpd(B) for any object c in C in context A ∈ B.

Proof. By item (2) of proposition 2.5.2, the map f is a left ﬁbration itself. Therefore f is an equivalence whenever the underlying map f0 : P0 → Q0 is one. The claim now follows from descent together with the

fact that C0 is canonically obtained as the colimit colimA→C0 A. 4.2. Slice categories. In this section we will discuss one particularly important example of left ﬁbrations in an ∞-topos B - that of slice categories.

Deﬁnition 4.2.1. Let f : D → C and g : E → C be two functors between categories in B. The comma category D ↓C E is deﬁned as the pullback

∆1 D ↓C E C

p (d1,d0) f×g D × E C × C. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 39

For any object c: A → C, we write C/c = C ↓C c and refer to this category as the slice category over c.

Dually we write Cc/ = c ↓C C and refer to this category as the slice category under c.

In the situation of deﬁnition 4.2.1, the slice category Cc/ comes along with a canonical map to A × C, and we will denote the composition of this map with the projection A × C → C by (πe)! : Cc/ → C. Furthermore, ∆1 note that the identity idc : A → C induces a map A → Cc/ over (idA, c): A → A × C that we will denote

by idc as well.

Remark 4.2.2. Let C be a category in B and let c: A → C be an object in C. The transpose of c with ∗ ∗ respect to the adjunction (πA)! a πA then determines a map 1 → πAC in Cat(B/A) that we will denote by c as well. Using proposition 3.3.5 and lemma 4.2.3 below, the map Cc/ → A × C is equivalent to the image of ∗ ∗ the projection (πc)! :(πAC)c/ → πAC along the forgetful functor (πA)! : Cat(B/A) → Cat(B).

Lemma 4.2.3. For any two categories C and D in B and any A ∈ B, the natural map

∗ ∗ ∗ πA[C, D] → [πAC, πAD] is an equivalence in Cat(B/A).

Proof. The natural map ∗ ∗ ∗ πA[C, D] → [πAC, πAD] is the component at D of the mate of the commutative square

Cat(B) C×− Cat(B)

∗ ∗ πA πA ∗ πAC×− Cat(B/A) Cat(B/A).

As a consequence, this map is an equivalence if and only if the transpose map

∗ (πA)!(πAC × −) → C × (πA)!(−) is an equivalence, which is obvious.

Hereafter, our goal is to prove that the projection (πc)! : Cc/ → C is a left ﬁbration for any category C in

B and any object c in C. We will achieve this by identifying (πc)! as the base change of a left ﬁbration that we will construct hereafter. Let − ? −: ∆ × ∆ → ∆ be the ordinal sum bifunctor. We may then deﬁne:

Deﬁnition 4.2.4. Let : ∆ → ∆ denote the functor [n] 7→ [n]op ? [n]. For any category C in B, we deﬁne the twisted arrow category Tw(C) to be the simplicial object given by the composition

op ∆op ∆op C B.

This deﬁnes a functor Tw: Cat(B) → B∆.

Note that the functor in deﬁnition 4.2.4 comes along with two canonical natural transformations

op (−) → ← id∆ which induces a map of simplicial objects

Tw(C) → Cop × C

that is natural in C. 40 LOUIS MARTINI

Proposition 4.2.5. For any category C in B, the simplicial object Tw(C) is a category as well, and the map Tw(C) → Cop × C is a left ﬁbration.

Proof. We will begin by showing that for any n ≥ 1 the square

op op Tw(C)n Cn × Cn

d{0} d{0} op Tw(C)0 C0 × C0 is a pullback diagram. Unwinding the deﬁnitions, this is equivalent to the diagram

C2n+1

d{0,...,n} d{n+1,...,2n+1} d{n,n+1}

Cn C1 Cn

d{n} d{0} d{1} d{0} C0 C0 being a limit diagram, which follows easily from the Segal conditions. Now to show that Tw(C) is a category, we will prove that more generally for any map P → C of simplicial objects in B one ﬁnds that P is a category whenever the square

Pn Cn

d{0} d{0}

P0 C0 is cartesian for all n ≥ 1. Regarding the Segal conditions, ﬁrst note that by induction and the pasting lemma for pullback squares, the square

Pn Cn

d{0,...,n−1} d{0,...,n−1}

Pn−1 Cn−1 must be cartesian as well, which implies that the square

Pn Pn−1 ×P0 P1

Cn Cn−1 ×C0 C1 that is induced by the diagram {n−1} [0] δ [n − 1]

δ{0} δ{0,...,n−1} {n−1,n} [1] δ [n] in ∆ is a pullback square. As C by assumption satisﬁes the Segal conditions, this shows that the map

Pn → Pn−1 ×P0 P1 must be an equivalence, which by induction on n proves that P satisﬁes the Segal conditions as well. By a similar argument, one shows that also the square

P0 P3 ×P1×P1 (P0 × P0)

C0 C3 ×C1×C1 (C0 × C0)

is a pullback and therefore that P must be univalent as well. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 41

Observe that the ordinal sum functor ?: ∆ × ∆ → ∆ ﬁts into the commutative square

∆ × ∆ ? ∆

Cat(S) × Cat(S) Cat(S)

in which denotes the bifunctor that sends a pair (C, D) of ∞-categories to the pushout

(d1,d0) (C × D) t (C × D) C × D × ∆1

pr0 t pr1 y C t D C D. In fact, the inclusions [m] ,→ [m] ? [n] ←- [n] in ∆ induce a map ∆m t ∆n → ∆m?n of 1-categories that is natural in m and n, and we may also deﬁne a map ∆n ×∆m ×∆1 → ∆n?m of 1-categories naturally in m and n by sending a triple (i, j, k) to i if k = 0 and to m + j otherwise. This construction gives rise to a natural map ∆m ∆n → ∆m?n that is an equivalence by [Lur09a, Proposition 4.2.1.2]. Combining this observation with proposition 3.1.2, we therefore conclude that for any category C in B the underlying simplicial object • op • of Tw(C) is obtained by applying the core functor to the simplicial object C(∆ ) ∆ in Cat(B).

Lemma 4.2.6. For any integer n ≥ 0, the canonical square

(∆n)op t ∆n (∆n)op ∆n

∆0 t ∆n ∆0 ∆n is a pushout in Cat(S).

Proof. By deﬁnition of the bifunctor and the pasting lemma for pushout squares, the commutative square in the statement of the lemma is a pushout if and only if the square

(d1,d0) ((∆n)op × ∆n) t ((∆n)op × ∆n) (∆n)op × ∆n × ∆1

(d1,d0) ∆n t ∆n ∆n × ∆1 is cocartesian. By making use of the fact that the functor − × ∆n commutes with colimits, we may assume n 0 n 1 1 ∆ ' ∆ . Moreover, in light of the decomposition ∆ ' ∆ t∆0 · · · t∆0 ∆ in Cat(S), we may asssume n = 1. We now have to show that the commutative square

(d1,d0) (∆1)op t (∆1)op (∆1)op × ∆1

(d1,d0) ∆0 t ∆0 ∆1

1 op 1 2 2 is a pushout, which is easily shown by making use of the equivalence (∆ ) × ∆ ' ∆ t∆1 ∆ and the fact that the diagram 0 ∆1 d ∆2

0 ∆0 d ∆1 is cocartesian. 42 LOUIS MARTINI

By making use of lemma 4.2.6, one now obtains a cartesian square

∆0∆• (C )0 Tw(C)

op C0 × C C × C. On the other hand, the deﬁning pushout for ∆0 ∆• gives rise to a cartesian square

(∆0)op∆• ∆1 (C )0 C

C0 × C C × C, which in particular shows:

Proposition 4.2.7. Let C be a category in B. For any object c: A → C, the canonical map Cc/ → A × C ﬁts into a cartesian square

Cc/ Tw(C) p

A × C c×id Cop × C in Cat(B).

Corollary 4.2.8. For any category C in B and any object c in C, the functor (πc)! :(C)c/ → C is a left ﬁbration.

Proof. This follows from proposition 4.2.7 together with the observation that the projection A × C → C is a left ﬁbration since the terminal map A → 1 is one (cf. proposition 4.1.11). 4.3. Initial functors. We will now focus on the left complement of the class of left ﬁbrations in Cat(B). The results in this section are heavily inspired by Cisinski’s book [Cis19].

Definition 4.3.1. A functor J → I between categories in B is said to be initial if it is left orthogonal to every left fibration in Cat(B). Dually, J → I is final if it is left orthogonal to every right fibration in Cat(B).

Remark 4.3.2. A functor J → I between categories in B is initial if and only if its opposite Jop → Iop is ﬁnal. Therefore all properties of initial functors carry over to ﬁnal functors upon taking opposite categories. We will therefore restrict our attention to the case of initial functors.

Example 4.3.3. By lemma 4.1.5, any functor between categories in B that is in the internal saturation of the map ∆1 → ∆0 deﬁnes both an initial and a ﬁnal functor.

Definition 4.3.4. Let C be a category in B. An object c: A → C is said to be initial if the transpose map ∗ ∗ 1 → πAC defines an initial functor in Cat(B/A). Dually, c is final if the transpose map 1 → πAC defines a final functor in Cat(B/A).

Remark 4.3.5. In corollary 4.3.20 we will see that initial and final objects satisfy the expected universal property, which in particular implies that an object in an ordinary ∞-category is initial or final in the sense of definition 4.3.4 precisely if it is initial or final in the usual sense.

Remark 4.3.6. In the context of deﬁnition 4.3.4, the object c is initial if and only if (c, id): A → C × A is ∗ an initial map in Cat(B). To see this, observe that (c, id) is precisely the image of c: 1 → πAC under the forgetful functor (πA)!, hence one direction follows from the observation that (πA)! preserves initial maps. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 43

The other direction follows from the fact that every map in Cat(B/A) arises as a pullback of a map that is ∗ in the image of the base change functor πA : Cat(B) → Cat(B/A), which implies that a map f in Cat(B/A) is initial if and only if (πA)!(f) is initial.

Remark 4.3.7. Note that an object c: A → C in a category C ∈ Cat(B) being initial is different from the condition that c is initial when viewed as a functor in Cat(B). In fact, remark 4.3.6 shows that the first condition is equivalent to (c, id): A → A × C being initial as a functor in Cat(B), hence either of the two conditions implying the other would imply that the projection A × C → C is initial as well, which is not true in general. In fact, since the projection A × C → C is a left fibration, this map is initial if and only if it is an equivalence.

Any functor between categories in B admits a unique factorisation into an initial map followed by a left ﬁbration. In what follows, our goal is to describe this factorization explicitly for the case where the domain is an object in B. To that end, recall that to any category C ∈ Cat(B) and any object c: A → C one can

associate the slice category Cc/ → A × C such that the identity map on c deﬁnes a section idc : A → Cc/ over

(idA, c): A → A × C.

