19 Oct 2020 (∞,1)-Categorical Comprehension Schemes

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The scheme states that every definable subclass of a set is again a set. Or in other words, that the set theoretic universe satisfies every set theoretical comprehension scheme. In very general non-rigorous terms, given a model M of some fragment of mathematics (this may be a model of set theory, some notion of category, some model of type theory, . . . ) and some logical structure S over M defined by external “meta-mathematical” means, we say that M satisfies the comprehension scheme associated to S if this structure applied to any mathematical object M in M can be internalized in M. What any of these words exactly mean is very much context dependent, we refer to Table 1 for the context at hand. In his critique of the foundations of naive category theory [4], Bénabou provided an intuition to define the notion of comprehension schemes not only in category theory (over the topos of sets), but in category theory over any other category in a syntax-free way.1 In this generality, comprehension schemes become properties of Grothendieck fibrations over arbitrary categories. Which particular comprehension schemes are satisfied by a given Grothendieck fibration E ։ C then depends on the categorical constructions available in E over C and in C itself. The notion has been made precise in considerable generality by Johnstone in [13, B1.3], tieing together the elementary examples given in the glossary of [4] to a structurally well behaved theory. This theory turns out to be even better behaved in the (∞, 1)-categorical set-up and in fact easier to manage, given the following motivation. For the moment being think of the Yoneda embedding of an ordinary category C as a canonical elementary embedding of C into its presheaf category Cˆ. As such it converts formal/synthetic structure in C to real-world structure over C. Here, “real-world” structure means structure in the background theory, that is, set theory. In this sense, if we think of naive mathematics as the background theory of formal set theory, we may as well think of the inclusion Set ֒→ Cls of sets into proper classes as a Yoneda embedding, mapping the formal membership relation to the naive membership relation. Following [4, 6], the ZF-comprehension scheme of a set X associated to a proper class S is then equivalent to representability of the “presheaf” X ∩ S. In a series of abstractions from set theory to fibred higher category theory, and hence from properties to structures, a comprehension scheme S for a cartesian fibration E ։ C is defined as a presheaf of diagrams of fixed shape in E defined over C, as illustrated in Table 1 and made precise in Lemma 3.5. We then say that S is satisfied by the fibration E ։ C if this collection of diagrams is “small” over C in the sense that its associated presheaf is representable. The expressive power of comprehension schemes encompasses important examples in category theory, most notably sufficient and necessary conditions to characterize elementary toposes (over other elementary toposes). The scope of Johnstone’s definition of comprehension schemes yields tools to discuss many other such notions defined in more specific contexts as well, including Jacob’s notion of comprehension categories ([11]), or even more specific Lawvere’s original notion of comprehension in the context of hyperdoctrines ([16], [11, 4]). It is therefore the (∞, 1)-categorical adaptation of Johnstone’s notion of comprehension which we will center this paper on. We will see that many characterizations of ordinary categorical constructions will carry over flawlessly, while others in fact will work out much better for the reason that “evil” equalities naturally arising in the context of ordinary category theory will be replaced by “good” instances of equivalences between (∞, 1)-categories. It eliminates the gymnastics with elementary fibrations in [28], and also affects the remarks on the “strangeness”

1It may be noteworthy to point out that B´enabou in this work purposefully did not even specify what exactly a “category” is ([4, (0.5)]), and that, on the basis of his eﬀectively meta-“meta-mathematical” analysis, the framework of (∞, 1)-category theory over the base ∞-topos of spaces appears to yield a suitable notion of such a “category theory” in his sense. In this reading one then is to replace the occurrences of “ZF” in [4] by a suitable univalent type theory.

2 Mathematics: Set Theory Higher Category Theory The naive set theory of (∞, 1)-Category Theory over Meta-mathematics: proper classes the ∞-topos S of spaces A model of a fragment A set theoretic universe An indexed (∞, 1)-category of mathematics: V E : Cop → S A mathematical ob- A set X in V A diagram X : I →E over C ject: Some external logical A functor of (∞, 1)-categories A class S structure: G: I → J The induced structure The presheaf G∗ ↓ X of J- over a mathematical S ∩ X shaped diagrams in E extend- object: ing X in E over C E has G-comprehension if the V has S-comprehension Meaning of Compre- presheaf G∗ ↓ X is repre- if S ∩ X is small for all hension: sentable for all I-diagrams X X ∈ V . in E.

Table 1: A short dictionary of comprehension of equality in [4, 8], because commutativity of squares in (∞, 1)-categories is a matter of equivalence, not of meta-mathematical equality. In this sense, the study of equality becomes a study of equivalence. The main sources for the ordinary categorical constructions and definitions we have adapted are Johnstone ([13]), Jacobs ([11], [12]) and Streicher ([28]). The results presented in this paper build to a large extent on the fundamental work on quasi-categories provided by Lurie in [17]. There are many alleys which lend themselves to be studied in the given context but were either only superficially glanced over or entirely omitted. An exhaustive treatment of fibred (∞, 1)-categories with a scope comparable to [13, B1] or [28] in the ordinary categorical context would naturally require (∞, 2)-categorical considerations, which are barely touched upon in this paper because the notions studied here are properties of one indexed (∞, 1)-category at a time. The collection of all C-indexed (∞, 1)-categories is only introduced in Section 2 to state its equivalence to the collection of fibred (∞, 1)-categories over C via Lurie’s Unstraightening construction, and therefore it is defined as an (∞, 1)- category itself. Partial treatments of fibred (∞, 1)-category theory in such generality can be found e.g. in [3], [17] and [23].

Outline of the paper and main results In Section 2 we recall the necessary material on cartesian fibrations from Lurie’s book [17]. We define the basic notions relevant to define and study comprehension schemes for fibred (∞, 1)-categories, give reference to the equivalence between indexed (∞, 1)-category theory and fibred (∞, 1)-category theory via the corresponding Grothendieck construction (“Unstraightening”) and discuss the most essential examples of indexed (∞, 1)-categories and their fibred counterparts. Section 3 introduces the definition of comprehension schemes and discusses many examples like smallness, local smallness and definability of equivalences between objects and parallel pairs of arrows for different cartesian fibrations. We provide tools for mu- tual reduction and verification of various instances, several of which are generalizations of notions and results from [13, B1.3]. In Section 4 we define the externalization construction of internal (∞, 1)-categories in left exact (∞, 1)-categories C. We will see that the cartesian fibrations arising from

3 internal (∞, 1)-categories in this way can be characterized by two comprehension schemes (smallness and local smallness). This is a result which fails for 1-category theory, because, in a nutshell, smallness defined relative to the base topos of sets is a fibrewise set-like property where it should be fibrewise groupoidal. We will revisit this matter in more detail in Remark 3.9. Here, the externalization construction

Ext: CS(C) → IC(C) from the (∞, 1)-category of internal (∞, 1)-categories in C to the (∞, 1)-category of indexed (∞, 1)-categories over C is a generalization of the synonymous ordinary categorical construction e.g. given in [12, 7.3]. It can be thought of as a Yoneda embedding for category objects, in the sense that when restricted to the (∞, 1)-category of internal ∞- groupoids, the externalization restricts exactly to the Yoneda embedding y : C → Cˆ. It furthermore is fully faithful as well, and allows the detection of the “representables” via a smallness criterion (Theorem 4.11). op ։ Two fundamental examples are given by the universal cartesian fibration Dat∞ op op ։ op Cat∞ and the universal right fibration S∗ S as constructed in [17, 3.3.2]. There, Lurie showed that the universal right fibration is represented by the terminal object in the (∞, 1)-category S of spaces. The universal cartesian fibration on the other hand cannot be representable in the same sense, but we will see that it is representable in the directed sense referred to above (Example 4.14). The associated internal (∞, 1)-category • is given by the “freely walking chain” ∆ ∈ Fun(N(∆), Cat∞). In fact, it will follow from general considerations about comprehension schemes that the universal right fibration must then be represented by the “freely walking invertible chain” I∆• in S in the same way (Example 3.26, Remark 4.14). But this is object is contractible in Fun(N(∆), S), which recovers Lurie’s original result by other means. Applied to comprehension (∞, 1)-categories over a fixed (∞, 1)-category C in the sense of Jacobs ([11]), we will relate smallness to univalent maps in C whenever C has enough structure to define univalence in the first place (Proposition 4.18). On the one hand, this will give a characterization of ∞-toposes in terms of smallness over themselves (Re- mark 4.21). On the other hand, this comparison suggests a general definition of univalent objects in fibred (∞, 1)-categories whose potential study we leave to future work. In Section 5 we take a look at indexed (∞, 1)-categories over an (∞, 1)-category C from a homotopical algebraic perspective and assume that C is the homotopy (∞, 1)-category of a model category M. Against the background of the results in Section 4, we will study the contravariant functors from M into the category S of simplicial sets. We show that the functors which arise as the externalization of a complete Segal object in M are exactly the right Quillen functors into S equipped with the Joyal model structure (Proposition 5.4). Hence, in the sense of Section 4, the right Quillen functors Mop → (S, QCat) can be thought of as the small and locally small (or the representable) indexed quasi-categories over M. We finish the section with a characterization of univalence (Corollary 5.15) and of the fibration extension property and the weak equivalence extension property as right Quillen conditions in this context (Proposition 5.18).

Notation. The notation will follow [17] in large. In particular, the exponential DC of two quasi-categories C, D in the category S of simplicial sets will be denoted by Fun(C, D). The large (∞, 1)-category of small (∞, 1)-categories as deﬁned and studied in [17, 3] will ≃ be denoted by Cat∞. Its hom-spaces Cat∞(C, D) are given by the core Fun(C, D) , that is, the largest Kan complex contained in the quasi-category Fun(C, D). The (∞, 1)-category of spaces will be denoted by S. Given an (∞, 1)-category C and an object C ∈C, we denote the associated slice (∞, 1)- category (called overcategory in [17]) by C/C . The standard n-simplex in the category of simplicial sets will be denoted by ∆n. Its non-degenerate edge from i to j for integers 0 ≤ i < j ≤ n will be denoted by ∆{i,j} ⊂ ∆n.

4 There are two model structures on the category S of particular importance for this paper. These are the Quillen model structure (S, Kan) for Kan complexes and the Joyal model structure (S, QCat) for quasi-categories. For the sake of readability only, whenever we say “∞-category” we mean (∞, 1)- category. While the results of this paper should not depend on any model of (∞, 1)- category theory (by the transition results in [23]), the author chose to work with quasi- categories for the presentation of computations and constructions in virtue of both the extensive use of the results in [17], as well as the model categorical material covered in Section 5. Here it appeared the most natural to consider simplicial sets with the Joyal model structure. The only other model for the theory of (∞, 1)-categories that will be used in this paper is the homotopy theory of complete Segal spaces in Section 4.

Acknowledgments. The author would like to thank John Bourke for multiple stimulat- ing discussions on this paper.

2 Preliminaries on indexed (∞, 1)-category theory

In order to define and study comprehension schemes for cartesian fibrations of ∞-categories, in this section we cover, first, the underlying basic definitions and examples of indexed ∞- categories and cartesian fibrations, second, the ∞-categorical Grothendieck construction in form of Lurie’s Straightening and Unstraightening pair, and, lastly, a few definitions and constructions regarding marked simplicial sets. The presentation of the material in this section is largely non-technical and mainly serves the purpose of introducing and organiz- ing the notions and examples to be studied in later sections. The underlying calculations and proofs are contained in [17] where all given ∞-categorical concepts are developed in detail.

Deﬁnition 2.1. Let C be an ∞-category. A C-indexed ∞-category is a functor I : Cop → Cat∞. If C has a terminal object 1 ∈ C, we say that I(1) is the underlying ∞-category op of I. We deﬁne the ∞-category IC(C) := Fun(C , Cat∞) of C-indexed ∞-categories. A functor of C-indexed ∞-categories I, J is simply an edge in IC(C), that is, a natural transformation from I to J . Likewise, a C-indexed ∞-groupoid is a functor X : Cop → S. The ∞-category of C- indexed ∞-groupoids is the ∞-category Cˆ ≃ Fun(Cop, S) of presheaves on C. The following examples are generalizations of some of the 1-categorical examples considered in [13, B1.2].

Examples 2.2. Let C be an ∞-category. 1. Our ﬁrst example is in fact not indexed over C itself, but rather represented by C in a (∞, 2)-categorical sense. Namely, the functor

op Fun( , C): Cat∞ → Cat∞

is a Cat∞-indexed ∞-category. Its restriction to the category S of spaces is the op “naive” S-indexed ∞-category Fun( , C): S → Cat∞. Its underlying ∞-category is C itself. op 2. If C has pullbacks, the slice functor C/ : C → Cat∞ is the “canonical indexing ∗ of C over itself”. It takes a map f : C → D to the pullback functor f : C/D → C/C . Again, its underlying ∞-category is C whenever C is left exact. When applied to op C ≃ S, the naive indexing Fun( , S): S → Cat∞ and the canonical indexing op S/ : S → Cat∞ are pointwise equivalent by the (unmarked) Straightening construction over Kan complexes ([17, Theorem 2.2.1.2]). The Straightening construction will be discussed below in more generality.

5 op 3. Given a C-indexed ∞-category I : C → Cat∞, each functor F : D→C induces a ∗ op op change of base F I : D → Cat∞ via precomposition with F . 4. If C has products, we obtain an action ( )(·) : Cop × IC(C) → IC(C) given on objects by IC(C′)= I(C′ × C). It is the currying of the change of base along the product op ˆ ×: C×C→C. This action induces a map [ , ]IC(C) : IC(C) × IC(C) → C as the curried adjoint of the composition

(·) 1×( ) HomIC(C) IC(C)op ×Cop × IC(C) −−−−−−→ (IC(C))op × IC(C) −−−−−−→S.

For C-indexed ∞-categories I, J one may thus form the C-indexed ∞-groupoid [I, J ]IC(C) of C-indexed functors between them. It is given pointwise by the mapping spaces C [I, J ]IC(C)(C) = IC(C)(I, J ), its underlying ∞-category is the mapping space IC(C)(I, J ). 5. The most fundamental examples of indexed ∞-groupoids are the representable presheaves C( ,C). In Bénabou’s foundational terms ([4]), they are the archetypes of “small” indexed ∞-groupoids over C, and as such they will serve as reference structures in the definition of comprehension schemes in Section 3. In ordinary category theory (that is the 2-category theory of 1-categories), the Grothendieck construction of pseudo-functors into the category Cat of small categories yields a 2- categorical equivalence between indexed category theory and fibred category theory (see e.g. [13, Theorem 1.3.6]). An (∞, 1)-categorical analogue of the Grothendieck construction was first given by Lurie’s (Un)Straightening construction ([17, 3.2]), which provides a very general framework to pass between cartesian fibrations over a simplicial set S and op functors from S into the ∞-category Cat∞. This construction yields an equivalence

/ op StC : Cart(C) o ≃ Fun(C , Cat∞): UnC between the ∞-category of cartesian fibrations and cartesian functors over C ([17, Def- inition 2.4.2.1]) and the ∞-category of C-indexed small ∞-categories. The fibres of the op fibration UnC(I):ΣI ։ C are equivalent to the values of the functor I : C → Cat∞ on the objects of C. Furthermore, the construction yields an equivalence

/ op StC : RFib(C) o ≃ Fun(C , S): UnC between the ∞-category of right ﬁbrations over C ([17, 2]) and the ∞-category of C- indexed ∞-groupoids (presheaves over C, that is). We will omit the index C and simply write St and Un to refer to the respective Straightening and Unstraightening functors whenever the base C is clear from context. In this sense, indexed ∞-category theory and ﬁbred ∞-category theory are equivalent notions.

≃ Definition 2.3. Let (·) : Cat∞ → S be the functor assigning to each ∞-category its core, i.e. C≃ is the largest ∞-groupoid contained in C. Given a cartesian fibration p: E ։ C, we obtain a presheaf ≃ op St(p) (·) C −−−→ Cat∞ −−→S whose unstraightening we denote by p× : E× ։ C and call the core of p. Given a cartesian fibration p: E ։ C, the domain E× is an ∞-category itself. It can × be constructed as a wide ∞-subcategory E ⊆E such that the restriction p|E× (which is a right fibration by [17, Corollary 2.4.2.5]) is equivalent to the right fibration p×. Here, the p×-cartesian arrows in E× are exactly the p-cartesian arrows in E, and the vertical arrows in E× are exactly the vertical equivalences in E.

6 Examples 2.4. Let C be an ∞-category. ։ 1. The forgetful functor s: (Cat∞)/C Cat∞ is the unstraightening of the repre- op sentable functor Cat∞( , C): Cat∞ → S. It therefore gives an example of a representable fibration as to be defined below. Similarly, the forgetful functor s: S/C ։ S op is the unstraightening of the restriction Cat∞( , C): S → S. More generally, we may consider the unstraightening of the naive indexing Fun( , C): Sop → ։ Cat∞ of C, and denote it in type theoretic fashion by D:S Fun(D, C) S. Its domain is the “∞-category of families of C”, generalizing the classic construction of the P 1-category Fam(C) of families over a 1-category C. In the same way, the cartesian fi- ։ op bration D:Cat∞ Fun(D, C) Cat∞ denotes the unstraightening of Fun( , C): Cat∞ → Cat . By construction, its core is exactly the representable fibration s: (Cat ) ։ ∞ P ∞ /C Cat∞. 2. The codomain map t: Fun(∆1, C) ։ C is a cartesian fibration whenever C has pullbacks. Its Unstraightening is the canonical indexing of C over itself and therefore will be called the canonical fibration over C. op 3. Change of base of an indexed ∞-category I : C → Cat∞ along a functor F : D→C unstraightens to the pullback of fibrations

Un(F ∗I) / Un(I) ·y D / C. F This follows directly from [17, Proposition 3.2.1.4]. Up to cartesian equivalence, every cartesian fibration p: E ։ C is the change of base of the universal cartesian fibration op op ։ op π : Dat∞ Cat∞ op op along its straightening St(p) : C → Cat∞. The construction of the universal cocartesian fibration π : Dat∞ ։ Cat∞ is presented in [17, 3.3.2]. In particular, whenever C has pullbacks, the canonical fibration t: Fun(∆1, C) ։ C is equivalent to the change of base of the universal cartesian fibration along the canonical indexing of C, so we obtain a homotopy cartesian square

1 / op Fun(∆ , C) Dat∞

t πop / op C Cat∞ C/

in the Joyal model structure (S, QCat). Change of base along some functor F : D→C of the canonical indexing of C over itself gives rise to the “Artin glueing” of F ,

C ↓ F / Fun(∆1, C) ·y gl(F ) t D / C. F One computes that the ∞-category of cartesian sections of the Artin glueing gl(F ) is the ∞-category of cartesian natural transformations over F , and as such it is ∗ op equivalent to the limit of the functor F (C/ ): D → Cat∞ ([17, 3.3.3])). The lo- calization of the ∞-category C ↓ F at the class of gl(F )-cartesian arrows is equivalent to the colimit of the very same functor ([17, 3.3.4]).

7 4. Similarly, up to cartesian equivalence, every right fibration is the change of base of op op the universal right fibration S∗ ։ S along its straightening. For example, every object C ∈ C yields a functor ev(C): Cˆ → S by evaluation of presheaves over C at C. If by y : C → Cˆ we denote the Yoneda embedding, the ôp ։ ôp associated right fibration is given by the forgetful functor s: C/y(C) C . That means the square ˆ / Cy(C)/ / S∗

Cˆ / S ev(C) is homotopy cartesian in the Joyal model structure. This is shown in [17, Theorem 5.1.5.2].

Definition 2.5. A right fibration over C is representable if its associated presheaf via Straightening is representable. Thus, up to equivalence, the representable fibrations over C are exactly the right ։ fibrations of the form s: C/C C for objects C in C.

Lemma 2.6 ([17, Proposition 4.4.4.5])). A right ﬁbration p: E ։ C is representable if and only if the domain F has a terminal object t ∈ F(C); then F≃C/C .

This subsumes the basic notions of indexed and ﬁbred ∞-category theory necessary for the subsequent sections to follow. Further constructions, e.g. concerning limits and colimits in this context can be found in [17, 4.3.1] and in more detail in [17, 6.1.1] in the case C is presentable. Other constructions have been studied in [3].

We end this section with a short recap of the category S+ of marked simplicial sets and the cartesian model structure as deﬁned in [17, 3.1]. A marked simplicial set is a pair X = (I, A) where I is a simplicial set and A ⊆ I1 is a subset which contains all degenerate edges. We say that A is the set of marked edges in X. A map f : (I, A) → (J, B) of marked simplicial sets is a functor f : I → J of simplicial sets such that f[A] ⊆ B. For every simplicial set I we obtain two cases of special interest. ♭ One, the minimally marked simplicial set I := (I,s0[I0]) and, two, the maximally marked simplicial set I♯ := (I,I). The category S+ of marked simplicial sets is cartesian closed, the underlying simplicial set of its internal hom-objects Y X is denoted by Map♭(X,Y ). The simplicial subset Map♯(X,Y ) ⊆ Map♭(X,Y ) consists of the simplices whose 1-boundaries are all marked in Y X . These two simplicial sets are characterized by the formulas S(I, Map♭(X,Y )) =∼ S+(I♭ × X,Y ), S(I, Map♯(X,Y )) =∼ S+(I♯ × X,Y ).

Whenever p: E ։ C is a cartesian fibration, the object E♮ ∈ S+ denotes the marked simplicial set (E, {p-cartesian edges}). Thus, E× is the largest subsimplicial set of E such × ♮ ♯ that (E ) = E with respect to the restriction p|E× . It follows that a cartesian fibration p: E ։ C is a right fibration if and only if E♮ = E♯. As a special case of [17, 2.1.4, 3.1.3], given an ∞-category C, Lurie constructs two S+ model structures on the slice category /C♯ . These are the contravariant model structure, whose fibrant objects are exactly maps of the form p: E♯ ։ C♯ such that p is a right fibration of underlying simplicial sets, and the cartesian model structure, whose fibrant objects are exactly of the form p: E♮ ։ C♯ such that p is a cartesian fibration on underlying simplicial sets.

8 3 (∞, 1)-Comprehension schemes

In this section we generalize the notion of comprehension schemes for indexed categories as treated in [13, B1.3] to the ∞-categorical framework and study fundamental examples with a view towards ∞-topos theory. In Section 3.2 we apply the discussion of comprehension schemes to cartesian subfibrations and consider the two special cases of definability in the sense of Bénabou ([4]) and comprehension ∞-categories in the sense of Jacobs ([11]). We fix an ∞-category C for the entire section.

3.1 Definitions and examples The following definitions are generalizations of the 1-categorical definitions given by John- stone in [13, B1.3]. The choice of these definitions which at first may appear rather opaque will be illuminated in Lemma 3.5. For now, as guiding motivation, we define comprehension schemes of a cartesian fibration p: E ։ C in such a way that any such scheme expresses the existence of some particular ∞-categorical structure S on all fibres p(C) collectively, such that all transition maps St(p)(f): p(C) → p(D) are S-homomorphisms.

Deﬁnition 3.1. Let p: E ։ C be a cartesian ﬁbration. For a simplicial set I ∈ S, let Rect(I, E) be given by the pullback Rect(I, E) / (EI )× ·y _ EI (1) pI C / CI . ∆

The vertical map pI : Map♭(I♭, E♮) ։ Map♭(I♭, C♯) is a cartesian fibration by [17, Re- mark 3.1.1.10, Proposition 3.1.2.3]. Its cartesian edges are exactly the edges in the wide ∞-subcategory (EI )× = Map♯(I♭, E♮) by [17, Proposition 3.1.2.1]. It follows that the vertical composite map (pI )× : (EI )× ։ CI in (1) is a right fibration, and so the pullback Rect(I, E) →C is a right fibration as well. In particular, Rect(I, E) is an ∞-category, and for every diagram X ∈ Rect(I, E), the induced map on slices Rect(I, E)/X → C/p(X) is a trivial Kan fibration. In less formal terms, Rect(I, E) ⊆ Fun(I, E) is the ∞-category of vertical diagrams and horizontal natural transformations over p: E ։ C. Its objects are the vertical diagrams contained in the ∞-category C∈C Fun(I, E(C)), its edges between two vertical diagrams X : I → E(C) and Y : I → E(C′) are the natural transformations α: X → Y in E which ` ′ are pointwise cartesian over one single arrow fα : C → C in C. At times it will be useful to consider the intermediate pullback Rect(I, E) / (EI )× _ ·y _ ∗ I / I ∆ (E )y E (2) · pI C / CI ∆ so that ∆∗(EI ) ։ C is a cartesian fibration with Rect(I, E)=(∆∗(EI ))×. Both pullbacks are homotopy pullbacks in the Joyal model structure for quasi-categories (because, for instance, the pullback of a span in which all vertices are fibrant and one leg is a fibration yields a homotopy pullback in any model category).