Proposition 4.3.8. For any category C in B and any object c in C in context A ∈ B, the section idc : A → Cc/ is initial as a map in Cat(B).

∗ Remark 4.3.9. In the situation of proposition 4.3.8, let us denote by c: 1 → πAC the global section in Cat(B/A) that arises as the transpose of c: A → C. Then the section idc : A → Cc/ arises as the image of the ∗ global section idc : 1 → (πAC)c/ under the forgetful functor (πA)! : Cat(B/A) → Cat(B). Since this functor creates initial maps (cf. remark 4.3.6), we conclude that the section idc : A → Cc/ is initial if and only if the ∗ ∗ associated global section idc : 1 → (πAC)c/ deﬁnes an initial object in the category (πAC)c/. The proof of proposition 4.3.8 requires a few preparations. We begin by describing how the comma category construction from deﬁnition 4.2.1 can be turned into a functor, and we will construct a left adjoint to this functor. To that end, let K denote the 1-category that is depicted in the following diagram: • • •

• •

/ 2 / Let i: K,→ K be the canonical inclusion, and let r :Λ0 → K be the span that is given by the two dashed arrows in the following depiction of K/: •

• • •

• •

2 Let furthermore ϕ: Fun(Λ2, Cat(B)) → Fun(K, Cat(B)) be the evident functor that sends a cospan E g f D C to the diagram E g 1 C∆ d0 C

d1 f D C. 44 LOUIS MARTINI

The composition

∗ 2 ϕ i∗ / r 2 Fun(Λ2, Cat(B)) Fun(K, Cat(B)) Fun(K , Cat(B)) Fun(Λ0, Cat(B))

∗ in which i∗ denotes the right adjoint of the restriction functor i then sends a cospan E g f D C

to the span D ← D ↓C E → E and therefore deﬁnes the desired comma category functor − ↓ −.

Deﬁnition 4.3.10. Let f : C → D and g : C → E be functors in Cat(B). The cocomma category D C E is the category in B that is deﬁned by the pushout square

(d1,d0) C t C ∆1 ⊗ C

ftg

D t E D C E.

. If A ∈ B is an arbitrary object and C → A is a map (i.e. if C is a category in B/A), we write C = C C A and / refer to this category as the right cone of C → A. Dually, we write C = A C C and refer to this category as the left cone of C → A.

Analogous to the comma category construction, the construction of the cocomma category from deﬁni- tion 4.3.10 can be turned into a functor − − as follows: Let L = Kop, let j : L,→ L. denote the canonical op 2 . 2 inclusion, and let s = r :Λ2 → L . If ψ : Fun(Λ0, Cat(B)) → Fun(L, Cat(B)) denotes the functor that sends a span g C E f D to the diagram g C E d0 1 C d ∆1 ⊗ C f D, the desired functor − − can be deﬁned by the composition

∗ 2 ψ j! . s 2 Fun(Λ0, Cat(B)) Fun(L, Cat(B)) Fun(L , Cat(B)) Fun(Λ2, Cat(B))

∗ in which j! denotes the left adjoint of the restriction functor j .

Proposition 4.3.11. The functor − − is left adjoint to the functor − ↓ −.

The proof of proposition 4.3.11 makes use of the following description of the mapping ∞-groupoids in functor ∞-categories, due to Glasman:

Lemma 4.3.12 ([Gla16, Proposition 2.3]). For any two locally small ∞-categories C and D, the mapping

∞-groupoid functor mapFun(C,D)(−, −) is equivalent to the composition

p∗map (−,−) Fun(C, D)op × Fun(C, D) Fun(Cop × C, Dop × D) D ∗ Fun(Tw(C), S) lim S, in which p: Tw(C) → Cop × C denotes the canonical projection. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 45

Remark 4.3.13. Quite generally, if I and J are arbitrary ∞-categories and if I → Fun(J, S) is any functor, the composition I Fun(J, S) lim S

is a limit of the diagram J → Fun(I, S) that corresponds to the map I → Fun(J, S) in light of the equivalence Fun(J, Fun(I, S)) ' Fun(I, Fun(J, S)).

In the situation of lemma 4.3.12, this observation implies that the mapping ∞-groupoid functor mapFun(C,D)(−, −) is equivalently obtained as the limit of the diagram

Tw(C) Fun(Fun(C, D)op × Fun(C, D), S)

that corresponds to the composition

p∗map (−,−) Fun(C, D)op × Fun(C, D) Fun(Cop × C, Dop × D) D ∗ Fun(Tw(C), S)

in light of the equivalence

Fun(Tw(C), Fun(Fun(C, D)op × Fun(C, D), S)) ' Fun(Fun(C, D)op × Fun(C, D), Fun(Tw(C), S)).

1 0 1 0 0 0 0 1 Proof of proposition 4.3.11. On account of the two adjunctions d a s : ∆ ∆ and s a d : ∆ ∆ , ∗ 2 / the functor r admits a left adjoint r! : Fun(Λ0, Cat(B)) → Fun(K , Cat(B)) that is given by precomposition 0 ∗ 2 . with s × id, and the functor s admits a right adjoint s∗ : Fun(Λ2, Cat(B)) → Fun(L , Cat(B)) that is given by precomposition with s0 × id. We therefore obtain equivalences

∗ map 2 (−, − ↓ −) ' map (i r!(−), ϕ(−)) Fun(Λ0,Cat(B)) Fun(K,Cat(B))

and

∗ map 2 (− −, −) ' map (ψ(−), j s∗(−)). Fun(Λ2,Cat(B)) Fun(L,Cat(B)) We complete the proof by constructing an equivalence

∗ ∗ mapFun(K,Cat(B))(i r!(−), ϕ(−)) ' mapFun(L,Cat(B))(ψ(−), j s∗(−)).

To that end, observe that the twisted arrow 1-categories Tw(K) and Tw(L) can both be identiﬁed with the poset that can be depicted by the diagram

• • • •

• • • • •

in which the vertices in the lower row correspond to the degenerate edges in K and L and the vertices in the upper row correspond to the non-degenerate edges in K and L, and in which the edges correspond to the assignment of each non-degenerate edge in K and L to its domain and its codomain, respectively. Let 2 2 us label the vertices and edges of the two 1-categories Λ2 and Λ0 as depicted in the following two diagrams

p c 1 b y

p0 q1 q a x 0 z. 46 LOUIS MARTINI

∗ By using lemma 4.3.12 and remark 4.3.13, the bifunctor mapFun(K,Cat(B))(i r!(−), ϕ(−)) is obtained as the limit of the diagram

∗ ∗ ∗ ∗ ∆1 ∗ ∗ mapCat(B)(a (−), x (−)) mapCat(B)(c (−), z (−) ) mapCat(B)(b (−), y (−)) ∗ ∗ (q0)∗p0 (d1)∗ (d0)∗ (q1)∗p1

∗ ∗ ∗ ∗ mapCat(B)(c (−), z (−)) mapCat(B)(c (−), z (−))

in which for i ∈ {0, 1} the maps pi and qi denote the morphisms of functors that are induced by the inclusion 1 2 1 2 ∗ of the two edges pi : ∆ ,→ Λ0 and qi : ∆ ,→ Λ2. Dually, the bifunctor mapFun(L,Cat(B))(ψ(−), j s∗(−)) is obtained as the limit of the diagram

∗ ∗ 1 ∗ ∗ ∗ ∗ mapCat(B)(a (−), x (−)) mapCat(B)(∆ ⊗ c (−), z (−)) mapCat(B)(b (−), y (−)) ∗ 1 ∗ 0 ∗ ∗ (q0)∗p0 (d ) (d ) (q1)∗p1

∗ ∗ ∗ ∗ mapCat(B)(c (−), z (−)) mapCat(B)(c (−), z (−)),

1 ∆1 hence the adjunction ∆ ⊗ − a (−) gives rise to the desired equivalence.

0 0 2 Remark 4.3.14. In the situation of proposition 4.3.11, let α: ∆ t ∆ ,→ Λ0 be the inclusion of the two 0 0 2 objects that are not initial, and let β : ∆ t∆ ,→ Λ2 be the inclusion of the two objects that are not ﬁnal. We ∗ 2 ∗ 2 obtain restriction functors α : Fun(Λ0, Cat(B)) → Cat(B) × Cat(B) and β : Fun(Λ2, Cat(B)) → Cat(B) × Cat(B) that commute with the comma and cocomma constructions, in that there are two commutative diagrams

2 −↓− 2 2 −− 2 Fun(Λ2, Cat(B)) Fun(Λ0, Cat(B)) Fun(Λ2, Cat(B)) Fun(Λ0, Cat(B))

β∗ α∗ β∗ α∗ Cat(B) × Cat(B) Cat(B) × Cat(B).

Moreover, the explicit construction of the equivalence

map 2 (− −, −) ' map 2 (−, − ↓ −) Fun(Λ2,Cat(B)) Fun(Λ0,Cat(B)) in the proof of proposition 4.3.11 shows that if η and denote unit and counit of the adjunction −− a − ↓ −, the induced morphisms α∗η and β∗ recover the identity transformations on α∗ and β∗, respectively.

Remark 4.3.15. In the situation of proposition 4.3.11, if C is a category in B, note that the cospan C C C in Cat(B) is given by C ,→ ∆1 ⊗ C ←- C in which the two inclusions are the cosimplicial maps d1 and d0. The explicit construction of the equivalence

map 2 (− −, −) ' map 2 (−, − ↓ −) Fun(Λ2,Cat(B)) Fun(Λ0,Cat(B))

in the proof of proposition 4.3.11 then shows that the transpose of the map (id, s0, id): (C = C = C) → C ↓C C 0 is given by the map (id, s , id): C C C → (C = C = C).

Remark 4.3.16. Let C be a category in B and let c: A → C be an object in C. Then the canonical 2 section idc : A → Cc/ can be viewed as a map (A = A = A) → A ↓C C in Fun(Λ0, Cat(B)) that ﬁts into a commutative square

(A = A = A) A ↓C C

(c,c,c) c↓idid

(id,s0,id) (C = C = C) C ↓C C. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 47

Using proposition 4.3.11 and remark 4.3.15, this diagram corresponds to a commutative square

A A A (A → C = C)

ccc (c,id,id) (id,s0,id) C C C (C = C = C),

hence the map A A A → (A → C = C) that corresponds to idc :(A = A = A) → A ↓C C is given by the triple (id, idc, c).