9 Definition 3.2. Let G: I → J be a map between simplicial sets such that J is connected and the ∞-category C has π0(I)-sized products. Say a cartesian fibration p: E ։ C has G-comprehension if the restriction G∗ : Rect(J, E) → Rect(I, E) has a right adjoint. Throughout this paper, the existence of a right adjoint to a given functor will by verified by the existence of universal arrows in its associated comma ∞-categories (see [19, Proposition 3.1.2] and [6, Proposition 6.1.11]). The two conditions on I and J in Definition 3.2 are inserted only to remedy a simpli- fication we applied in Definition 3.1 when compared to Johnstone’s original 1-categorical definitions. We explain the difference and its irrelevance in the following remark. Anyway, Definition 3.2 is clearly well-typed for functors G: I → J between any pair of simplicial sets, only that it will not yield a faithful generalization of Johnstone’s 1-categorical notion of G-comprehension if the two conditions are not satisfied.

Remark 3.3. Definition 3.1 is slightly unfaithful to Johnstone’s 1-categorical definition given in [13, B1.3]. The reason for the choice of Definition 3.1 is, first, that it eliminates repetitive arguments throughout this paper, and second, that this rectangular diagram ∞-category yields a presheaf over C by construction. Following [13] on the nose, one would define Rect(I, E) ⊆ (EI )× to be the full ∞- subcategory generated by the diagrams contained in the set of vertices S+(I♯, (E, {vertical edges in E}). This notion of “rectangular diagram” comes with a less strict notion of both vertical diagram and horizontal natural transformation. Indeed, if we denote these ∞-categories of less strictly rectangular diagrams by Rect+(I, E) for the moment being, we obtain inclusions as follows.

ι+ I × Rect(I, E) / Rect+(I, E) / (E ) ·y ·y (3) C / Cπ0(I) / CI ∆ We ignore this “+”-fragment of additional generality, because the whole idea of rectangular functor ∞-categories is solely introduced to deﬁne and study the notion of G- comprehension. In practice – that is, in all examples considered here and, in the 1- categorical case, in [13] or [28] otherwise – the codomain J is a connected ∞-category and the ∞-category C has π0(I)-sized products. In this case, one can show that the two induced notions of G-comprehension are equivalent as follows. The restriction along any functor G: I → J yields the following commutative square.

∗ Rect (J, E) G / Rect (I, E) +O +O ι+ ι+ (4) ? ? Rect(J, E) / Rect(I, E) G∗

The two vertical inclusions have a right adjoint each whenever C has both π0(I)- and π0(J)-indexed products. That is, because for every diagram X ∈ Rect+(I, E) we obtain pullbacks of the form / ι+ ↓ X / Rect+(I, E)/X ·y s Rect(I, E) / Rect+(I, E) ·y ι+ C / Cπ0(I). ∆ All vertical arrows are right ﬁbrations, and the upper right corner comes with an acyclic π0(I) ﬁbration into the slice ∞-category (C )/p(X). Consequently, the comma ∞-category

10 ι+ ↓ X is equivalent to the comma ∞-category ∆ ↓ p(X). Since we assume that C has

π (I)-indexed products, this comma ∞-category has a terminal object given by the counit 0 ⊢ of the adjunction ∆ lim. It follows that ι+ has a right adjoint as well. In fact, if J is connected, the left vertical arrow in (4) is an equivalence. In this case, by the above, the existence of π0(I)-sized products in C implies that the upper horizontal arrow in (4) admits a right adjoint if and only if the lower horizontal arrow admits a right adjoint. Indeed, one direction is immediate. The other direction follows since the vertical inclusions are full (it is easy to show that a functor which admits a right adjoint after postcomposition with a fully faithful inclusion admits a right adjoint itself). Thus, whenever J is connected, the two notions of G-comprehension are equivalent. In this sense, Definition 3.2 yields a generalization of the comprehension schemes for indexed 1-categories treated in [13, B1.3]. In the terms provided by Bénabou’s original intuition, validation of G-comprehension means that, given a vertical I-diagram X in E, the ∞-category of horizontal J-extensions of X is “small” over C in that it is equivalent to a representable fibration over a slice of C. We explain this in more detail in the following paragraph. Let G: I → J be a map of simplicial sets, p: E ։ C be a cartesian fibration and X ∈ Rect(I, E). Assume for the moment being that the functor G is a monomorphism.

Lemma 3.4. If G: I → J is a monomorphism, then the restriction

G∗ : Rect(J, E) → Rect(I, E) is a right ﬁbration.

Proof. The restriction G∗ : Rect(J, E) → Rect(I, E) commutes with the two right fibrations into C, and so it suffices to show that it is an inner fibration (all edges in Rect(J, E) are automatically G∗-cartesian). For integers 0

(hn)♯×1 n ♯ ♭ i / n ♯ ♭ (Λi ) × I (∆ ) × I ◆◆ ❍❍ ¯ ◆◆ 1×G ❍ f2 ◆◆◆ ❍❍ ◆ π1 ❍❍ ◆◆◆& ❍ f¯ ❍❍$ π n ♯ ♭ 1 / ♮ 1 (Λi ) × J E

π1 (hn)♯ n ♯ i / n ♯ p (Λi ) (∆ ) ◆◆◆ ❍❍ ◆◆◆ ❍❍ g2 ◆◆◆ ❍❍ ◆◆◆ ❍❍ ◆◆ ❍❍$ (Λn)♯ / C♯ i g1

together with a lifting problem

(f¯ ,f¯ ) n ♯ ♭ n ♯ ♭ 1 2 / ♮ ((Λi ) × J ) ∪(Λn)♯×J♭ ((∆ ) × I ) ❣3 E i _ ❣ ❣ ❣ ❣ ❣ ((hn)♯,G♭) ❣ ❣ p (5) i ❣ ❣ ❣ ❣ ❣ ❣ ❣ (∆n)♯ × J ♭ / (∆n)♯ / C♯ π1 g2

11 n ♯ of marked simplicial sets. The horn inclusion (hi ) is marked anodyne by an application of [17, Proposition 3.1.1.6] and the fact that cartesian morphisms are closed under composition ([17, Proposition 2.4.1.7]). It follows by [17, Proposition 3.1.2.3] that the smash n ♯ ♭ product ((hi ) , G ) on the left hand side of (5) is marked anodyne as well. We hence obtain the desired lift in (5) again by [17, Proposition 3.1.1.6].

Now, consider the comma ∞-category G∗ ↓ X which is given by the pullback

∗ / G ↓ X / Rect(I, E)/X ·y s (6) Rect(J, E) / Rect(I, E). G∗

The upper right corner of this square comes with a trivial ﬁbration to the slice C/p(X), and so we obtain a composite right ﬁbration

∗ (G ↓ X) →C/p(X) (7) which takes a rectangular natural transformation into X to its underlying arrow in C. Intuitively, the fact that its fibres are ∞-groupoids holds because if two objects α, β are contained in the same fiber, then their underlying vertical diagrams are vertical over the same object in C, and so rectangular arrows between those two are both cartesian and vertical, hence equivalences. Now, if the fibration p has G-comprehension, let R be a right adjoint to the restriction G∗ : Rect(J, E) → Rect(I, E). Then, given a vertical diagram X : I → E, the object p(R(X)) ∈ C classifies vertical J-diagrams extending X, in that we have categorical equivalences

∗ G ↓ X ≃ Rect(J, E)/RX ≃C/p(RX). (8)

The counit of this adjunction gives an object p(R(X)) ∈C/p(X) which represents the right fibration in (7). Vice versa, a choice of objects representing (7) for every diagram X ∈ Rect(I, E) yields equivalences as in (8) and hence a right adjoint to G∗ (since these equivalences determine a terminal object in the comma ∞-categories G∗ ↓ X). This shows the following characterization which was motivated in Table 1 and which provides a general strategy to verify specific G-comprehension schemes for a given cartesian fibration p: E ։ C.

Lemma 3.5. Let G: I → J be a map of simplicial sets and p: E ։ C a cartesian fibration. 1. If G is a monomorphism, then p has G-comprehension (Definition 3.2) if and only if for every diagram X ∈ Rect(I, E) the presheaf G∗ ↓ X of vertical J-structures extending X horizontally is representable over C/p(X). 2. Generally, p has G-comprehension if and only if for every diagram X ∈ Rect(I, E) ∗ the map G ↓ X →C/p(X) is categorically equivalent to a representable fibration over C/p(X).

Every functor G: I → J can be factored into an epimorphism followed by a monomorphism, and so every comprehension scheme can be decomposed into a pair of comprehension schemes for a monomorphism and an epimorphism. Comprehension schemes for monomorphisms have a direct interpretation as a smallness criterion for presheaves over slices of C as explained above. Comprehension schemes along epimorphisms can be explained in terms of “deﬁnability” of subﬁbrations over C which we will study later in this section (Remark 3.30). In fact all examples considered in this paper are either monomorphisms or epimorphisms.

12 Remark 3.6. Given a cartesian ﬁbration p: E ։ C, it will be useful for later examples to note that the functor Rect( , E): Sop → S/C preserves limits. That is, because the hom-functors Map♯( )♭, E♮) and Map♭(( )♭, C♯) preserve limits, and so for every diagram F : I → S we have pullback squares

Rect(limF, E) / limRect(F , E) / limMap♯((F )♭, E♮) ·y ·y C / Cπ0(I) / limMap♭((F )♭, C♯). ∆

Example 3.7. If G: I → J is a weak categorical equivalence and p: E ։ C is any cartesian ﬁbration, then the restriction

G∗ : Rect(J, E) → Rect(I, E) (9) is a categorical equivalence as well. In particular, the ﬁbration p satisﬁes G-comprehension in a trivial way. Indeed, via Diagram (2), the map (9) is a categorical equivalence if the restriction ∗ ∆∗(EJ ) G / ∆∗(EI )

) ) C u u is a cartesian equivalence. By [17, Proposition 3.1.3.5] this is the case if and only if G∗ induces a categorical equivalence between the ﬁbres of the two cartesian ﬁbrations. But for C ∈C these are exactly the maps

G∗ : E(C)J →E(C)I which are categorical equivalences by [17, Proposition 1.2.7.3.(3)].

Examples 3.8. Let p: E ։ C be a cartesian fibration. The following are direct generalizations of the most fundamental comprehension schemes in the case of indexed 1-categories. 1. Say p is small if it has (∅ → ∆0)-comprehension. That is if and only if the core E× has a terminal object. Or in other words, if and only if the right fibration p× is representable. 2. Say p is locally small if it has δ1-comprehension, where δ1 : ∂∆1 → ∆1 is the boundary inclusion. That is if and only if for any two points t1,t2 ∈ E(C), the right fibration 1 ∗ (δ ) ↓ (t1,t2) ։ C/C is representable. Since the vertical arrows and the cartesian arrows generate a factorization system in E ([17, Example 5.2.8.15]), this right fibration is the ∞-categorical generalization of the elementary fibration Homc(t1,t2) ։ C/C of ordinary categories in [28, Definition 10.1]. 3. Say p is well-powered if it has (d0 : ∆0 → ∆1)-comprehension restricted to monomorphic diagrams. In the sense of [13, B1.3.14], by that we mean the following generalized notion of comprehension.

For any ∞-category I, let Rect−1(I, E)0 ⊆ Rect(I, E)0 be the subcollection of pointwise monomorphic functors, and let Rect−1(I, E) ⊆ Rect(I, E) be the thereby gener- ∗ 1 ated full ∞-subcategory. Then p is well-powered if the restriction (d0) : Rect−1(∆ , E) → Rect(∆0, E) has a right adjoint. Informally, a ﬁbration p is well-powered if for every

13 × ∗ -object t ∈ E there is a vertical monomorphism s ֒→ p(εt) (t) which classifies ver tical subobjects of those elements which come equipped with a cartesian arrow into t. In analogy to the 1-categorical case, if C has pullbacks, the canonical fibration 1 ։ t: Fun(∆ , C) C is well-powered if and only if all slices C/C have power objects (and hence subobject classifiers).

Remark 3.9. As noted in [13, B1.3.13] (and [28, 9], respectively), in the 1-categorical setting, smallness of a Grothendieck fibration p: T ։ C implies rigidity of the objects in its fibres. That is because the 1-category Rect([0], F) is not enriched in the category Cat of small categories but in Set only. This means that existence of a terminal object t ∈ T× requires the set of cartesian morphisms with fixed domain into t to be a singleton on the nose, not leaving any room for automorphisms in the fibres of p. Inthe(∞, 1)-categorical context however, smallness merely implies contractibility of the automorphism-spaces of the fibres, and is a “good” categorical notion as it is stable under equivalence.

Example 3.10. Let C ∈ C be an object and consider its associated representable right 1 1 fibration C/C ։ C. This fibration satisfies (∂∆ → ∆ )-comprehension if and only if the diagonal ∆: C/C →C/C ×C C/C has a right adjoint. This in turn holds if and only if C has all equalizers for parallel maps with codomain C. Thus, all representable fibrations over C are locally small if and only if the ∞-category C has all equalizers. In particular, if C has finite products, the representable fibrations over C are locally small if and only if C has pullbacks.

Example 3.11. The terminal map C ։ ∆0 is a cartesian fibration, its cartesian edges are exactly the equivalences in C. For any simplicial set I, we obtain an equivalence Rect(I, C) ≃ Fun(I, C)≃. It follows that C ։ ∆0 is small if and only if the core C≃ is contractible. It is locally small if and only if all hom-spaces of C are contractible. These two characterizations allow easy constructions of cartesian fibrations which are small but not locally small. For instance, we may take the free monoid on one generator and consider it as an ordinary category M with one object. Its nerve yields a cartesian fibration N(M) ։ ∆0 which is small but not locally small.

Example 3.12 (On smallness and size). The ∞-category Cat∞ as introduced in Section 2 is the ∞-category of small ∞-categories. That means its objects are exactly the ∞- categories C which are simplicial sets (opposed to proper classes). The ∞-category Cat∞ itself is a proper class, so formally we have to invoke some set theoretical gymnastics to justify its use, e.g. by assuming the existence of an inaccessible cardinal. + For the moment being, let Cat∞ denote the ∞-category of large ∞-categories (via + .∞assumption of two inaccessible cardinals). We then obtain an inclusion ι: Cat∞ ֒→ Cat Given a large ∞-category C, we can consider its associated large indexing from Exam- ples 2.2.1 restricted to small ∞-categories given as follows.

Fun( ,C) ι / + / + Cat∞ Cat∞ Cat∞ (10)

+ The core of the indexing Fun( , C) over Cat∞ is representable as explained in Exam- + ple 2.4.1. That means, its straightening over Cat∞ is a small cartesian ﬁbration in the sense of Example 3.8.1. The ∞-category C is small if and only if the straightening of the restricted indexing (10) over Cat∞ is a small cartesian ﬁbration as well. Similarly, recall from [17, 5.4.1] that an ∞-category C is locally small if all its associated hom-spaces are essentially small (i.e. categorically equivalent to small spaces). For example, the large ∞-category Cat∞ itself is locally small. Then, similarly to the ordinary categorical case, one can show that a large ∞-category C is locally small if and only if the

14 straightening of its restricted indexing (10) is a locally small cartesian ﬁbration. In order to show this, one computes that the latter holds if and only if for all small ∞-categories D and all pairs of functors F, G: D→C the homotopy-pullback

∗ + / ι Cat∞/(F ↓G) Cat∞/(D×D) (F,G)∗ ∗ + / ∗ + ι Cat∞ 1 ι Cat∞ /Fun(∆ ,C) (s,t)∗ /(C×C) is representable over Cat∞. Here, the lower left corner is obtained from Proposition 3.23. Representability of these pullbacks directly implies essential smallness of the hom-space C(C, D) ≃ C ↓ D for all pairs of objects C, D : ∆0 → C. For the converse it is easy to see that local smallness of C and smallness of D implies essential smallness of the comma category F ↓ G for all functors F, G: D→C.

Remark 3.13. The size considerations in Example 3.11 arise because we implicitly are assuming a cartesian fibration E ։ C to have small fibres whenever we consider its associated indexed ∞-category over C. In particular, we are assuming the ∞-category C to be small with respect to Cat∞ whenever we refer to its naive indexing or the canonical indexing over itself. This either means that C is a simplicial set or that Cat∞ is chosen large enough. Either way, the results in this paper will not depend on the choice of (Cat∞)|κ if multiple inaccessibles κ are available, so that size issues of this sort will consistently be suppressed. Recall that an ∞-category C is defined to be locally cartesian closed if and only if for every object C ∈ C, the slice C/C is cartesian closed. That means that for every pair x,y ∈C/C , the presheaf op C/C (x ×C ,y): (C/C ) → S is representable. We have the following characterization of local cartesian closedness, generalizing the corresponding result for 1-categories in [28, Theorem 10.2].

Proposition 3.14. Suppose C has pullbacks. Then the canonical ﬁbration t: Fun(∆1, C) ։ C is locally small if and only if C is locally cartesian closed.

Proof. We show that the right ﬁbration (x,y) := Un(C/C (x ×C ,y)) is equivalent to the right ﬁbration (δ1)∗ ↓ (x,y) ։ C from Example 3.8.2. We do so by using that the /C Q class of squares which are vertical up to equivalence and the class of cartesian squares 1 yield a factorization system on Fun(∆ , C). For visualization, a map f : x ×C z → y in ∗ 1 the slice C/C corresponds to a square z x → y in Fun(∆ , C), which in turn yields an essentially unique tuple (z, v) in (δ1)∗ ↓ (x,y) by factorization through the pullback z∗y.

❣❣3 X ❣❣❣❣❣ ❣❣❣❣❣ ❣❣❣❣❣ ❣❣❣❣❣ x ❣❣❣❣❣ X ×C Z ❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴ /4 Y ❙ f ✐✐✐✐ ❙❙❙ ✐✐✐✐ ❙❙❙ ✐✐✐✐ v❙❙❙ ✐✐✐ ❙❙❙ ) ✐✐✐✐ y ❙) ✐✐✐ " w Y ×C Z 3 C ❣❣❣❣❣ ❣❣❣❣❣ ❣❣❣❣❣ ❣❣❣❣❣ z % ❣❣❣❣ Z ❣u ❣❣❣

Thus, representability of one implies representability of the other. In other words, the 1 ։ slices C/C are cartesian closed if and only if the ﬁbration t: Fun(∆ , C) C is locally small.

15 To prove equivalence of the two right ﬁbrations, on the one hand, remember from Example 2.4.3 that the ﬁbration (x,y) is given by the pullback

Q / (x,y) / (C/C )/y ·y Q s (11) / C/C / C/C . x×C

On the other hand, by deﬁnition we have a pullback square

1 ∗ / 1 1 (δ ) ↓ (x,y) / Rect(∂∆ , Fun(∆ , C))/(x,y) ·y s (12) Rect(∆1, Fun(∆1, C)) / Rect(∂∆1, Fun(∆1, C)). (δ1)∗

In the following we decompose the bottom map of the square (11) to see that (δ1)∗ ↓ (x,y) is the homotopy pullback of the same diagram. Some of the technical arguments will be sketched only since the presentation of the proof is already rather lengthy as it is. Thus, first, we note that the bottom map of the square (11) factors into a sequence 1 × ։ of functors defined as follows. The acyclic fibration Fun(∆ , C)/x C/C has a section σ unique up to homotopy which picks for every arrow z : Z → X a cartesian square over z into x. The inclusion l : ∆2 ֒→ ∆1 × ∆1 determined by the three vertices (0, 0), (1, 0) and (1, 1) (which we picture to span the lower of the two triangles when the vertical edges are downwards directed) induces a map

∗ 1 × 1 1 l 2 (Fun(∆ , C)/x ֒→ Fun(∆ × ∆ , C) −→ Fun(∆ , C) (13 which takes a cartesian square of the form z∗x → x to the lower triangle with 2-boundary z∗x and 0-boundary z. For a subsimplex S ∈ ∆n with maximal vertex s ∈ S, consider the pullback

0 Fun(S, C)C / / ∆ (14) _ ·y C Fun(S, C) / / C. ev(s)

2 Then the map (13) factors through Fun(∆ , C)C . Lastly, we have the evaluation at the 2 1 1-boundary d1 : Fun(∆ , C)C → Fun(∆ , C)C whose codomain is isomorphic to the slice C/C . So the bottom map of diagram (11) can be factored as follows.

∗ σ / 1 × l / 2 d1 / C/C ∼ Fun(∆ , C)/x Fun(∆ , C)C C/C . (15)

∆1 ։ In fact, the map d1 on the right hand side is isomorphic to the ﬁbration d1 : (C/C ) C/C , and furthermore there is a pullback square of the form

1 / Fun(∆ , C)/y / (C/C )/y ·y s ∆1 / / (C/C ) / / C/C . d1

Both facts can be seen by computing the n-simplices in each case, applying elementary simplicial combinatorics on joins and pushouts of standard simplices. The diagram simply

16 expresses that squares in C into y : Y → C are the same as pairs of suitable triangles glued together at their 1-boundary. The square (11) therefore factors as follows.

/ / 1 / (x,y) / P / Fun(∆ , C)/y / (C/C )/y ·y ·y ·y Q s (16) ∼ / 1 × / 2 / C/C / Fun(∆ , C) / Fun(∆ , C)C / C/C . σ /x d1

Here, the object P is defined via (16). So, the map (x,y) → P is the pullback of a categorical equivalence along a right fibration, and hence a categorical equivalence itself. Q On the other hand, in diagram (12), the vertical fibration on the right hand side is isomorphic to the pair

1 × 1 × ։ 1 × 1 × (s,s): Fun(∆ , C)/x ×C/C Fun(∆ , C)/y Fun(∆ , C) ×C Fun(∆ , C) by Remark 3.6. It follows that the square

× 1 ∗ / 1 1 × 1 (δ ) ↓ (x,y) Rect(∆ , Fun(∆ , C)) ×Fun(∆1,C) Fun(∆ , C)/y

(d1,t/y) (17) 1 × / 1 × Fun(∆ , C)/x Fun(∆ , C) ×C C/C (s,t/x) is both a strict pullback and a homotopy pullback square in the Joyal model structure, since the pullback of a limit of spans is the limit of the pullbacks, both 1-categorically and ∞-categorically. Thus, we can ﬁnish the proof by showing that this square (17) is equivalent to the pullback square in the center of (16). Therefore, let E ⊂ Fun(∆1 × ∆1, C) be the class of squares whose bottom map is an equivalence in C, and let Fun(∆2, Fun(∆1, C))EC ⊂ Fun(∆2, Fun(∆1, C)) be the full ∞-subcategory of triangles such that the edge ∆{0,1} maps to an E-square and the edge ∆{1,2} maps to a cartesian square. By [17, 5.2.8], the class of E-squares and the class of cartesian squares form a factorization system in Fun(∆1, C) and we obtain acyclic ﬁbrations as follows.

EC 2 1 EC 2 1 Fun (∆ , Fun(∆ , C))y / Fun (∆ , Fun(∆ , C)) ·y d d ∼ 1 ∼ 1 1 / 1 1 Fun(∆ , C)/y / Fun(∆ , Fun(∆ , C)) ·y t 0 / 1 ∆ y Fun(∆ , C)

The upper left corner is given by the notation in (14). Furthermore, since Fun(∆1, C) is 2 2 2 an ∞-category, the inner horn inclusion h1 : Λ1 → ∆ induces an acyclic ﬁbration 2 ∗ EC 2 1 ։ EC 2 1 (h1) : Fun (∆ , Fun(∆ , C))y Fun (Λ1, Fun(∆ , C))y as well. One can show that the inclusion

1 1 × 1 × EC 2 1 Rect(∆ , Fun(∆ , C)) ×Fun(∆1,C) Fun(∆ , C)/y → Fun (Λ1, Fun(∆ , C))y is a homotopy equivalence in the Joyal model structure, using that every E-square yields an essentially unique vertical map by pullback along the underlying equivalence in C, EC 2 1 and that every arrow in Fun (Λ1, Fun(∆ , C))y necessarily yields a cartesian morphism

17 between associated vertical maps on the ∆{0,1}-component since the ∆{1,2}-component is cartesian. Eventually, we obtain a map between the pullbacks in (12) and (16) as follows.