Lemma 4.3.17. For any category C in B and any object c in C in context 1 ∈ B, the map c: 1 → C is initial if the canonical projection (πc)! : Cc/ → C has a section s: C → Cc/ such that s(c) is equivalent to the canonical section idc.

Proof. The section s deﬁnes a commutative diagram

(1 = 1 = 1) (id,idc,id)

(c,c,id) (id,s,id) (1 ← C = C) A ↓C C

2 in Fun(Λ2, Cat(B)). Using the adjunction between the comma and the cocomma construction as well as remark 4.3.16, this diagram corresponds to a commutative diagram

(id,idc,c) 1 1 1

(id,r,id) 1 C C (1 → C = C)

2 in Fun(Λ0, Cat(B)), which is explicitly given by the commutative diagram

1 id 1 id 1

1 d idc c ∆1 C/ r C

d0 id 1 c C id C

in Cat(B). As a consequence, the map c: 1 → C is seen to be a retract of the map ∆1 → C/, hence it suﬃces to show that the latter is initial. Since both d1 and the map 1 → C/ are initial, this follows immediately from item (2) in proposition 2.5.2.

Proof of Proposition 4.3.8. The canonical section idc : A → Cc/ is obtained as the image of the canonical ∗ section idc : 1 → (πAC)c/ in Cat(B/A) under the forgetful functor (πA)! : Cat(B/A) → Cat(B) when viewing ∗ c as a an object of πAC (cf. remark 4.2.2). As the functor (πA)! preserves initial maps, we may replace B with B/A and can therefore assume without loss of generality A ' 1. We would like to apply lemma 4.3.17

to the pair (Cc/, idc). By using the adjunction between the comma and the cocomma construction, it suﬃces / to construct a map r :(Cc/) → Cc/ such that the triple (id, r, id) deﬁnes a map 1 Cc/ Cc/ → (1 → Cc/ = Cc/) 2 1 1 1 in Fun(Λ2, Cat(B)). To that end, let d: ∆ × ∆ → ∆ be the projection onto the diagonal that is given 1 1 2 2 0 0 2 2 1 by composing the equivalence ∆ × ∆ ' ∆ t∆1 ∆ with s t∆1 s : ∆ t∆1 ∆ → ∆ . Then the map 48 LOUIS MARTINI

1 1 1 d∗ : C∆ → C∆ ×∆ ﬁts into the two commutative squares

1 ∗ 1 1 1 ∗ 1 1 C∆ d C∆ ×∆ C∆ d C∆ ×∆

1 ∗ 1 ∗ d1 (id ×d ) d1 (d ×id)

1 1 C s0 C∆ C s0 C∆ .

1 1 1 Transposing d∗ along the adjunction ∆1 ⊗ − a (−)∆ thus determines a map e: ∆1 ⊗ C∆ → C∆ together with two commutative squares

1 1 1 1 1 ∆1 ⊗ C∆ e C∆ C∆ d ⊗id ∆1 ⊗ C∆

id ⊗d1 d1 d1 e

0 1 ∆1 ⊗ C s ⊗id C C s0 C∆ .

By pasting the right square with the pullback diagram that deﬁnes the slice category Cc/, we obtain a map / ∆1 h:(Cc/) → C that ﬁts into the commutative diagram

1 id ⊗πC d ⊗id 1 c/ 1 Cc/ ∆ ⊗ Cc/ ∆

id ⊗c 1 1 1 C∆ d ⊗id ∆1 ⊗ C∆ id ⊗d1 ∆1 ⊗ C

e s0⊗id / d1 1 (Cc/) 1

c h c 1 C s0 C∆ . d1 C. / The horizontal square on the bottom right in this diagram now gives rise to the desired map r :(Cc/) → Cc/. / By inspection of the above commutative diagram, it is clear that r sends the section 1 → (Cc/) to the

canonical section idc : 1 → Cc/. Lastly, the observation that the composition

1 0 1 1 C∆ d ⊗id ∆1 ⊗ C∆ e C∆

∆1 / recovers the identity on C implies that r is a retract of the map Cc/ → (Cc/) , which ﬁnishes the proof. Corollary 4.3.18. Let C be a category in B and let c: A → C be an object in C. The factorisation of c into an initial map and a left ﬁbration is given by the composition (πc)! idc : A → (C)c/ → C. Lemma 4.3.17 can furthermore be used to derive the following characterization of initial objects in a category:

Proposition 4.3.19. Let C be a category in B. For any object c: A → C, the following are equivalent: (1) c is an initial object;

(2) the projection Cc/ → A × C is an equivalence;

(3) for any object d: B → C the map mapC(cπ0, dπ1) → A × B is an equivalence in B. Proof. If c is initial, the map (id, c): A → A×C is an initial functor in Cat(B), hence corollary 4.3.18 implies

that the left ﬁbration Cc/ → A × C must be initial as well and therefore an equivalence. Conversely, if this map is an equivalence, corollary 4.3.18 implies that (id, c): A → A × C is initial, hence c deﬁnes an initial

object by remark 4.3.6. Lastly, since the map Cc/ → A × C is a left ﬁbration, proposition 4.1.13 implies

that this map is an equivalence whenever the induced map (Cc/)|(s,d) → B is an equivalence for any map (s, d): B → A × C. Since any such map factors as

(s,id) (id,d) B A × B A × C, YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 49

the map Cc/ → A × C is an equivalence if and only if the induced map (Cc/)|(id,d) → A × B is an equivalence.

As the latter recovers the morphism mapC(cπ0, dπ1) → A × B, the claim follows.

Corollary 4.3.20. Let C be a category in B and let c and d be objects in C in context A ∈ B such that c is initial. Then there is a unique map c → d in C in context A that is an equivalence if and only if d is initial as well.

Proof. By item (3) of proposition 4.3.19, the map mapC(cπ0, dπ1) → A × A is an equivalence. Since the

pullback of this map along the diagonal (id, id): A → A × A recovers mapC(c, d) → A, the latter is an equivalence as well. Therefore, there is a unique map f : c → d in context A that corresponds to the unique

section A → mapC(c, d). If d is initial, then by the same argumentation there is a unique map g : d → c, and by uniqueness this must be an inverse of f. Hence f is an equivalence. Conversely, if f is an equivalence, ∗ then c and d are equivalent as objects in C, which implies that the two maps c, d: 1 ⇒ πAC in Cat(B/A) are equivalent, which shows that d must be initial.

4.4. Covariant equivalences. Let L: Fun(∆1, Cat(B)) → LFib denote a left adjoint to the inclusion (see

the discussion in section 2.5) and let L/C : Cat(B)/C → LFib/C be the induced functor on the ﬁbre over a category C in B.

Deﬁnition 4.4.1. Let C be a category in B and let f : P → Q be a functor over C, i.e. a map in Cat(B)/C.

Then f is said to be a covariant equivalence if L/C(f) is an equivalence in LFib/C.

Remark 4.4.2. In the context of deﬁnition 4.4.1, the functor L/C(f) is constructed by means of the unique commutative diagram

f P Q

L/C(f) L/C(P) L/C(Q)

C in which the two vertical maps are initial and the two diagonal maps are left ﬁbrations. In particular, if f is initial then f is a covariant equivalence over C. The converse implication is true whenever the map Q → C is already a left ﬁbration.

If P → C is a functor in Cat(B) and if c is an arbitrary object in C, let us use the notation P/c = P ×C C/c. The main goal of this section is to prove the following characterisation of covariant equivalences in B:

Proposition 4.4.3. Let

f P Q

p q C be a commutative triangle in Cat(B). Then the following are equivalent: (1) f is a covariant equivalence over C;

(2) for any object c: A → C the induced map f/c : P/c → Q/c is a covariant equivalence over C/c; gpd gpd gpd (3) for any object c: A → C the induced map (f/c) :(P/c) → (Q/c) is an equivalence in Grpd(B). 50 LOUIS MARTINI

The proof of proposition 4.4.3 is based on the concept of a proper functor. Observe that for any functor p: D → C in B the commutative square

LFib/C Cat(B)/C

∗ p∗ p

LFib/D Cat(Bb)/D

gives rise to a left lax square

L/C LFib/C Cat(Bb)/C

∗ p∗ p

L/D LFib/D Cat(Bb)/D by means of the mate construction. As L does not preserve pullbacks, this square does not commute in general.

Deﬁnition 4.4.4. A functor p: D → C between categories in B is said to be proper if for any cartesian square F D q p E C in Cat(B) the left lax square

L/E LFib/E Cat(Bb)/E

∗ q∗ q

L/F LFib/F Cat(Bb)/F commutes. Dually, a functor p: D → C is smooth if Dop → Cop is proper.

Unwinding the deﬁnitions, the left lax square from deﬁnition 4.4.4 is commutative if and only if for any f : P → E the lower square in the commutative diagram

q∗P P

q∗f L(q∗P) L(P) f

q F E is cartesian. Here the vertical maps are given by the factorisation of P → E and q∗P → F into an initial map and a left fibration. This is equivalent to the condition that base change along q : F → E preserves initial maps. In fact, if this condition is satisfied, then in particular the map L(q∗P ) → q∗L(P) is initial. But since this map must also be a left fibration, it is necessarily an equivalence. The converse direction follows from chasing an initial map through the commutative square that is provided in the definition of proper maps. By definition, proper functors preserve covariant equivalences:

Proposition 4.4.5. If p: D → C is a proper functor between categories in B, then the base change functor ∗ p : Cat(B)/C → Cat(B)/D carries covariant equivalences over C to covariant equivalences over D. Proposition 4.4.6. For any two categories C and D in B the projection C × D → C is proper. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 51

Proof. This follows immediately from the fact that for any initial map i: J → I the map i×idD : J×D → I×D is initial as well (since initial maps are internally left orthogonal to left ﬁbrations).

Proposition 4.4.7. Any right ﬁbration between categories in B is proper.