× 1 ∗ / 1 1 × 1 (δ ) ↓ (x,y) Rect(∆ , Fun(∆ , C) ×Fun(∆1,C) Fun(∆ , C)/y ♥♥♥ ❣❣❣E e ♥♥ ❣❣❣❣ ♥♥ ❣❣❣∼❣ ♥♥♥ ❣❣❣❣ ♥♥♥ ❣s ❣❣❣ ♥w ♥ / 1 P / Fun(∆ , C)/y

1 × / 1 × Fun(∆ , C)/x Fun(∆ , C) ×C C/C ❤ f ♦♦ ❤❤❤❤ F ♦♦♦ ❤❤❤❤ ♦♦♦ ❤❤❤❤ ♦♦♦ ❤❤❤❤ ♦♦♦ ❤s ❤❤❤ 1 × / 2 Fun(∆ , C)/x Fun(Λ1, C)C (18)

2 Here, we replaced the bottom right corner Fun(∆ , C)C of the center pullback in 2 (16) with Fun(Λ1, C)C , so the front square in (18) is a homotopy pullback in the Joyal model structure only. The diagonal map on the bottom right corner is a (−1)-truncated monomorphism, and it follows that the induced map

(δ1)∗ ↓ (x,y) → P is a categorical equivalence of ∞-categories. Composition with the categorical equivalence (x,y) −→∼ P from (16) yields a categorical equivalence

Q (x,y) ≃ (δ1)∗ ↓ (x,y) Y that commutes with the respective right ﬁbrations over C/C up to equivalence.

It follows for instance that the ∞-category S of spaces is locally small over itself (i.e. its canonical fibration is locally small). Since it is a locally small ∞-category as well, but its slice ∞-categories are generally large, one can show that it cannot be small over itself. Hence, Proposition 3.14 and Example 3.11 together show that, in general, smallness – which can be thought of as horizontal smallness – and local smallness – which can be thought of as vertical smallness – are mutually independent properties. In order to deduce various comprehension schemes from a given set of comprehension schemes for a fixed cartesian fibration p: E ։ C, we note that the class of functors

Compp := {G: I → J | p satisﬁes G-comprehension} satisﬁes the following stability properties.

Lemma 3.15. Let p: E→C be a cartesian ﬁbration. 1. Suppose p is a right ﬁbration and D is an ∞-category. Then pullback along p preserves left adjoints, i.e. for every pullback square of the form

∗ p F / P y E · p D / C, F

the functor p∗F : P→E has a right adjoint whenever F : D→C has a right adjoint.

2. The class Compp contains all weak categorical equivalences. It is closed under composition and under pushouts along monomorphisms.

18 3. The class of monomorphisms in Compp contains all acyclic coﬁbrations in the Joyal model structure and is closed under compositions and arbitrary pushouts.

4. If C has pullbacks, the class Compp furthermore has the right cancellation property. Proof. For Part 1. assume that p is a right ﬁbration and D is an ∞-category. For any object E ∈E, we obtain the following diagram of pullbacks. ∗ / (p F ) ↓ E / E/E ▼ ❊ ▼▼▼ ❊❊ ▼▼ ❊❊ ·y ▼▼▼ s ❊❊ ▼▼& ❊❊" " / F ↓ p(E) / C/p(E) ·y ∗ p F P / E y s · p { { D / C F

Since p is a right fibration, the fibration E/E → C/p(E) is acyclic. It follows that the pullback (p∗F ) ↓ E → F ↓ p(E) is an acyclic fibration as well. Thus, the comma ∞- category (p∗F ) ↓ E has a terminal object whenever F ↓ p(E) has such. But the latter holds because F has a right adjoint. This proves Part 1. For Part 2. closure under composition is easily seen. It contains the weak categorical equivalences by Example 3.7. Furthermore, we have seen that the functor Rect( , E): Sop → S/C preserves pullbacks in Remark 3.6, and that this functor also takes monomorphisms to right fibrations in Lemma 3.4. Thus the statement follows from Part 1. For Part 3. we only are left to show that arbitrary pushouts of monomorphisms in Compp are contained in Compp again. Therefore, let j : A ֒→ B be a monomorphism in Compp and consider a pushout square of the form

f /

A _ C

· j y B / D in S. We can factor the map f : A → C into a monomorphism i: A → C′ followed by an acyclic Kan ﬁbration q : C′ ։ C, and consider the induced factorization of pushouts

i / ′ q / /

A _ C _ ∼ C _

· · y j y B / D′ / D ′ ′ ′ Then the map C → D is contained in Compp by Part 2. The map D → D is a weak categorical equivalence, because the Joyal model structure is left proper. By Example 3.7, ′ ′ it follows that the map C → D is contained in Compp if and only if C → D is contained in Compp. For Part 4. let G: I → J and H : J → K be maps of simplicial sets such that p has both G- and (H ◦ G)-comprehension. Let X ∈ Rect(J, E) and consider the following diagram. ∗ / ∗ H ↓ X Rect(J, E)/X G/X P PPP PPP y PPP PPP · PPP s PPP PPP( PPP' ( ∗ ∗ / ∗ ∗ / (HG) ↓ G (X) / G ↓ G (X) / Rect(I, E)/G∗(X) ·y ·y s Rect(K, E) r r / Rect(J, E) r r / Rect(I, E) H∗ G∗

19 Both the front and the back square on the left hand side are (homotopy) pullback squares (in the Joyal model structure) and hence so is the top square inbetween. It is a homotopy pullback square of maps over C/p(X) and all three corners of the underlying span are rep- ∗ resentable over C/p(X) by assumption. It follows that the pullback H ↓ X is categorically equivalent to a representable right ﬁbration over C/p(X) as well, because C has pullbacks (and the Yoneda embedding is fully faithful). By Lemma 3.5, this ﬁnishes the proof.

Example 3.16. Following [13, B1.3.15], we say that a cartesian fibration p: E ։ C satisfies definability of invertibility (for arrows) if it has inv-comprehension, where inv: ∆1 → I∆1 is the embedding of ∆1 into the free groupoid generated by it. It has definability of equivalence (of parallel morphisms) if it has ∇1-comprehension, 1 1 1 where the map ∇1 : (∆ ∪∂∆1 ∆ ) → ∆ identifies the two parallel non-degenerate arrows. / The domain can be described as the nerve of the parallel arrow category • // • . Furthermore, say p has definability of equivalence of parallel n-cells if it has ∇n- n n n comprehension, where the map ∇n : (∆ ∪∂∆n ∆ ) → ∆ is the codiagonal identifying the two parallel non-degenerate n-cells. The following lemma can be thought of as an ∞-categorical generalization of [13, Lemma 1.3.15] extended by its natural higher analogues.

Proposition 3.17. Let p: E ։ C be a cartesian fibration. 1. Let n ≥ 1. If p has definable equivalence of parallel n-cells, it has δn+1-comprehension. 2. Suppose C has all equalizers. (a) Let n ≥ 1. If p has δn-comprehension, it has definable equivalence for parallel n-cells. (b) If p is locally small, it has δn-comprehension for all n ≥ 1. (c) If p is locally small, it has definability of invertibility.

Proof. For Part 1. let n ≥ 1 and 0

n+1 hi

Λn+1 / ∆n+1

i _

ι2 · ι2 y i n n+1 n n+1(id,d ) _ n n ∆ ∪ n Λ / ∆ ∪ n ∆ o ? ∆ ∪ n ∆ ∂∆O i n+1 ∂∆ ∂∆ O (id,hi ) i ∇ ∼ = (d ,id) n

· y n+1 / n+1 o n ∂∆ n+1 ∆ ∆ δ di

n+1 n+1 n+1 The inner horn inclusion hi : Λi → ∆ is contained in Compp by Example 3.7, the n n n codiagonal ∇n : ∆ ∪∂∆n ∆ → ∆ is contained in the class Compp by assumption. It thus follows from Lemma 3.15.(b) that the composition δn+1 : ∂∆n+1 → ∆n+1 is contained in the class Compp as well. For Part 2.(a) assume that p has δn-comprehension. We want to show that for every C ∈ C and every pair of vertical diagrams α, β : ∆n → E(C) with common boundary, ∗ ¯ the ∞-category (∇n) ↓ (α, β) is presentable over C/C . Let C be the object in C/C n n ∗ n ∗ classifying rectangular ∆ -diagrams extending (δ ) (α) = (δ ) (β) in E, and let fα : C → C¯, fβ : C → C¯ be the arrows classifying α and β themselves, respectively. Since n n ∼ n n Rect(∆ ∪∂∆n ∆ , E) = Rect(∆ , E) ×Rect(∂∆n,E) Rect(∆ , E), we obtain an isomorphism between slices

n n n n n ∼ n Rect(∆ ∪∂∆ ∆ , E)/(α,β) = Rect(∆ , E)/α ×Rect(∂∆ ,E)/(δn)∗α Rect(∆ , E)/β

20 ∗ as well. Thus, by Diagram (6), the ∞-category (∇n) ↓ (g, h) ﬁts into the pullback square ∗ / n n (∇n) ↓ (α, β) Rect(∆ , E)/α ×Rect(∂∆n,E) n ∗ Rect(∆ , E)/β ·y /(δ ) α n n n Rect(∆ , E) / Rect(∆ , E) × n Rect(∆ , E). ∆ Rect(∂∆ ,E) Since pullbacks commute with each other, we obtain equivalences ∗ ∼ n n (∇n) ↓ (α, β) = Rect(∆ , E)/α ×(δn)∗↓(δn)∗α Rect(∆ , E)/β

≃C/C ×C/C¯ C/C

≃C/Eq(fα,fβ) over C/C . This ﬁnishes Part (a). Part 2.(b) follows immediately from Parts 1. and 2.(a) since local smallness is δ1- comprehension. For Part 2.(c) let J (2) ⊂ I∆1 be the subsimplicial set given by exactly one of the two non-degenerate 2-simplices. It can be thought of as the free left (or right) invertible map, depicted as follows. [0] − ? ❅ f 1 ⑧ ❅ f ⑧⑧ ❅❅ ⑧⑧ ❅❅ ⑧⑧ ❅ [1] / [1] s0([1]) (2) (2) 0 2 The pushout K := J ∪∆1 J along the boundaries d and d is the free biinvertible map, the inclusion inv: ∆1 → I∆1 factors through K. The natural map K → I∆1 is a weak categorical equivalence (both are interval objects in the Joyal model structure), and hence by Example 3.7 and Lemma 3.15 we can reduce inv-comprehension to (∆1 → K)- comprehension. Since the boundaries di : ∆1 → J (2) for i = 0, 2 are monomorphisms, we can reduce (∆1 → K)-comprehension further to (d0 : ∆1 → J (2))-comprehension again by Lemma 3.15. Therefore, we factor the inclusion d0 : ∆1 → J (2) through subobjects 1 (2) (2) (2) ∆ → ∂J → (∂J )+ → J such that each component is the pushout of a map in Compp along a monomorphism. Then we can use Lemma 3.15 once more to ﬁnish the proof. First, consider the circle given by the pushout

1 δ / 1

∂∆ _ ∆

¯1 · (19) δ y ∆1 / ∂J (2) where δ¯1 : ∂∆1 → ∆1 swaps the endpoints. Second, consider the pushout

Λ2 1 / ∆2

1 _

· (20) y (2) (2) ∂J / (∂J )+ where the left vertical map picks out the two non-degenerate edges added in (19) concate- nated at [0]. Third, we obtain a pushout

1 1 ∇1 1

∆ ∪ 1 ∆ / ∆

∂∆ _

· y (2) (2) (∂J )+ / J

21 where the left vertical map picks out the pair consisting of the arrow [1] → [1] freely added in (20) and the identity on [1]. The composition of the bottom maps of the three pushout squares is exactly the inclusion d0 : ∆1 → J (2). The cartesian fibration p has δ1-comprehension by assumption, 2 and thus it has ∇1-comprehension by Part 2.(a). It has h1-comprehension by Example 3.7 and thus the statement follows by Lemma 3.15 as claimed. Remark 3.18. The reader familiar with Johnstone’s original proof of Proposition 3.17.2.(c) in the 1-categorical setting may note that he assumes C to have all pullbacks in order to show the statement (this is [13, Lemma 1.3.15]), while the proof above only requires the existence of equalizers. But this apparently stronger statement is not stronger at all, since Johnstone implicitly assumes the existence of (at least certain) finite products and hence the existence of (at least certain) pullbacks in C. And indeed, to equate his notion of local smallness with the one given in Example 3.16 via Definition 3.2, we need to require that 1 C has π0(∂∆ )-sized products (as explained in Remark 3.3). But equalizers and finite products yield pullbacks ([17, Proposition 4.4.3.2]).

Corollary 3.19. A cartesian fibration p: E ։ C is small and locally small if and only if it has j-comprehension for every monomorphism j between finite simplicial sets. Proof. The monomorphisms in S are exactly the “free” cofibrations generated by the class of boundary inclusions ∂∆n ֒→ ∆n for n ≥ 0. That means, every monomorphism between finite simplicial sets is the composition of pushouts of boundary inclusions. Thus, the corollary follows from Lemma 3.15.3 and Proposition 3.17. We will prove an “internal” characterization of the small and locally small cartesian fibrations in Section 4. In that context it will be useful to compute the straightening of the rectangular diagram ∞-category Rect(I, E) explicitly in terms of an indexed ∞-category op I : C → Cat∞ when p is given as the unstraightening of I. Therefore, given a simplicial set I and a C-indexed ∞-category I, in the following we show that the composition

I ≃ op I (·) (·) C −→ Cat∞ −−→ Cat∞ −−→S (21) is equivalent to the straightening of Rect(I, Un(I)). We start with an observation about the universal cocartesian ﬁbration

π : Dat∞ ։ Cat∞. To simplify notation, we note that Definition 3.1 can be dualized to define for every cocartesian fibration p: E ։ C a left fibration Rect(I, E) ։ C such that Rect(I, Eop) =∼ Rect(Iop, E)op.

Proposition 3.20. The left fibration Rect(I, Dat∞) ։ Cat∞ is corepresentable for every op op ։ op simplicial set I. In other words, the cartesian fibration π : Dat∞ Cat∞ has (∅→ I)- comprehension for every simplicial set I. op Proof. Since the functor Rect( , Dat∞) preserves weak categorical equivalences by Ex- ample 3.7, we may assume without loss of generality that I is an ∞-category. The set of vertices of Rect(I, Dat∞) is in 1-1 correspondence to the set of pairs of vertices C ∈ Cat∞ and functors I → C. Indeed, it is easy to see that the fiber of the 0 left fibration Rect(I, Dat∞) ։ Cat∞ over an ∞-category C : ∆ → Cat∞ is equivalent to the hom-space Cat∞(I, C). In the following we show that the tuple (I, idI ) is initial in Rect(I, Dat∞), which proves that I corepresents the left fibration Rect(I, Dat∞) ։ Cat∞. That means, for every n ≥ 1 and for every map f in a diagram of the form

n f ∂∆ / Rect(I, Dat∞) _ 7 ♣ ♣ δn ♣ ♣ ♣ f¯ ∆n ♣

22 such that f([0]) = (I, idI), we need to construct a dotted lift as depicted. By deﬁnition of the rectangular diagram ∞-category given via Diagram (1), such a lift corresponds exactly to a diagram extension of the map f = (u(f),s) as follows.

I ▼ rrrr ▼▼ rrr ▼▼ rrr y ▼▼▼ rrr · ▼▼ ∗ rrr ▼▼ {0} s rrrr ▼& ♮ ♮ rrrrr 0 Q / Dat rrr ■ ; ∞ rrr y■ ✇ rrr s · ■ ✇ rrr ■ rrrr ■ ✇ rrrr ■$ ✇ rrrr 0 $ ♮ ✇ I rr / / ∆ ❜0 Q¯ π ❍❍ y ▲ ❤ ❢ ❞ (22) ❍❍ · ▲▲▲ ❦ ✤ y ❍❍ ▲▲{0} ♣ ♠ ✤ · ❍❍ ▲▲ t 1×{0} ❍❍$ ①▲▲% ✤ $ % u(f) n ♯ ⑤ / / n ♯ ✤ / ♯ I × (∂∆ t) π (∂∆ ) r Cat∞ ◆ 2 ☎ ❍ ✤ ; ◆◆ ❍❍ ✇ ◆◆◆ ✟ s¯ ❍❍ ✤ ✇ n◆◆ n ❍ ✇ 1×δ ◆◆ ☛ δ ❍❍ ✤ ✇ u(f¯) ◆& ✍ ❍$ ✇ I × (∆n)♯ / / (∆n)♯ π2

In fact, following the deﬁnition in Diagram (1) on the nose, the instances of the ∞-category I in Diagram (22) should be ♭-marked. But since every functor out of I preserves equivalences, and equivalences in Dat∞ are π-cartesian, we may mark the set of all equivalences in I instead without changing the lifting problem at hand. This implies that the horizontal projections π2, and hence all double headed arrows in diagram (22), are cocartesian ﬁbrations (on unmarked underlying simplicial sets) and have a ♮-marked domain. n Now, via Straightening over ∂∆ we obtain a natural transformation St∂∆n (s) between n ♯ the functors St∂∆n (I × (∂∆ ) ) = CI (the constant functor with value I ∈ Cat∞) and ♮ St∂∆n (Q )= u(f). That is, a map 1 n St∂∆n (s): ∆ × ∂∆ → Cat∞

n n which restricts to CI on {0}× ∂∆ and to u(f) on {1}× ∂∆ . Precisely because CI is 1 n n ∼ constant, we obtain an extension of St∂∆n (s) to the quotient ∆ × ∂∆ /({0}× ∂∆ ) = 0 n ∼ n+1 ∆ ∗ ∂∆ = Λ0 . The resulting map n+1 St∂∆n (s): Λ0 → Cat∞ {0,1} takes the initial edge ∆ to the equivalence idI in Cat∞, and hence yields a lift of the form St n (s) n+1 ∂∆ / Λ0 Cat; ∞ _ ✈; n+1 ✈ h ✈ 0 ✈ n+1 ✈ s ∆n+1 by [17, Proposition 1.2.4.3]. This induces a composition as follows.

u(f)

{1}×id St n (s) ' ∂∆n / ∆1 × ∂∆n ∂∆ / Cat _ 9 ∞ δn sn+1 ∆n / ∆n+1 d0

The map u(f¯) := sn+1 ◦ d0 yields the dotted cartesian square on the right hand side of Diagram (22). Since ∆n+1 is the join ∆0 ∗ ∆n, the map sn+1 yields a natural transfor- n mation from CI to u(f¯), which by unstraightening over ∆ yields the dotted maps ¯ in Diagram (22). This ﬁnishes the proof.

23 For a (co)cartesian ﬁbration p: E ։ C and a simplicial set I, recall the pullback

∆∗(EI ) / EI ·y pI C / CI . ∆ from Diagram (2). It may be thought of as a parametrized ∞-category of functors out of I, its core is the parametrized ∞-groupoid Rect(I, E) of functors out of I.

∗ op I ։ op Corollary 3.21. The cartesian fibration ∆ ((Dat∞) ) Cat∞ is small for every simplicial set I. Its core is represented by the opposite of the fibrant replacement of the simplicial set I in the Joyal model structure. op Proof. By Proposition 3.20, the left fibration Rect(I , Dat∞) ։ Cat∞ is corepresented op op ։ op by the fibrant replacement of I . That means its opposite Rect(I, Dat∞) Cat∞ is rep- ∗ op I ։ resented by the same object. But this right fibration is exactly the core of ∆ ((Dat∞) ) op Cat∞, which means the latter is small.

op Lemma 3.22. Let I : C → Cat∞ be an indexed ∞-category over C and I be a simplicial set with ﬁbrant replacement J (I) in the Joyal model structure. Then there is a pullback square of the form / op Rect(I, Un(I)) / ((Cat∞)J (I)/) ·y t C / Catop. I ∞ Proof. Let p: Un(I) ։ C denote the Unstraightening of I. By deﬁnition of the rectangular diagram category, we have pullback squares Rect(I, Un(I)) / (Un(I)I )× _ ·y _ ∆∗(Un(I)) / Un(I)I / ((Dat )op)I y y ∞ · I · p (πop)I I op I C / C / ((Cat∞) ) . ∆ II Since II ◦ ∆=∆ ◦I, we obtain cartesian squares as follows. ∗ / ∗ op / op I ∆ (Un(I)) ∆ (Dat∞) ((Dat∞) ) ·y ·y pI (πop)I C / Catop / (Catop)I I ∞ ∆ ∞ ∗ op ∗ op The pullback ∆ (Dat∞) is isomorphic to the pullback ∆ (Dat∞) for the diagonal Iop ∆: Cat∞ → (Cat∞) . By Lemma 3.21, its core is represented by J (I). We hence obtain pullbacks as follows. / op Rect(I, Un(I)) / ((Cat∞)J (I)/) _ ·y _ ∗ ∗ op ∆ (Un(I)) / ∆ ((Dat∞) ) ·y C / Catop I ∞

24 Proposition 3.23. Let I ∈ Ind(C). Then for every simplicial set I, the straightening of the right ﬁbration Rect(I, Un(I)) ։ C is equivalent to the composition

I ≃ op I (·) (·) C −→ Cat∞ −−→ Cat∞ −−→S given in (21).

Proof. Since the functor Cat∞(J (I), ): Cat∞ → S is equivalent to the straightening of the representable left fibration ((Cat∞)J (I)/) ։ Cat∞, it follows from Lemma 3.22 and [17, Proposition 3.2.1.4] that the straightening of the right fibration Rect(I, Un(I)) ։ C is equivalent to the composition Cat∞(J (I), (·)) ◦I. The map I → J (I) is a weak categorical equivalence, and so it induces a natural I J (I) weak equivalence between the functors (·) , (·) : Cat∞ → Cat∞. But the representable ≃ ≃ J (I) Cat∞(J (I), (·)) is equivalent to Fun(J (I), (·)) = (·) ◦(·) by definition. In particular, the diagram (·)I Cat∞ / Cat∞ ■■ ■■ ≃ ■■ (·) Cat∞(D(I),(·))■■ ■■$ S commutes up to equivalence.

We end this section with a very useful transition result for comprehension schemes. The following is an ∞-categorical generalization of [13, Proposition 1.3.17].

Lemma 3.24. Let p: E ։ C be a cartesian ﬁbration and suppose D is an ∞-category with pullbacks. If F : D→C is a functor with a right adjoint, then CompF ∗p ⊆ Compp.

Proof. Let p: E ։ C be a cartesian ﬁbration, D an ∞-category with pullbacks and F : D→C be a left adjoint. Since exponentiation with simplicial sets preserves pullbacks, via Diagram (1) we obtain a pullback square of the following form for every simplicial set K. p∗F Rect(K, F ∗E) / Rect(K, E) ·y D / C F Since the vertical maps are right ﬁbrations, the upper horizontal map is a left adjoint as well by Lemma 3.15.1. For every functor G: I → J we hence obtain a diagram of the form

∗ Rect(J, E) G / Rect(I, E) 4 4 ✐✐4 ❥❥4 ✐✐✐✐ ❥❥❥ ✐✐✐ ❥❥❥❥ ✐✐✐✐ ❥❥❥ ✐✐✐ ∗ ❥❥❥ Rect(J, F ∗E) G / Rect(I, F ∗E) ·y ·y (23) ! ! ~ ~ ❤4 C ❤❤❤❤ ❤❤❤❤ ❤❤❤❤ ❤❤❤❤ F " " | | ❤❤❤❤ D ❤❤❤

25 where all three diagonal arrows pointing to the upper right are left adjoints. For any X ∈ Rect(I, F ∗E) this furthermore induces a sequence of the following form.

∗ / ∗ ∗ G ↓ X Rect(I, F E)/X p F/X ❙❙❙ ❙❙ ❙❙❙ ❙❙❙ ·y ❙❙❙ ❙❙❙ ❙❙❙ s ❙❙❙ ❙❙❙) ❙❙❙) ) ∗ ∗ ∗ / ∗ ∗ / (p F )G ↓ (p F )(X) / (p F ) ↓ (p F )(X) / Rect(I, E)/(p∗F )(X) ·y ·y s Rect(J, F ∗E) r r / Rect(I, F ∗E) r r / Rect(I, E) G∗ p∗F (24)

Now, suppose p has G-comprehension. It follows that the composition of p∗F and G∗ on the bottom of Diagram (24) has a right adjoint since the top square in Diagram (23) commutes. But the map p∗F has a right adjoint as well, and so, along the lines of the proof of Lemma 3.15.4, it follows that the ∞-category G∗ ↓ X is categorically equivalent to a representable right ﬁbration over D. It follows that the change of base F ∗p has G-comprehension as well.

Remark 3.25. More generally, one can adapt the proof of Lemma 3.24 to show that whenever S ⊆ D is a class of morphisms such that F −1[F [S]] = S, then F ∗p has generalized G-comprehension for a functor G with respect to S whenever p has generalized G-comprehension with respect to F [S].