Proof. Since right fibrations are stable under pullbacks, it suffices to show that the base change along a right fibration preserves initial maps. To that end, let S be the set of maps in Cat(B) whose base change along right fibrations results in an initial map. We claim that S is saturated. In fact, it is obvious that S is closed under composition and contains all equivalences, and the stability of S under pushouts and small colimits follows from the fact that the Cat(bS)-valued presheaf that classifies the cartesian fibration RFib → Cat(B) is a sheaf on Cat(B) (cf. the dual version of lemma 4.5.7 below). As a consequence, it suffices to show that the initial map d1 : D ,→ D ⊗ ∆1 is contained in S for any category D in B. Let therefore

f Q P p p

1 D d D ⊗ ∆1

be a cartesian square in Cat(B) such that p is a right ﬁbration. Let τ : ∆1 × ∆1 → ∆1 be the map that sends the ﬁnal vertex (1, 1) to 1 and any other vertex to 0. We then obtain a commutative diagram

∆1 s0 ∆0

id ×d1 d1 0 ∆1 d ×id ∆1 × ∆1 τ ∆1 id id ×d0 ∆1

1 of ∞-categories such that the composition of the two horizontal arrows is the identity. Now let (Q⊗∆ )tQ P 0 1 1 1 be the pushout of f along the map d : Q → Q⊗∆ , and observe that the induced map (Q⊗∆ )tQ P → P⊗∆ is ﬁnal (by making use of item (2) of proposition 2.5.2). Therefore the lifting problem

0 1 (f⊗s ,id) (Q ⊗ ∆ ) tQ P P h p

(id ⊗τ)◦(p⊗id) P ⊗ ∆1 D ⊗ ∆1 admits a unique solution h. Let r : P → Q be deﬁned as the unique functor that makes the diagram

P d1 P ⊗ ∆1

r p f h Q P p⊗id p id ⊗d1 q D ⊗ ∆1 (D ⊗ (∆1 × ∆1)

0 1 id ⊗τ D s d D ⊗ ∆1 52 LOUIS MARTINI

commute. By construction, this map satisﬁes rf ' id and moreover ﬁts into the commutative diagram P fr d1 P ⊗ ∆1 h P

d0 id P. Hence the commutative diagram

0 1 (s ,r) Q Q ⊗ ∆ tQ P Q

f f

1 0 P d P ⊗ ∆1 s P

1 1 exhibits f as a retract of the initial map Q ⊗ ∆ tQ P → P ⊗ ∆ (in which the domain is the pushout of f 1 1 along the inclusion d : Q → Q ⊗ ∆ ) and therefore as an initial map itself. Proof of proposition 4.4.3. Suppose ﬁrst that f is a covariant equivalence over C. Since the projection

(πc)! : C/c → C is a right ﬁbration and since by proposition 4.4.7 any right ﬁbration is proper, proposition 4.4.5 implies that the map f/c must be a covariant equivalence over C/c.

Suppose now that f/c is a covariant equivalence over C/c, i.e. that L/(C/c)(f) is an equivalence. Quite ∗ generally, note that for any category D in Cat(B) the base change functor πD : Cat(B) → Cat(B)/D admits a left adjoint (πD)! that is given by the forgetful functor, which implies that the base change functor ∗ πD : Grpd(B) → LFib(B)/D admits a left adjoint (πD)! as well that is explicitly given by the composition

gpd (πD)! (−) LFib/D Cat(B)/D Cat(B) Grpd(B), cf. remark 4.1.12. One consequently obtains a commutative square

(πD)! Cat(B)/D Cat(B)

gpd L/D (−)

(πD)! LFib/C Grpd(B).

gpd Applying this observation to D = C/c, one finds that the map (f/c) arises as the image of L/(C/c)(f) along the functor (πC/c )! and is therefore an equivalence. gpd Lastly, assume that (f/c) is an equivalence in Grpd(B) for any object c: A → C. We need to show that the map L/C(f): L/C(P) → L/C(Q) in LFib/C is an equivalence. By proposition 4.1.13, it suffices to show that the map L/C(f)|c : L/C(P)|c → L/C(Q)|c that is induced on the fibres over c: A → C is an equivalence of groupoids for all objects c in C. By making use of the factorisation of c into the canonical final map idc : A → C/c followed by the right fibration (πc)! : C/c → C, one obtains a pullback square

L/C(P)|c L/C(P)|c L/C(Q)|c

(L/C(f))/c L/C(P)/c L/C(P)/c in which the vertical maps are final since they arise as pullbacks of the final map idc along left fibrations and since the dual of proposition 4.4.7 implies that left fibrations are smooth. Note that the fibres L/C(P)|c and

L/C(Q)|c are groupoids. Hence the groupoidiﬁcation functor sends the two vertical maps in the above square to equivalences in Grpd(B) while leaving the upper horizontal map unchanged. We conclude that L/C(P)|c YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 53

gpd gpd is an equivalence whenever ((L/C(f))/c) is one, and since the latter recovers the map (f/c) (again using properness of the right ﬁbration (πc)! : C/c → C), the result follows.

Proposition 4.4.3 can be used to derive an internal version of Quillen’s theorem A:

Corollary 4.4.8. A functor f : J → I between categories in B is initial if and only if for every object i in I gpd in context A ∈ B the canonical map (J/i) → A is an equivalence.

Proof. On account of remark 4.4.2, the map f is initial if and only if it is a covariant equivalence over I. By gpd gpd proposition 4.4.3, this is the case if and only if for every object i: A → I the map (J/i) → (I/i) is an equivalence. By construction, there is a commutative diagram

gpd gpd (J/i) (I/i)

gpd Therefore, the proof is finished once we show that the map (I/i) → A is an equivalence. But this follows from the observation that the map I/i → A is final as it is a retraction of the final section idc : A → I/c.

We end this section with yet another characterisation of covariant equivalences that will be useful later:

Proposition 4.4.9. Let

f P Q

p q C be a commutative triangle in Cat(B) in which both p and q are smooth. Then f is a covariant equivalence gpd gpd over C if and only if for any object c in C the induced map f|c :(P|c) → (Q|c) is an equivalence.

Proof. f is a covariant equivalence over C if and only if L/C(f) is an equivalence. Let

f P Q

i j

L/C(f) L/C(P) L/C(Q)

be the canonical square in which the two vertical functors are obtained from the adjunction unit and are

therefore initial. Since L/C(f) is a map in of left fibrations over C, proposition 4.1.13 implies that this map is an equivalence if and only if the induced map L/C(f)|c on the fibres over c is one for every object c: A → C. It therefore suffices to show that in the induced commutative diagram

f|c P|c Q|c

i|c j|c

L/C(f)|c L/C(P)|c L/C(Q)|c

of the ﬁbres over c the maps i|c and j|c are initial. We will show this for i|c, the case of j|c is analogous. 54 LOUIS MARTINI

0 Let p : L/C(P) → C be the structure map, and consider the commutative diagram

i| P|c P/c P c i/c i id id id L/C(P)|c L/C(P)/c L/C(P)

0 0 0 p |c p/c p P|c P/c P p|c p/c p

idc (πc)! A C/c C.

The projection (πc)! is a right ﬁbration, hence so are the maps P/c → P and L/C(P)/c → L/C(P). Since right

fibrations are proper by proposition 4.4.7, we conclude that the map i/c must be initial. Moreover, since p 0 0 and p are smooth the maps p/c and p/c must be smooth as well, which implies that the maps P|c → P/c and L/C(P)|c → L/C(P)/c must be final since idc is final. We therefore obtain a pullback square

P|c P/c

i|c i/c

L/C(P)|c L/C(P)/c

in which the horizontal maps are final and the vertical map on the right is initial. Since the groupoidification functor (−)gpd carries both final and initial maps to equivalences in Grpd(B) ' B (cf. remark 2.5.3 and gpd remark 4.1.12), the map (i|c) must be an equivalence. But as L/C(P)|c is already a groupoid, the map i|c is equivalent to the composition of an initial map with an equivalence and therefore initial itself.

4.5. The Grothendieck construction. Let p: P → C be a functor between large categories in B. We will say that p is small if for every functor D → C such that D is a small category in B, the pullback ∗ p D = P ×C D is small as well. The collection of small functors defines a cartesian subfibration of the codomain fibration Fun(∆1, Cat(Bb)) → Cat(Bb). Therefore the full subcategory LFibU ⊂ Fun(∆1, Cat(Bb)) of small left fibrations determines a Cat(bS)-valued presheaf on Cat(Bb) that we will denote by LFibU as well.

Proposition 4.5.1. Let p: P → C be a left ﬁbration between large categories in B. Then p is small if and only if for all objects c: A → C with A ∈ B the ﬁbre P|c = P ×C A is a small groupoid.

Proof. The condition is clearly necessary. For the converse direction, it suﬃces to show that if p: P → C

is a left fibration in Cat(Bb) such that C is a small category in B and such that the fibre P|c is small for every object c in C in context A ∈ B, the category P is small as well. To see this, note that since p is a left fibration, it suffices to show that P' is small. But this follows from the fact that P' arises as the fibre of p over the object C0 → C.

For any large category C in B, let LFibC be the presheaf on B that is given by the assignment A 7→ U LFib (C × A). This deﬁnes a functor C 7→ LFibC from Cat(Bb) into the ∞-category of Cat(bS)-valued presheaves on B.

Theorem 4.5.2. For any large category C in B, the presheaf LFibC deﬁnes a large category in B, and there is a canonical equivalence

[C, Ω] ' LFibC that is natural in C. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 55

Note that the statement of theorem 4.5.2 can be reformulated as follows: By making use of the embedding Cat(B) ,→ PSh (B), one sees that the functor [−, Ω] is equivalent to the bifunctor Cat(Sb)

Fun (− × −, Ω): Bop × Cat(B)op → Cat(S). Bb b b

Similarly, the functor LFib(−) corresponds to the bifunctor

LFibU(− × −): Bop × Cat(Bb)op → Cat(bS)

under this identification. It therefore suffices to show that the presheaf LFibU takes values in Cat(bS) and that there is an equivalence Fun (−, Ω) ' LFibU Bb of Cat(bS)-valued presheaves on Cat(Bb). The proof of theorem 4.5.2 requires a few preparatory steps. We begin by describing canonical generators for the ∞-category Cat(Bb). To that end, note that since Cat(Bb) arises as a localisation of the presheaf ∞- category PSh (∆ × B), the composition of the localisation functor L with the Yoneda embedding h defines Sb a functor Lh: ∆ × B → Cat(Bb). Note that by its very construction this is precisely the restriction of the tensoring bifunctor − ⊗ − along the inclusion ∆ × B ,→ Cat(bS) × Cat(Bb).

Lemma 4.5.3. The identity on Cat(Bb) is a left Kan extension of the functor Lh along itself.

Proof. Let i: Cat(B) ,→ PSh (∆ × B) denote the inclusion. In the diagram b Sb

∗ L∗ i Fun(PSh (∆ × B), PSh (∆ × B)) Fun(PSh (∆ × B), Cat(B)) Fun(Cat(B), Cat(B)) Sb Sb Sb b ∗ b b i∗ L ∗ h! h∗ h! h

L∗ Fun(∆ × B, PSh (∆ × B)) Fun(∆ × B, Cat(B)) Sb b i∗ the square of right adjoints is commutative, hence the the square of left adjoints must commute as well. By [Lur09a, Lemma 5.1.5.3] the identity on PSh (∆×B) arises as the left Kan extension of h along itself, i.e. Sb as the functor h (h). Since moreover the composite i∗L sends the identity on PSh (∆ × B) to the identity ! ∗ Sb ∗ on Cat(Bb), a simple diagram chase shows that the identity on Cat(Bb) is equivalent to the functor i h!(Lh). ∗ ∗ ∗ We conclude by observing that since the adjunction L a i induces an adjunction i a L , the functor i h! is the functor of left Kan extension along Lh.

Our next step towards the proof of theorem 4.5.2 will be to establish an equivalence

Fun (∆• ⊗ −, Ω) ' LFibU(∆• ⊗ −) Bb of Cat(bS)-valued presheaves on ∆ × B. To that end, for any n ≥ 0 an any A ∈ B let us denote by ∆n × A the presheaf on ∆ × B that is represented by the pair ([n],A). We thus obtain L(∆n × A) ' ∆n ⊗ A.

Lemma 4.5.4. Assining to a small left ﬁbration P → ∆n ⊗ A in Cat(Bb) its pullback along the adjunction unit ∆n × A → ∆n ⊗ A in PSh (∆ × B) deﬁnes an embedding Sb U • LFib (∆ ⊗ −) ,→ PShS(∆ × B)/∆•×− of presheaves on ∆ × B. 56 LOUIS MARTINI

U • Proof. Since the presheaf LFib (∆ ⊗ −) embeds into the presheaf Cat(Bb)/∆•⊗−, it suﬃces to prove that one can deﬁne an embedding

Cat(B) • ,→ PSh (∆ × B) • b /∆ ⊗− Sb /∆ ×− in the desired way. To see this, ﬁrst observe that the inclusion Cat(B) ,→ PSh (∆ × B) can be decomposed b Sb into the composition Cat(B) ,→ B ,→ PSh (∆ × B) b b ∆ Sb in which the second arrow is induced by the inclusion B ,→ PSh (B). Either of these functors admits a left b Sb adjoint, and the left adjoint of the functor on the right-hand side is left exact by [Lur09a, Lemma 6.3.5.28]. Moreover, note that the composition of the Yoneda embedding h: ∆×B ,→ PSh (∆×B) with the localisation Sb PSh (∆ × B) → B already takes values in Cat(B). Therefore we may deﬁne a functorial map Sb b ∆ b

Cat(Bb)/∆•⊗− ,→ (Bb ∆)/∆•⊗− on the level of the associated cartesian ﬁbrations as the pullback of the natural map

Fun(∆1, Cat(B)) → Fun(∆1, B ) × B b b ∆ Cat(Bb ) b ∆

along the functor Lh: ∆ × B → Cat(Bb). As a second step, we will deﬁne a functorial map

(B ) • ,→ PSh (∆ × B) • b ∆ /∆ ⊗− Sb /∆ ×− as follows: Since the localisation functor PSh (∆ × B) → B is left exact, there is a functorial map Sb b ∆

PSh (∆ × B) • → (B ) • Sb /∆ ×− b ∆ /∆ ⊗− that is given on the level of cartesian ﬁbrations by the pullback of the natural map

1 1 Fun(∆ , PSh (∆ × B)) → Fun(∆ , Bb ∆) ×PSh (∆×B) Bb ∆ Sb Sb along the Yoneda embedding ∆ × B ,→ PSh (∆ × B). On the ﬁbre over ([n],A), this functor is given by the Sb map that is naturally induced by the localisation functor PSh (∆×B) → B upon taking slice ∞-categories. Sb b ∆ Hence there are ﬁbrewise right adjoints that are given by composing the natural map

(B ) n → PSh (∆ × B) n b ∆ /∆ ⊗A Sb /∆ ⊗A that is induced by the inclusion B ,→ PSh (∆ × B) with the pullback functor along the adjunction unit b ∆ Sb ∆n × A → ∆n ⊗ A in PSh (∆ × B). Note that each of these ﬁbrewise right adjoints is fully faithful since the Sb localisation functor commutes with pullbacks. We conclude by observing that these ﬁbrewise right adjoints assemble to a map of Cat(bS)-valued presheaves on ∆ × B. By [GHN17, Proposition 9.8] there is a functorial equivalence

PShS(∆ × B)/∆•×− ' PShS((∆ × B)/∆•×−)

in which the right-hand side can furthermore be functorially identiﬁed with PShS(∆/∆• × B/−). Combining this result with lemma 4.5.4, we conclude that there is an embedding

U • LFib (∆ ⊗ −) ,→ PShS(∆/∆• × B/−) that sends a left ﬁbration P → ∆n ⊗ A to the presheaf that maps a pair (τ :[k] → [n], s: B → A) to the n ﬁbre of the map Pk(B) → const(∆k ) × A(B) over the point that is determined by (τ, s). Note that this embedding factors through the inclusion

• • • PShB/− (∆/∆ ) ,→ PShPShS(B/−)(∆/∆ ) ' PShS(∆/∆ × B/−), YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 57 which implies that we end up with a functorial embedding

U • • LFib (∆ ⊗ −) ,→ PShB/− (∆/∆ ). Our next goal is to characterize the essential image of this embedding:

n Lemma 4.5.5. For any A ∈ B and any n ≥ 0, a presheaf F ∈ PShB/A (∆/∆ ) is contained in the essential U n n image of the inclusion LFib (∆ ⊗ A) ,→ PShB/A (∆/∆ ) if and only if the following conditions hold: (1) For any k ≥ 2 and any map τ :[k] → [n] in ∆, the map

{0,1} {k−1,k} F (τ) → F (τδ ) ×F (τδ{1}) · · · ×F (τδ{k−1}) F (τδ ) is an equivalence. (2) For any map τ : [0] → [n] in ∆, the commutative square

F (τ) F (τσ{0,...,3})

(id,id) (F (δ{0,2}),F (δ{1,3})) F (τ) × F (τ) F (τσ{0,1}) × F (τσ{0,1})

is cartesian. (3) For any k ≥ 1 and any map τ :[k] → [n] in ∆, the inclusion δ{0} : [0] → [k] induces an equivalence F (τ) ' F (τδ{0}).

n Proof. Let F be a B/A-valued presheaf on ∆/∆n and let P → ∆ × A be the map in PShS(∆ × B) that n n n corresponds to F in view of the inclusion PShB/A (∆/∆ ) ,→ PShS(∆ × B)/∆ ×A. Then P → ∆ × A is in the essential image of the inclusion (B∆)/∆n⊗A ,→ PShS(∆ × B)/∆n×A. To see this, let (L a i) denotes the adjunction B∆ PShS(∆ × B). We need to show that the commutative square P iL(P )

∆n × A i(∆n ⊗ A)

that is induced by the adjunction unit id → iL is a pullback square in PShS(∆ × B). It suﬃces to show this n for each k ∈ ∆ individually. In this case, the map Pk → const(∆k ) × A is given by the coproduct G G F (τ) → A, τ :[k]→[n] τ :[k]→[n] hence it suﬃces to show that for each map τ :[k] → [n] in ∆ the square

F (τ) iL(F (τ))

A iL(A)

is cartesian, which follows from F (τ) being contained in B. Hence F is contained in LFibU(∆n ⊗A) precisely if the simplicial object L(P ) is a category in B and the map L(P ) → ∆n ⊗ A is a left ﬁbration. Let k ≥ 2 be an integer and consider the commutative square

L(Pk) L(P1 ×P0 · · · ×P0 P1)

n n n const(∆ ) × A const(∆ × n · · · × n ∆ ) × A k 1 ∆0 ∆0 1 58 LOUIS MARTINI

in B. Since ∆n ⊗ A is a category, the lower horizontal map is an equivalence, hence L(P ) satisﬁes the Segal conditions precisely if the above square is cartesian. As L is left exact and as L(P ) → ∆n ⊗ A is a pullback of P → ∆n × A, this is in turn equivalent to the condition that the commutative square

Pk P1 ×P0 · · · ×P0 P1

n n n ∆ × A ∆ × n · · · × n ∆ × A k 1 ∆0 ∆0 1

n in PShS(B) is cartesian. Since the map Pk → ∆k × A is equivalent to the coproduct G G F (τ) → A, τ :[k]→[n] τ :[k]→[n] this square is seen to be a pullback square if and only if the map

{0,1} {k−1,k} F (τ) → F (τδ ) ×F (τδ{1}) · · · ×F (τδ{k−1}) F (τδ )

n is an equivalence for all τ ∈ ∆k , which shows that L(P•) satisfying the Segal conditions is equivalent to the ﬁrst condition in the lemma. Analogously, univalence of L(P ) is equivalent to the second condition in the lemma. In fact, L(P ) is univalent if and only if the commutative square

s{0,...,3} L(P0) L(P3)

(id,id) (d{0,2},d{1,3})

s0×s0 L(P0 × P0) L(P1 × P1)

is cartesian. By the same argument as above, this is equivalent to the condition that the commutative square

s0 P0 P3

(id,id) (d{0,2},d{1,3})

s0×s0 P0 × P0 P1 × P1 F is cartesian. Using the decomposition P0 ' τ : [0]→[n] F (τ) as above, this is precisely condition (2). Lastly, proposition 4.1.3 and the same argumentation as above show that the map L(P ) → ∆n ⊗ A being a left ﬁbration is equivalent to the square

d{0} Pk P0

d n {0} n ∆k × A ∆0 × A

being a pullback diagram for all k ≥ 1. On account of the commutative diagrams

F (τ) F (τδ{0})

d{0} Pk P0 A id A τ d {0} n {0} n τδ ∆k × A ∆0 × A

for all τ :[k] → [n], in which the squares on the left and on the right are cartesian, this is seen to be equivalent to the third condition in the lemma. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 59

Recall that there is an equivalence Fun (∆• ⊗ −, Ω) ' Fun(∆•, Ω(−)) ' Fun(∆•, B ) Bb /− of Cat(bS)-valued presheaves on ∆ × B. Let n op λ: ∆ → ∆/∆n denote the functor that sends k ≤ n to the inclusion d{k,...,n} : ∆n−k ⊂ ∆n. This functor admits a right adjoint op n : ∆/∆n → ∆ that sends τ :[k] → [n] to τ(0). One easily checks that is natural in n. Moreover, since λσ is the identity n n op functor on ∆ , this adjunction exhibits ∆ as a localisation of ∆/∆n . By precomposition, we therefore obtain a functorial embedding ∗ • • : Fun(∆ , B/−) ,→ PShB/− (∆/∆ ) n n that exibits each ∞-category Fun(∆ , B/A) as a colocalisation of PShB/A (∆/∆ ), with the right adjoint given by λ∗.