Example 3.26. The universal left fibration S∗ ։ S is the pullback of the universal ,cocartesian fibration π : Dat∞ ։ Cat∞ along the canonical inclusion S ֒→ Cat∞ ([17 3.3.2]). This inclusion comes with a left adjoint F : Cat∞ → S which assigns to every ∞-category the free ∞-groupoid generated by it. Taking opposites all over this pullback op op square, it follows by Lemma 3.24 that the universal right fibration (S∗) ։ S satisfies all comprehension schemes satisfied by the universal cartesian fibration πop.

Remark 3.27. One might expect that the notion of elementary ∞-topos, however defined, should be recoverable from suitable comprehension schemes. Such a presentation would be useful to define and study elementary ∞-toposes over arbitrary base ∞-toposes. For example, in 1-category theory, a left exact 1-category C is an elementary 1-topos if and only if the canonical fibration t: C[1] ։ C is well-powered (local smallness can be derived, see [28, Theorem 11.1]). But the naive ∞-categorical generalization of this scheme does not characterize ∞-toposes. Left exact ∞-categories C whose canonical indexing is well-powered do not necessarily yield object classifiers. But “super-poweredness” in form of unrestricted (d0 : ∆0 → ∆1)-comprehension is too strong, since it would require each slice of C to have an object classifier with no size restriction whatsoever. Whenever C is presentable however, for any cardinal κ one may consider the generalized comprehension scheme of super-poweredness restricted to pointwise relatively κ-compact diagrams. That is, the existence of a right adjoint to the functor

0 ∗ 1 1 0 1 (d ) : Rectκ(∆ , Fun(∆ , C)) → Rect(∆ , Fun(∆ , C))).

Then once can show that all slices of C have object classifiers for relatively κ-compact maps if and only if the canonical indexing of C is “κ-super powered”. It follows that a presentable ∞-category is an ∞-topos if and only if it its associated canonical indexing is locally small and κ-super powered for an infinite chain of inaccessible cardinals κ (via [17, Theorem 6.1.6.8] and [17, Proposition 6.3.5.1.(1)]). We will give a more conceptual characterization of ∞-toposes which also captures the definition of elemenatry ∞-toposes as defined in [22, 3] in Remark 4.21.

26 3.2 Definability and comprehension (∞, 1)-categories A notion of comprehension for ordinary categories of particular interest to type theorists was studied by Jacobs in [11], generalizing ideas of Lawvere and others. Therefore, in the following, we pass to the definition of subfibrations in the sense of [13, B1.3] and introduce the notion of definability and the notion of comprehension ∞-categories in the sense of [11] as special cases.

Definition 3.28. Let p: E ։ C be a cartesian fibration and E′ ⊆E be an ∞-subcategory. If the restriction E′ ι / E ❄ ❄❄ ❄❄ p′ ❄❄ p ❄ Ð Ð C is a morphism of cartesian fibrations such that an edge of E′ is p′-cartesian if and only if it is p-cartesian, we say that p′ is a subfibration of p. A subfibration p′ of p is definable if the inclusions (E′)× →E× (25) Rect(∆1, E′) → Rect(∆1, E)

have a right adjoint each. The notion of cartesian subfibration in the ∞-categorical setting has been defined and studied in [3]. The examples we will be interested in are subfibrations p′ ⊆ p whose ∞- subcategory E′ ⊆ E is full and replete. In that case we say that p′ is a full and replete subfibration of p. By [3, Lemma 4.5], given a cartesian fibration p: E ։ C and a full and replete ∞- subcategory E′ ⊆E, the restriction of p to E′ yields a (full and replete) subfibration of p if and only if for every p-cartesian edge f in E, the edge f is contained in E′ whenever its codomain is contained in E′. In fact, even when p′ ⊆ p is not full, but replete, then the inclusions (25) are full anyway. Definability of subfibrations is a generalized comprehension scheme similar to Exam- ′ ple 3.8.3, in that p is a definable subfibration of p if and only if p satisfies id∆0 - and ′ id∆1 -comprehension restricted to the class of diagrams which are contained in E . Less formally, the scheme expresses that every object t and every vertical arrow f : t1 → t2 ′ in E have a best approximation R0(t) and R1(f) in E . In all examples we will consider it suffices to have a best approximation of objects in order to obtain definability of both objects and vertical arrows, using the following definability criteria.2

Lemma 3.29. Let p: E ։ C be a cartesian fibration and p′ : E′ ։ C be a subfibration. 1. Suppose C has pullbacks and p′ is a full subfibration. Then p′ is definable if and only .if the inclusion (E′)× ֒→E× has a right adjoint 2. Suppose p is a right fibration. Then p′ is a right fibration as well and, again, is .definable if and only if the inclusion (E′)× ֒→E× has a right adjoint

Proof. For Part 1. assume that the inclusion ι: (E′)× ֒→ E× has a right adjoint R. It follows that the product

′ × ′ × × × ι ×C ι: (E ) ×C (E ) ֒→E ×C E

2Our definition of definability slightly differs from Johnstone’s definition in [13, 1.3] since he claims that definability of arrows implies definability of objects ([13, Lemma 1.3.16.(1)]). Nevertheless, Definition 3.28 conforms with Streicher’s notion in [28, Definition 12.5].

27 has a right adjoint as well, since for every pair of objects t1,t2 ∈E(C) the pullback

/ × × (ι ×C ι) ↓ (t1,t2) / (E ×C E )/(t ,t ) ·y 1 2 s ′ × ′ × × × (E ) ×C (E ) / E ×C E ι×Cι is isomorphic to the pullback (ι ↓ t1)×C (ι ↓ t2). This, by assumption, in turn is equivalent to the representable

C/p(R(t1)) ×C C/p(R(t2)) ≃C/p(R(t1 ))×C p(R(t2))

1 ′ 1 over C/C . But the map ι ×C ι is just the inclusion Rect(∂∆ , E ) → Rect(∂∆ , E), and since E′ ⊆E is full, the square

Rect(∆1, E′) / Rect(∆1, E) ·y (δ1)∗ (δ1)∗ Rect(∂∆1, E′) / Rect(∂∆1, E) is a pullback and the vertical maps are right fibrations. Thus, the top horizontal arrow has a right adjoint by Lemma 3.15. For Part 2. we simply note that if p is a right fibration then the ∞-category Rect(∆1, E) is a path object for p in the sliced model category (S, QCat)/C. That is, because the vertical arrows in the fibres of p are equivalences, so that Rect(∆1, E) ≃ Rect(I∆1, E) is exactly the ∞-category of fibrewise equivalences. Thus, the vertical maps in the diagram Rect(∆1, E′) / Rect(∆1, E) O O (s0)∗ (s0)∗ ? ? Rect(∆0, E′) / Rect(∆0, E) are categorical equivalences, and it follows that the top horizontal arrow has a right adjoint if and only if the bottom horizontal arrow does.

Remark 3.30. In the paragraph subsequent to Lemma 3.5 we stated that G-comprehension for surjective functors G: I → J may be explained in terms of deﬁnability. Indeed, note that for surjective functors G and for every cartesian ﬁbration p: E ։ C the restriction

∗ Rect(J, E) G / Rect(I, E)

* * C t t yields a subfibration of the right fibration Rect(I, E) ։ C. Then, by Lemma 3.29.2, this ×(subfibration is definable if and only if the induced inclusion Rect(J, E)× ֒→ Rect(I, E has a right adjoint. But since both fibrations are right fibrations, this inclusion is just the restriction G∗ : Rect(J, E) → Rect(I, E) itself. This means that the subfibration Rect(J, E) ։ C is definable if and only if p has G-comprehension.

Remark 3.31. In [4] (and in [28] accordingly), definability is a property of classes of objects in a cartesian fibration E ։ C. The conditions thereby imposed on such classes K ⊆ E of objects are equivalent to the assertion that the full ∞-subcategory K⊆E generated by K yields a full and replete subfibration of p. Then it is easy to show that the class K is definable in the sense of [4] if and only if the subfibration K× ⊆E× ։ C is definable in the sense of Definition 3.28.

28 Definition 3.32. Suppose C has pullbacks. A (full) comprehension ∞-category of C is a (full) subfibration of the canonical fibration t: Fun(∆1, C) ։ C. (Full) comprehension ∞-categories correspond to those indexed categories F : Cop → Cat∞ such that the values F (C) ⊆C/C are (full) ∞-subcategories.

Proposition 3.33. 1. A definable subfibration of a locally small fibration is again locally small. 2. Every definable comprehension ∞-category over a locally cartesian closed ∞-category C is locally small.

Proof. For Part 1. let p: E ։ C be locally small and p′ : E ։ C be a deﬁnable subﬁbration of p. Consider the square

∗ Rect(∆1, E) δ / Rect(∂∆1, E) O O

? ? Rect(∆1, E′) / Rect(∂∆1, E′). δ∗

By assumption, the top arrow has a right adjoint. If p is deﬁnable, the left vertical arrow has a right adjoint, and thus so has the bottom arrow, because the right vertical arrow is fully faithful. Part 2. follows immediately from Part 1. and Example 3.14.

We will come back to comprehension ∞-categories and their comprehension schemes in Section 4.2.

4 Externalization of internal (∞, 1)-categories

As explained in [13, Example 1.3.13] and alluded to in Remark 3.9, the notion of smallness of a fibration E ։ C between 1-categories is “evil”, because it implicitly requires that the objects in the fibres FC are rigid – a notion not invariant under categorical equivalence. 3 This is remedied in (∞, 1)-category theory where on-the-nose uniqueness and hence rigidity is replaced by equivalence and hence by contractibility. In type theoretic terms, one might say that an extensional identity type principle is replaced by a univalent inten- sional identity type principle. Because of this flaw in the definition of the comprehension scheme, Johnstone defines essentially small indexed categories to be those arising from internal categories by the way of externalization ([13, B2.3], also [12, 7.3]). In this section we generalize the externalization construction to internal ∞-categories and show that the indexed ∞-categories which arise as the externalization of an internal ∞-category are exactly the small and locally small ones. As an example we will see that the universal op op ։ op cartesian fibration π : Dat∞ Cat∞ is the externalization of the internal ∞-category • op ∆ in Cat∞. In the special case of comprehension ∞-categories over locally cartesian closed ∞- categories C we will find that such are small if and only if they are the externalization of the nerve of a univalent map in C. Against this background we will end the section with a short discussion on higher elementary toposes and univalent objects in arbitrary fibred ∞-categories.

Remark 4.1. In this section we will sometimes refer to notions and results of [21], since the cartesian fibrations which arise as the externalization of some internal ∞-category as to be defined in this section are exactly the “representable cartesian fibrations” in the sense of [21, 3]. Since we work with the quasi-categorical model for (∞, 1)-category theory

3Streicher introduces the notion of elementary ﬁbrations for that reason in [28, 9].

29 while the author of [21] works with complete Segal spaces, these references will implicitly use the transition results of [23] to move between these two models via the horizontal projection ph : (sS, CS) → (S, QCat) which is part of a Quillen equivalence by [14, 4]. Here, the pair (sS, CS) denotes Rezk’s model structure for complete Segal spaces on the category sS of simplicial spaces. Most importantly, as stated in [20], given a complete Segal space C and a fibration p: E ։ C in the model structure (sS, CS) for complete Segal spaces, the horizontal projection takes p to a cartesian fibration if and only if p is cartesian in the sense of [21, Definition 3.10], using [20, Theorem 7.43]. In fact, it follows directly from the proof of [20, Theorem 7.43] that whenever a cartesian fibration p: E ։ C is presented by a “Reedy right fibration” R ։ C ([20, Definition 7.15, Definition 4.14]), then the right fibration Rn ։ C in (sS, CS) translates exactly to n the right fibration Rect(∆ , E) ։ C in (S, QCat) via the Quillen equivalence ph for every n ≥ 0.

Notation. For an ∞-category C, we denote the ∞-category Fun(N(∆op), C) of simplicial objects in C by sC.

4.1 Definition and characterization via (∞, 1)-comprehension In the following, let C be a fixed left exact ∞-category. We recall the definition of internal ∞-categories in C from [21, 2], going only into as much technical detail as necessary to define and study their externalization.