Lemma 4.5.6. For any pair ([n],A) ∈ ∆ × B, the essential image of the functor

∗ n n : Fun(∆ , B/A) ,→ PShB/A (∆/∆ ) coincides with the essential image of the embedding

U n n LFib (∆ ⊗ A) ,→ PShB/A (∆/∆ ).

n op Proof. Let us ﬁrst show that for any σ : ∆ → B/A the associated presheaf σ: (∆/∆n ) → B/A satisﬁes the conditions of lemma 4.5.5. Let therefore k ≥ 2 be an integer and let τ :[k] → [n] be a map in ∆. Then the map in condition (1) of lemma 4.5.5 is given by the natural map of σ(τ) = σ(τ(0)) into the limit of the diagram σ(τ(0)) σ(τ(1)) σ(τ(k − 1)) σ(τ(k))

id id id σ(τ(1)) ··· σ(τ(k)) which is just the identity on σ(τ(0)) and therefore in particular an equivalence. Therefore the ﬁrst condition of lemma 4.5.5 is satisﬁed. Similarly, for any τ : [0] → [n] the square in the second condition of the lemma is given by σ(τ(0)) id σ(τ(0))

σ(τ(0)) × σ(τ(0)) id σ(τ(0)) × σ(τ(0)) which is trivially cartesian. Lastly, if τ :[k] → [n] is a map in ∆ with k ≥ 1, then the map in B/A that is induced by the inclusion δ{0} : [0] → [k] is simply the identity σ(τ(0)) ' σ(τ(0)), hence the third condition of the lemma is satisﬁed as well.

n To finish the proof, it now suffices to show that for any presheaf F ∈ PShB/A (∆/∆ ) the adjunction counit ∗λ∗F → F is an equivalence. Since the counit of the adjunction ∗ a λ∗ is given by precomposition with the ∗ ∗ adjunction counit of λ a , the map λ F → F is defined on each object τ :[k] → [n] in ∆/∆n by applying F to the map [k] → [n − τ(0)] over [n], where the structure map [k] → [n] is given by the inclusion δ{τ(0),...,n}. Since F satisfies the first condition in lemma 4.5.5, we need only show that F sends this map to an equivalence in the case that k ∈ {0, 1}. In either of these cases this is an immediate consequence of the third condition in lemma 4.5.5. 60 LOUIS MARTINI

Lemma 4.5.7. The Cat(bS)-valued presheaf LFibU on Cat(Bb) sends V-small colimit diagrams in Cat(Bb) to limit diagrams in Cat(bS).

Proof. Suppose that I is a V-small ∞-category. Let i: Cat(B) ,→ PSh (B) denote the inclusion and b Cat(Sb) let L be the left adjoint of i. Since Li ' id holds, we may restrict our attention to diagrams of the form Lα, where α: I → PSh (B) is an arbitrary functor. Letα ¯ be the colimit diagram associated with α and Cat(Sb) note that Lα¯ is a colimit diagram in Cat(Bb) and that moreover any colimit diagram in Cat(Bb) is of this form. By [Lur09b, Proposition 1.2.4], it suffices to show that if β¯ → Lα¯ is a map of diagrams I. → Cat(Bb) such that for each i ∈ I the associated map in Cat(Bb) is a small left fibration and such that the restriction of β → Lα to I is cartesian, then β¯ is a colimit diagram if and only if the transformation β¯ → Lα¯ is cartesian and the map β¯(∞) → Lα¯(∞) is a small left fibration, where ∞ denotes the cone point of I.. Suppose therefore that we are given such a map β¯ → Lα¯ of diagrams in Cat(Bb), and consider the cartesian square of diagrams γ¯ iβ¯ p

α¯ iLα¯ in PSh (B), in which the horizontal map on the bottom is induced by the adjunction unit id → iL. Let Cat(Sb) γ be the restriction ofγ ¯ to J. Note that for any i ∈ I the components of the map γ(i) → α(i) are small left ﬁbrations in Cat(S). Let δ¯ be the colimit diagram in PSh (B) that is associated with γ, and note that b Cat(Sb) there is a commutative triangle δ¯ γ¯

α¯ of diagrams such that the horizontal map gives rise to the identity map upon restriction to I. Applying L to this diagram produces the commutative triangle

Lδ¯ β¯

Lα¯ in which Lδ¯ is a colimit diagram and in which the vertical arrow on the right recovers the original transformation. By [Lur09b, Proposition 1.4.7] (and by adapting the arguments in the in the proofs of [Lur09b, Lemma 1.4.10] and [Lur09b, Lemma 1.4.14] to replace cocartesian fibrations by small left fibrations in the proposition), the map δ¯ → γ¯ is an equivalence if and only ifγ ¯ → α¯ is a cartesian transformation and γ(∞) → α¯(∞) is objectwise given by a small left fibration of ∞-categories. Now if β¯ → Lα¯ is a cartesian transformation and β¯(∞) → Lα¯(∞) is moreover a small left fibration in Cat(Bb), thenγ ¯ → α¯ is a cartesian transformation as well, and the mapγ ¯(∞) → α¯(∞) is objectwise given by a small left fibration of ∞-categories. Therefore the map δ¯ → γ¯ is an equivalence, which implies that the diagram β¯ must be a colimit diagram. Conversely, if β¯ is a colimit diagram, then the map Lδ¯ → β¯ must be an equivalence, which implies that the transformation β¯ → Lα¯ is cartesian. We still need to show that the functor β¯(∞) → Lα¯(∞) is a small left fibration between categories in Bb. Since this map is obtained from the objectwise small left fibrationγ ¯(∞) → α¯(∞) by applying the localisation functor L, this follows from the left exactness of L and proposition 4.1.3. Proof of theorem 4.5.2. By lemma 4.5.6, there is an equivalence LFibU(∆• ⊗ −) ' Fun (∆• ⊗ −, Ω) Bb YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 61

of functors Bop × ∆op → Cat(B)op → Cat(bS). Note that lemma 4.5.7 and lemma 4.5.3 already imply that LFibU takes values in Cat(S). Together with the fact that Fun (−, Ω) sends V-small colimits in Cat(B) b Bb b to limits in Cat(bS), these two lemmas furthermore imply that the above equivalence can be extended to an equivalence LFibU ' Fun (−, Ω) Bb of presheaves on Cat(Bb). Remark 4.5.8. The proof of theorem 4.5.2 shows that the restriction of the equivalence Fun (−, Ω) ' Bb LFibU along the inclusion B ,→ Cat(Bb) recovers the equivalence Fun (−, Ω) ' B Bb /− of Cat(bS)-valued sheaves on B.

Remark 4.5.9. By Yoneda’s lemma, the equivalence map (−, Ω) ' (LFibU)' Cat(Bb ) that is induced by the equivalence in theorem 4.5.2 on the underlying bS-valued sheaves is induced by assigning each functor C → Ω to the left ﬁbration P → C that is determined by the pullback square

P Ωb p C Ω

in which the map Ωb → Ω is the universal left fibration that is defined as the image of the identity functor Ω → Ω along the above equivalence. The inverse functor can also be described quite explicitly: Given a small left fibration P → C of categories in Bb, the associated functor C → Ω is given on the level of Cat(bS)-valued sheaves by the map Fun (−, C) → Fun (−, Ω) ' B , (c: A → C) 7→ P| Bb Bb /− c

in which the object P|c ∈ B/A is determined by the cartesian square

P|c P p A c C. This is a straightforward consequence of remark 4.5.8.

4.6. Yoneda’s lemma. We may use theorem 4.5.2 to construct a functorial version of the mapping groupoid construction for categories in B.

Deﬁnition 4.6.1. A large category C in B is said to be locally small if the left ﬁbration Tw(C) → Cop × C is small.

As a consequence of proposition 4.2.5, the fibre of the functor Tw(C) → Cop × C over any pair of objects (c, d): A → Cop × C in context A ∈ Bb is a groupoid in Bb, hence the fibre can be computed as the fibre of the induced map of core groupoids, which is simply the pullback

mapC(c, d) C1

(c,d) A C0 × C0. Using proposition 4.5.1, we may therefore deduce: 62 LOUIS MARTINI

Proposition 4.6.2. A large category C in B is locally small if and only if for any pair of objects (c, d) in

C × C in context A ∈ B the mapping groupoid mapC(c, d) is contained in B.

Example 4.6.3. The universe Ω for groupoids in B is locally small as its mapping groupoids can be identiﬁed with the internal mapping objects in the slice ∞-topoi over B (cf. proposition 3.7.2).

Proposition 4.6.4. A locally small category C in B is small if and only if C' is a small groupoid.

Proof. The condition is clearly necessary, so let us assume that C is locally small and that C' is a small

groupoid. By making use of the Segal conditions, we need only show that C1 is contained in B. Since C0 ×C0

is an object of B, this follows from the observation that C1 is recovered as the mapping groupoid of the pair (π0, π1): C0 × C0 → C0 × C0.

Lemma 4.6.5. Let C be a small category and D be a locally small category in B, and let f : C → D be a functor. Then the essential image E of f is small.

Proof. Being a full subcategory of D, the category E is locally small (using proposition 3.8.7), hence propo- ' sition 4.6.4 implies that E is small whenever E is a small groupoid. By corollary 3.9.5, E0 is the image of the map f0 : C0 → D0, hence one finds E0 ' colimn C0 ×D0 · · · ×D0 C0. As ∆ is a small category, it suffices to show that for each n ≥ 0 the (n + 1)-fold fibre product C0 ×D0 · · · ×D0 C0 ∈ Bb is contained in B. We may n+1 n+1 n+1 identify this object as the pullback of the map C0 → D0 along the diagonal D0 → D0 . Since the map D0 → Dn is a monomorphism in Bb, we obtain a monomorphism

C0 ×D0 · · · ×D0 C0 ,→ mapD(f0 pr0, . . . , fn prn)

n+1 where pri : C0 → C0 denotes the ith projection. Since D is by assumption locally small, the codomain of this map is contained in B, hence the result follows.

Proposition 4.6.6. For any small category C and any locally small category D in B, the functor category [C, D] is locally small as well.

Proof. Using proposition 4.6.2, we need to show that for any pair (f, g): A → [C, D] × [C, D] the mapping

groupoid map[C,D](f, g) is contained in B. Let E be the essential image of f when viewed as a functor A × C → D, and let E0 be the essential image of g when viewed as a functor A × C → D. Then both E and E0 are small categories by lemma 4.6.5. By the same argument, the image E00 of the functor E t E0 → D that is induced by the two inclusions is a small category that embeds fully faithfully into D. By construction, the map (f, g): A → [C, D] × [C, D] factors through the full embedding [C, E00] × [C, E00] ,→ [C, D] × [C, D]. As 00 [C, E ] is small, the mapping groupoid map[C,D](f, g) ' map[C,E00](f, g) must be contained in B.

If C is an arbitrary locally small category in B, applying theorem 4.5.2 to the left ﬁbration Tw(C) → Cop×C gives rise the mapping groupoid functor

op mapC(−, −): C × C → Ω.

By transposing across the adjunction Cop × − a [Cop, −], this functor determines the Yoneda embedding

h: C → [Cop, Ω].

Remark 4.6.7. Let C be a locally small category in B, let A ∈ B be an arbitrary object and let c be an object in C in context A. Let us furthermore ﬁx a map f : d → e in C in context A. Applying the mapping groupoid

functor mapC(−, −) to the pair (idc, f) of maps in C then results in a morphism f! : mapC(c, d) → mapC(c, e) YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 63

U 1 1 in B/A. Explicitly, this map is given by applying the chain of equivalences LFib (∆ ⊗ A) ' FunB(∆ ⊗ 1 1 A, Ω) ' FunS(∆ , B/A) to the left ﬁbration P → ∆ ⊗ A that arises as the pullback

P Cc/ p

(π ,f) ∆1 ⊗ A 1 A × C

1 in which π1 : ∆ ⊗ A → A denotes the projection. By construction of the equivalence of ∞-categories U 1 1 LFib (∆ ⊗ A) ' FunS(∆ , B/A), one now sees that the map f! ﬁts into the commutative diagram

' mapC(c, d) Z mapC(c, e)

d1 d0 (Cc/)0 (Cc/)1 (Cc/)0

A id A id A

d f e d1 d0 C0 C1 C0.

Let g : c → d be an arbitrary map in C in context A, and let σ : ∆2 ⊗ A → C be the 2-morphism that is encoded by the commutative diagram g c d f fg e. Let furthermore τ : ∆1 ⊗ A → C be the 2-morphism that is determined by the commutative diagram

c id c fg fg e.

1 1 2 2 1 1 On account of the decomposition ∆ ×∆ ' ∆ t∆1 ∆ , the pair (τ, σ) gives rise to a map (∆ ×∆ )⊗A → C

that by construction deﬁnes a section A → Z. Furthermore, the composition A → Z ' mapC(c, d) recovers

g and the composition A → Z → mapC(c, e) recovers fg. Therefore, the map f! acts by sending a map ∗ g : c → d to the composition fg : c → d → e. By a dual argument, the map f : mapC(e, c) → mapC(d, c) that is determined by applying the mapping groupoid functor to the pair (f, idc) sends a map g : e → c to the composition gf : d → e → c.

If C is a category in B, let us denote by ev: Cop × [Cop, Ω] → Ω the evaluation functor, i.e. the counit of the adjunction Cop × − a [Cop, −].

Theorem 4.6.8 (Yoneda’s lemma). For any small category C in B, there is a commutative diagram

Cop × [Cop, Ω] h×id [Cop, Ω]op × [Cop, Ω]

map[Cop,Ω](−,−) ev Ω in Cat(Bb). 64 LOUIS MARTINI

The proof of theorem 4.6.8 employs a strategy that is similar to the one used by Cisinski in [Cis19] for a proof of Yoneda’s lemma for ∞-categories. We begin with the following lemma:

Lemma 4.6.9. Let f P Q

p q id ×g C × C C × D be a commutative diagram in Cat(B) such that the maps p and q are left ﬁbrations, and suppose that for any object c: A → C the induced map f|c : P|c → Q|c is initial. Then f is initial.

Proof. By remark 4.4.2, q being a left ﬁbration implies that it suﬃces to show that f is a covariant equivalence over C×D. Using proposition 4.4.3, it is moreover enough to show that for any object d: A → D the induced

map f/d (that is obtained by pulling back f along id ×(πd)! : C×D/d → C×D) is a covariant equivalence over C. In fact, if this is the case, then proposition 4.4.3 implies that for any object c: A → C the induced map

(f/d)/c becomes an equivalence after applying the groupoidiﬁcation functor. Now it is straightfoward to see

that this map is equivalently given by the pullback f/c×d of f along the right ﬁbration (C × D)/c×d → C × D

that is determined by the object c × d: A × A → C × D. Since the natural map C × D/(c,d) → C × D/c×d is a gpd map of right ﬁbrations over C × D and therefore a right ﬁbration itself, one concludes that (f/(c,d)) is an equivalence for any object (c, d): A → C × D, hence another application of proposition 4.4.3 implies that f is a covariant equivalence over C × D.

Now since the projections C × C/d → C and C × D/d → C are smooth, the diagonal maps in the induced commutative diagram

f/d P/d Q/d

C are smooth (as left ﬁbrations are smooth by the dual of proposition 4.4.7). As the induced map f/d |c on

the ﬁbres over c: A → C is a pullback of the initial functor f|c along a proper map, this functor must be initial as well, hence we may apply 4.4.9 to deduce that f/d is a covariant equivalence over C, as required.

∗ ∗ ∗ Remark 4.6.10. In the situation of lemma 4.6.9, let πA(f)|c : πAP|c → πAQ be the functor in Cat(B/A) ∗ ∗ ∗ ∗ that arises as the ﬁbre of the map πA(f): πA(P) → πA(Q) over the object c: 1 → πAC that corresponds to ∗ c: A → C by transposition. Then f|c is obtained as the image of πA(f)|c under the forgetful functor (πA)!, ∗ hence remark 4.3.6 implies that f|c is initial if and only if πA(f)|c is initial. Hence lemma 4.6.9 implies ∗ that f is an initial map if and only if for every object A ∈ B the ﬁbre of πA(f) over every global object ∗ c: 1 → πAC is initial. Lemma 4.6.11. Let C be a category in B, let c: 1 → C be an object and suppose that there is a map h: ∆1 ⊗ C → C such that (1) the composite hd1 : C → C is equivalent to the constant map with value c, i.e. to the composite C → 1 → C in which the second arrow is given by c; (2) the composite hd0 : C → C is equivalent to the identity; 1 (3) the map h ◦ (id ⊗c): ∆ → C is equivalent to the identity map idc. Then c is an initial object.

Proof. Let us abuse notation and denote by c the constant map C → C with value c. The map h can be ∆1 ∆1 regarded as an object 1 → [C, C] that is sent to the constant map c by the projection d1 :[C, C] → [C, C]. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 65

Therefore the map h deﬁnes an object in the slice category [C, C]c/. Observe, moreover, that the commutative diagram

1 ∆ ' ∆1 [C, C]c/ [C, C] [C, C ]

1 c [C, C] id [C, C] in which both squares are cartesian gives rise to an equivalence

[C, C]c/ ' [C, Cc/].

With respect to this equivalence, the map h corresponds to a section C → Cc/ of the functor (πc)! : Cc/ → C that sends the object c: 1 → C to idc : 1 → Cc/. By lemma 4.3.17, this implies that c is initial.

Lemma 4.6.12. Let g : D → C be a functor between large categories in B and let pi: D → P → C be its factorisation into a final map and a right fibration. Suppose that the right fibration p is small, and let f : Cop → Ω be the associated functor. Consider the left fibration π that is defined by the cartesian square

Z [Dop, Ωb] p π g∗ [Cop, Ω] [Dop, Ω].

Then there is an initial object z : 1 → Z whose image along π is f.

Proof. Since i: Dop → Pop is initial and Ωb → Ω is a left ﬁbration, the pullback square in the statement of the lemma decomposes into a pasting of cartesian squares

Z [Pop, Ωb] [Dop, Ωb] π −◦p [Cop, Ω] [Pop, Ω] −◦i [Dop, Ω].

By applying the Yoneda embedding Cat(B) ,→ PSh (Cat(B)) to the left square, we obtain a pullback square b Sb b

map (−, Z) map (− × Pop, Ω) Cat(Bb ) Cat(Bb ) b

−◦(id ×p) map (− × Cop, Ω) map (− × Pop, Ω) Cat(Bb ) Cat(Bb )

of bS-valued presheaves on Cat(Bb). Hence maps E → Z correspond to pairs (h, k) together with a commutative diagram h E × Pop Ωb id ×p E × Cop k Ω, such that precomposition with a map F → E corresponds to pasting with the square

F × Pop E × Pop

F × Cop E × Cop. 66 LOUIS MARTINI

As a consequence, the cartesian square Pop Ωb p f Cop Ω gives rise to an object z : 1 → Z whose image along π is f. We still need to show that z is initial. To that end, recall that if we ﬁx a map k : E × Cop → Ω and view op the category E as an object in Cat(Bb)/[Cop,Ω] via the map E → [C , Ω] that is given by the transpose of k, op we may identify the ∞-groupoid map op (E, Z) with the ﬁbre of map (E, Z) → map (E×C , Ω) /[C ,Ω] Cat(Bb ) Cat(Bb ) op over k. Using the above cartesian square, this ∞-groupoid is in turn equivalent to map/Ω(k!(E × P ), Ωb), ∗ hence the adjunction k! a k gives rise to an equivalence of ∞-groupoids

op ∗ map/[Cop,Ω](E, Z) ' map/E×Cop (E × P , k Ωb)

that is natural in E in the sense that for any map s: F → E between categories in Bb there is a commutative square

' op ∗ map/[Cop,Ω](E, Z) map/E×Cop (E × P , k Ωb)

' op ∗ ∗ map/[Cop,Ω](F, Z) map/F×Cop (F × P , (s × id) k Ωb) in which the left vertical map is given by precomposition with s and the right vertical map is induced by the action of the base change functor (s × id)∗. By theorem 4.5.2 and the observation that the left ﬁbration id ×p: E × Pop → E × Cop classiﬁes the functor that is given by the composition

pr f E × Cop 1 Cop Ω, we therefore end up with an equivalence of ∞-groupoids ˆ map/[Cop,Ω](E, Z) ' Γ(map[E×Cop,Ω](f pr1, k))

that is natural in E in that for any map s: F → E in Cat(Bb) there is a commutative square

' ˆ map/[Cop,Ω](E, Z) Γ(map[E×Cop,Ω](f pr1, k))

' ˆ map/[Cop,Ω](F, Z) Γ(map[F×Cop,Ω](f pr1, k(s × id))).

op Let ϕ: f pr1 → k be the map in [Z × C , Ω] that corresponds to the identity Z → Z under the above equivalence, and consider the evident commutative square

id f pr1 f pr1

id ϕ ϕ fπ1 k

1 op as a map τ : id → ϕ in the category [∆ ⊗ (Z × C ), Ω]. As the identity map on f pr1 corresponds to the composition pr f (∆1 ⊗ Z) × Cop 1 Cop Ω,

the map τ deﬁnes a global section of the groupoid map[∆1⊗(Z×Cop),Ω](f pr1, ϕ) and therefore corresponds to a functor ∆1 ⊗ Z → Z. By construction, the map Z → Z that is induced by precomposition with the inclusion YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 67

d1 : Z → ∆1 ⊗ Z corresponds to the outer square in the commutative diagram

pr Z × Pop 1 Pop Ωb id ×p p pr f Z × Cop 1 Cop Ω, hence this map is equivalent to the constant functor z : Z → 1 → Z. Precomposing the map ∆1 ⊗Z → Z with the inclusion d0 : Z → ∆1 ⊗ Z, on the other hand, produces the identity on Z. As moreover the restriction of 1 the map ∆ ⊗ Z → Z along z : 1 → Z recovers the identity on z, lemma 4.6.11 implies that z is initial. Lemma 4.6.12 has the following immediate consequence:

Corollary 4.6.13. Let g : D → C be a functor between large categories in B and let pi: D → P → C be its factorisation into a final map and a right fibration. Suppose that the right fibration p is small, and let f : Cop → Ω be the associated functor. Then there is a cartesian square

op op [C , Ω]f/ [D , Ωb] p

g∗ [Cop, Ω] [Dop, Ω].

Proof. By lemma 4.6.12, the category Z that is deﬁned by the cartesian square

Z [Dop, Ωb] p

g∗ [Cop, Ω] [Dop, Ω]. admits an initial object z : 1 → Z whose image in [Cop, Ω] is f. By corollary 4.3.18 we obtain a commutative square 1 z Z f

op (πf )! op [C , Ω]f/ [C , Ω] in which the two maps starting in the upper left corner are initial and the two maps ending in the lower right corner are left fibrations. When regarded as a lifting problem, the above square thus admits a unique op filler [C , Ω]f/ → Z that is both initial and a left fibration and therefore an equivalence. Corollary 4.6.14. The global section 1: 1 → Ω that is determined by the final object 1 ∈ B defines a final object in the universe Ω, and there is an equivalence Ω1/ ' Ωb that fits into the commutative diagram

' Ω1/ Ωb

(π1)! Ω. Proof. Applying corollary 4.6.13 to D ' C ' 1, the object 1: 1 → Ω gives rise to a cartesian square

Ω1/ Ωb p

Ω id Ω, which implies that the upper horizontal map must be an equivalence. Moreover, if a: A → Ω is an object

that classiﬁes a map p: P → A in B, the equivalence mapΩ(a, 1πA) ' [P,A]/A from proposition 3.7.2 shows 68 LOUIS MARTINI

together with the dual version of proposition 4.3.19 that 1: 1 → Ω is also ﬁnal when viewed as an object in

Ω (as [P,A]/A is ﬁnal in B/A).

Remark 4.6.15. Corollary 4.6.14 shows that the Cat(bS)-valued sheaf that corresponds to Ωb is given by the assignment

A 7→ BA//A in which the right-hand side denotes the ∞-category of pointed objects in B/A.

∗ ∗ Lemma 4.6.16. For any object A ∈ B, the base change πÂΩb → πÂΩ of the universal left fibration in Cat(Bb) is the universal left fibration in Cat(Bb/A).

Proof. Corollary 4.6.14 implies that the universal left ﬁbration ΩbB → ΩB in Cat(B) is equivalent to the slice ∗ projection (π1)! :(ΩB)1/ → ΩB. Moreover, by lemma 3.7.6 there is a natural equivalenceπ ˆAΩB ' ΩB/A . Together with lemma 4.2.3, this shows that there is a commutative diagram

1 1 πˆ∗ Ω πˆ∗ (Ω∆ ) ' Ω∆ A b A B B/A

(d1,d0) (d1,d0) ∗ (1,id) ∗ ∗ ' πÂΩ πÂΩ × πÂΩB ΩB/A × ΩB/A

in Cat(Bb/A) in which both squares are cartesian, which proves the desired result. Proof of theorem 4.6.8. Let R ev → Cop × [Cop, Ω] be the left fibration that classifies the evaluation functor ev. Using theorem 4.5.2, it suffices to show that there is a cartesian square R ev Tw([Cop, Ω])

Cop × [Cop, Ω] h×id [Cop, Ω]op × [Cop, Ω]. Note that by deﬁnition of the evaluation functor, there is a cartesian square

f Tw(C) R ev

Cop × C id ×h Cop × [Cop, Ω]. Moreover, using functoriality of the twisted arrow category construction, we may construct a commutative diagram Tw(h)

g Tw(C) P Tw([Cop, Ω])

Cop × C id ×h Cop × [Cop, Ω] h×id [Cop, Ω]op × [Cop, Ω]

h×h

in which the right square is cartesian. As a consequence, one obtains a commutative square

f Tw(C) R ev

g P Cop × [Cop, Ω]. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 69

To complete the proof, it therefore suﬃces to produce a lift P → R ev in the previous square and to show that this map is an equivalence. This is possible once we verify that the two maps f and g are initial. In order to show that the map g is initial, note that we are in the situation of lemma 4.6.9, which means

op Cc/ → [C , Ω]h(c)/. op This map is initial as it sends the initial section idc : A → Cc/ to the initial section 1h(c) : A → [C , Ω]h(c)/. In order to prove that the map f : Tw(C) → R ev is initial, we employ lemma 4.6.9 once more to conclude

that it will be suﬃcient to show that the map f|c in the induced cartesian square

f|c R Cc/ ev |c

A × C id ×h A × [Cop, Ω]

∗ ∗ ∗ op ∗ is initial. Note that by lemma 4.2.3 there is an equivalence πA Tw(C) ' Tw(πAC) over πAC × πAC, and ∗ ∗ ∗ op by furthermore using lemma 3.7.6 one easily verifies that πA(h): πAC ,→ πA[C , Ω] recovers the Yoneda ∗ embedding of πAC. An application of lemma 4.6.16 moreover shows that the base change of the left fibration R op op ∗ ∗ op ∗ op ev → C × [C , Ω] along πA classifies the evaluation functor πAC × [πAC , ΩB/A ] → ΩB/A . Together with remark 4.6.10, these observations allow us to assume without loss of generality A ' 1. op op Applying lemma 4.6.12 to the factorisation 1 → Cc/ → C of c into a final map followed by a right fibration (cf. corollary 4.3.18), we obtain the cartesian square

op [C , Ω]h(c)/ Ωb

[Cop, Ω] −◦c Ω.

Observe that the functor − ◦ c is equivalent to the composition ev ◦(c × id). Hence there is an equivalence op R op [C , Ω]h(c)/ ' ev |c over [C , Ω]. In order to show that f|c is initial, it therefore suﬃces to show that the canonical square op Cc/ [C , Ω]h(c)/

C h [Cop, Ω]

that is induced by functoriality of the slice construction is a pullback square, as in this case the map f|c op is given as a composition of an equivalence with the initial map Cc/ → [C , Ω]h(c)/. Now the equivalence of functors (− ◦ c) ◦ h ' mapC(−, −) ◦ (c × id) implies that the above square is cartesian whenever the composition op Cc/ → [C , Ω]h(c)/ → Ωb is equivalent to the composition

Cc/ → Tw(C) → Ωb in which the ﬁrst map is the one that is determined by the pullback square from proposition 4.2.7. Note

that two maps Cc/ ⇒ Ωb over Ω must be equivalent whenever they send an initial object in Cc/ to equivalent objects in Ωb, as in this case the two maps present two solutions to a lifting problem between an initial map and the left ﬁbration Ωb → Ω. Therefore, it suﬃces to compare the image of idc under these two functors. 70 LOUIS MARTINI

Now the proof of lemma 4.6.12 shows that the image of idc along the ﬁrst map is given by the composition of the two upper horizontal maps in the diagram

idc op 1 C/c Ωb

h(c) Cop Ω

in which the square is the pullback diagram that is induced by proposition 4.2.7. Therefore the claim follows op from the observation that both the map idc : 1 → C/c → Tw(C) and the map idc : 1 → Cc/ → Tw(C) correspond to the composite s0c: 1 → C0 → C1 and are therefore equivalent.

Corollary 4.6.17. For any small category C in B, the Yoneda embedding h: C → [Cop, Ω] is fully faithful.

Proof. By theorem 4.6.8 the canonical square

Tw(C) Tw([Cop, Ω])

op Cop × C h ×h [Cop, Ω]op × [Cop, Ω]

that is obtained by functoriality of the twisted arrow construction is cartesian, which proves the claim upon applying the core groupoid functor.

Corollary 4.6.18. Let C and D be categories in B and let α: ∆1 ⊗ A → [C, D] be a morphism in [C, D]. Then α is an equivalence if and only if for all c: B → C the map α(c): ∆1 ⊗ (A × B) → D is an equivalence in D.

Proof. The condition is clearly necessary, so suppose that α(c) is an equivalence for every object c: B → C.

By replacing B with B/A, we may assume without loss of generality A ' 1. By corollary 4.6.17, the functor op op h∗ :[C, D] ,→ [C, [D , Ω]] ' [C × D , Ω] is fully faithful and therefore in particular conservative. It therefore op suﬃces to show that the map h∗α is an equivalence. For any (c, d): B → C × D , the map h∗α(c, d) corresponds to the image of α(c) under the map

D ,→ [Dop, Ω] → [B, Ω]

in which the second arrow is obtained by precomposition with d. As a consequence, the map h∗α(c, d) must be an equivalence in Ω. By replacing C with C × Dop, we may therefore assume without loss of generality D ' Ω. In this case, the desired result follows from proposition 4.1.13.

Deﬁnition 4.6.19. Let C be a category in B. Then a presheaf f : A × Cop → Ω is said to be representable by an object c: A → C if there is an equivalence f ' h(c), where h: C ,→ [Cop, Ω] denotes the Yoneda embedding.

Remark 4.6.20. If p: P → A × C is a right fibration between categories in B (i.e. an object A → RFibC), we may say that p is representable if the associated presheaf A × Cop → Ω that classifies p is representable in the sense of definition 4.6.19. Equivalently, this means that there is an object c: A → C and an equivalence

C/c ' P over A × C.

Proposition 4.6.21. A left ﬁbration p: P → C × A between categories in B is representable by an object c: A → C if and only if there is an initial section A → P over A. YONEDA’S LEMMA FOR INTERNAL HIGHER CATEGORIES 71

Proof. If p is representable by an object c: A → C then there is an equivalence P ' Cc/ over C × A, hence proposition 4.3.8 implies that there is an initial section A → P over A. Conversely, if there is such an initial section s: A → P and if c: A → C denotes the image of s along the functor π0p: P → C, the lifting problem

idc A Cc/

s p P C × A

admits a unique solution which is necessarily an equivalence since idc is initial and p is a left ﬁbration.

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