Deﬁnition 4.2. A Segal object in C is a simplicial object X ∈ sC such that its associated Segal maps

ξn : Xn → X1 ×X0 ···×X0 X1 are equivalences in C. A Segal object X in C is complete if the degeneracy

s0 : X0 → Equiv(X) is an equivalence in C, where Equiv(X) ⊆ X3 is the object of internal equivalences in X as defined in [21, Definition 2.37, Definition 2.47]. We obtain full ∞-subcategories CS(C) ⊆ S(C) ⊆ sC of Segal objects and complete Segal objects in C, respectively. The definition of completeness in Definition 4.2 is directly derived from Rezk’s original definition of completeness of Segal spaces.

Deﬁnition 4.3. A Segal groupoid in C is a Segal object X in C such that the natural map Equiv(X) → X1 is an equivalence in C. The full ∞-subcategory of Segal groupoids in S(C) is denoted by SGpd(C).

Remark 4.4. This definition of a Segal groupoid is equivalent to Lurie’s definition of a groupoid object in C as given in [17, Definition 6.1.2.7]. This can be shown using [17, Proposition 6.2.1.6.(3)] and [26, 3]. In [18, Proposition 1.1.14], Lurie constructs the “core” Core(X) of a Segal object X in C whenever C is left exact. That is a right adjoint Core: S(C) → SGpd(C) to the inclusion of Segal groupoids into Segal objects in C. He then defines a Segal object X to be complete if the degeneracy s0 : Core(X)0 → Core(X)1 is an equivalence in C. That is, in Lurie’s terms, if the groupoid object Core(X) is constant ([18, Remark 1.1.5]). This definition is equivalent to the one given in Definition 4.2 essentially by [18, Proposition 1.1.13.(2)].

The externalization of complete Segal objects in C – in other words, internal ∞- categories in C - can be given as follows. The ∞-category C comes equipped with a

30 op hom-space functor HomC : C ×C→S and an associated Yoneda embedding y : C → Cˆ ([17, 5.1.3]). The Yoneda embedding can be exponentiated to give the pushforward of simplicial objects y∗ : sC → sCˆ. Under currying, it yields associated functors

op [ , ]C : C × sC → sS and

op op op HomC ◦ (1Cop × ev): (C × N(∆) × Fun(N(∆) , C)) → S. (26)

Given a simplicial object X in C, the map [ , X]C is equivalent to the composite

y ∗ Cop −→ Fun(C, S) −−→X sS.

If X ∈ sC is a complete Segal object, then for every object C ∈C, the presheaf [C, X]C ∈ sS is a complete Segal space, because the corepresentables HomC(C, ) preserve limits and equivalences. Indeed, they therefore preserve the Segal conditions, and also completeness is preserved since the object Equiv(X) is deﬁned as the limit of a ﬁnite cone of structure maps, so that Equiv([C, X]C ) is equivalent to HomC(C, Equiv(X)). Thus the restriction

op [ , ]C : C × CS(C) → CS(S)

≃ can be postcomposed with the right derived horizontal projection Ho∞(ph): CS(S) −→ Cat∞ ([14, 4]) to give a functor

Ext: CS(C) → IC(C) (27) by currying back. For a given complete Segal object X, we denote the corresponding unstraightened cartesian fibration by Ext(X) ։ C as well. By construction, its fibres Ext(X)(C) are naturally equivalent to the composition Ho∞(ph) ◦ [C, X]C . They can be described as the ∞-categories given by the 0-th row of any Reedy fibrant replacement of the complete Segal object [C, X]C = C(C, X(·)) in the category sS of simplicial spaces.

Remark 4.5. Completeness is not a necessary condition to deﬁne the externalization of a Segal object X in C. One can construct as well a functor

Ext: S(C) → IC(C) in the very same way. The only detail that requires additional care is the horizontal projection from Segal spaces to simplicial sets. But, on the level of model categories, the horizontal projection (sS, Reedy) → (S, QCat) preserves trivial ﬁbrations and takes Segal spaces to quasi-categories. It therefore takes weak equivalences between Segal spaces to categorical equivalences between quasi-categories. It thus induces a derived functor on underlying quasi-categories. We therefore obtain a functor Ext: S(C) → Cat∞ which restricts to the same externalization of complete Segal objects.

Remark 4.6. If C is a 1-category, then CS(C) is the 1-category of internal categories in C and Ext: CS(C) → Ind(C) is exactly the 1-categorical externalization as described in [13, B2.3] and [12, 7.3]. Explicitly, the morphisms between vertices x,y : C → X0 in Ext(X)(C) are maps f : C → X1 such that d1f = x and d0f = y. Given a morphism f : C → D in C, we obtain the morphism f ∗ : Ext(X)(D) → Ext(X)(C) by precomposition with f.

31 Remark 4.7. The cartesian fibrations corresponding to an externalization Ext(X): Cop → Cat∞ of a complete Segal object X in C are exactly the “representable” Segal cartesian fibrations C/X in [21, Definition 3.25] by [21, Remark 3.31]). Despite the clash of notation, note that these are not the cartesian fibrations given by the ∞-overcategories C/X ։ C as defined in [17, Proposition 1.2.9.2]. It follows from [21, Corollary 3.26] that the externalization construction (27) is a fully faithful embedding.

Remark 4.8. A Segal object X in C is a Segal groupoid if and only if its externalization is an indexed ∞-groupoid. Indeed, the Yoneda embedding is conservative and so the map Equiv(X) → X1 is an equivalence if and only if for every object C ∈ C, the presheaf [C, X]C is a Segal space such that C(C, Equiv(X)) →C(C, X1) is an equivalence. But this map is equivalent to the map Equiv([C, X]C ) → ([C, X]C )1. Hence, by an application of [17, Proposition 6.1.2.6.(3)], X is a Segal groupoid if and only if [C, X]C is a Bousﬁeld- Segal space in terms of [26, Corollary 5.4]. But this in turn holds if and only if its right derived horizontal projection Ext(X)(C) is a Kan complex by [26, Remark 5.5]. For the following, recall that Joyal and Tierney constructed in fact two Quillen equivalences between the model structure for complete Segal spaces and the model structure for quasi-categories. On the one hand, we have already considered the right Quillen functor

ph : (sS, CS) → (S, QCat).

On the other hand, in [14, 4] they construct a right Quillen functor

t! : (S, QCat) → (sS, CS) which is part of a Quillen equivalence as well. If by I∆n ∈ S we denote the nerve of the free groupoid on [n], the functor t! is given by the formula

! m n t (J)mn = S(∆ × I[∆ ], J) m ≃ = (Fun(∆ , J) )n for simplicial sets J. In particular, for every quasi-category C we have

! • • op t (C) = Cat∞(∆ , C) = [C, ∆ ]Cat∞ .

! In the proof of [14, Theorem 4.12] the authors show that the composition ph ◦ t : S → S is isomorphic to the identity, and so it follows that the composition

• Ho∞(ph) ◦ Cat∞(∆ , ): Cat∞ → Cat∞ is naturally equivalent to the identity on Cat∞.

Example 4.9 (The universal complete Segal object). The Yoneda embedding y : ∆ → S • op yields a simplicial object ∆ in Cat∞ which is a complete Segal object precisely because n the spine inclusions Sn → ∆ (i.e. the inclusions of the maximal chain of non-degenerate 1- simplices into ∆n) and the endpoint inclusion ∆0 → I∆1 are weak categorical equivalences in the Joyal model structure. We therefore can construct the externalization

• Ext(∆ ): Cat∞ → Cat∞

• given by Ho∞(ph) ◦ Cat∞(∆ , ) via the definition in (27). By the above, it follows that the given externalization is naturally equivalent to the identity on Cat∞. In other words, the universal cocartesian fibration π : Dat∞ → Cat∞ – defined as the unstraightening of • the identity on Cat∞ – is “represented” by the co-cocomplete co-Segal object ∆ in Cat∞ • op in that it is equivalent to the externalization of ∆ ∈ Cat∞. Under the translation referred to in Remark 4.1, the following proposition is equivalent to a corresponding sequence of results in [21, 3.3].

32 Proposition 4.10. 1. Let X be a complete Segal object in C. Then for every n ≥ 0, n the right fibration Rect(∆ , Ext(X)) ։ C is represented by Xn ∈ C. In particular, × the right fibration Ext(X) ։ C is represented by X0 ∈C. 2. Let p: E ։ C be a cartesian fibration such that for each n ≥ 0 the right fibration n Rect(∆ , E) ։ C is represented by some object Xn ∈ C. Then there is a complete Segal object X such that p ≃ Ext(X). Proof. For Part 1. let X be a complete Segal object in C and n ≥ 0. Then, by Proposi- tion 3.23, the straightening of Rect(∆n, Ext(X)) is given by the composite

n ≃ op Ext(X) Fun(∆ ,(·)) (·) C / Cat∞ / Cat∞ /4 S. (28)

Cat∞(∆n,(·))

It therefore suﬃces to show that the following diagram commutes up to equivalence.

op op 1∆×[ , ]C op 1∆×Ho∞(ph) op N(∆) ×C × CS(C) / N(∆) × CS(S) / N(∆) × Cat∞ ❱❱ ❱❱❱ ev • op ❱❱❱ 1C ×ev ❱❱❱❱ Cat∞(∆ ,(·)) ❱❱❱❱ ❱❱❱❱ Cop ×C ❱❱*/ S HomC (29) Towards homotopy-commutativity of the top right triangle, note that under currying with N(∆op) the counit ev: N(∆)op × CS(S) → S is mapped to the identity on CS(S) while the composition of the top horizontal arrow and the right vertical arrow is mapped to the composition

• Cat∞(∆ , ) ◦ Ho∞(ph): CS(S) → CS(S). (30) We have seen that both components of (31) are equivalences, and that their converse composition is equivalent to the identity. It follows that they are mutually inverse to each

other, so that the composition (31) is equivalent to the identity on CS(S) as well. Since ⊢ op the adjunction N(∆op) × ( )N(∆ ) is a simplicially enriched adjunction, it follows • that the two maps ev and Cat∞(∆ , (·)) ◦ 1∆ × Ho∞(ph) are equivalent as well. The bottom square commutes strictly, simply by the deﬁnition of the upper horizontal op arrow given by [ , ]C. If we denote multiplication with ∆ in S by F , the respective exponentiation by G, and the map (26) by f : F (Cop×C) → S, then we obtain the following naturality square for the counit applied to f.

op FG(f) F GF (Cop ×C∆ ) / F G(S) D ✌ F η ✤ ε ε ✶ op F (Cop ×C∆ ) / S f The map F η is a section to the counit on the right hand side by the triangle identities. op But the composition F G(f) ◦ F η : F (Cop ×C∆ ) → F G(S) is exactly the top horizontal map in the square in (29). This ﬁnishes the proof of Part 1. For Part 2. recall from Remark 3.6 and Remark 3.7 that the functor Rect( , E): Sop → S/C preserves limits and weak categorical equivalences. It follows that the complete Segal object (∆•)op ∈ (S, QCat)op yields a complete Segal object Rect(∆•, E) in the ∞-category RFib(C) of right ﬁbrations over C.

sRFib(C)

St∗ ≃ / ˆ sC y∗ sC

33 n The assumption that the right ﬁbration Rect(∆ , E) ։ C is represented by some Xn ∈C • for all n ≥ 0 just means that the complete Segal object St∗(Rect(∆ , E)) is contained in the image of y∗. Since the Yoneda embedding is fully faithful, we obtain a complete • ˆ Segal object X in C such that C( , X(·)) ≃ St∗(Rect(∆ , E)) in sC. Under currying, by • Proposition 3.23, the simplicial presheaf St∗(Rect(∆ , E)) corresponds to the composite

• op St(p) Cat∞(∆ , ) C −−−→ Cat∞ −−−−−−−−−→ sS. (31)

Its image factors through CS(S). In the same way, the simplicial presheaf C( , X(·)) corresponds to [ , X]C by deﬁnition, and we have equivalences

Ho∞(ph) ◦ [ , X]C ≃ Ext(X) and • Ho∞(ph) ◦ Cat∞(∆ , ) ◦ St(p) ≃ St(p). This ﬁnishes the proof.

Generally, the existence of an internal presentation of a given C-indexed ∞-category can be captured via (∞, 1)-comprehension in the following way.

Theorem 4.11. Let p: E ։ C be a cartesian fibration and suppose C has pullbacks. 1. The fibration p is both small and locally small if and only if there is a complete Segal object X in C such that Ext(X) is equivalent to E over C. 2. If p is a right fibration, then p is small (and hence the unstraightening of a representable presheaf) if and only if there is a complete Segal groupoid X in C such that Ext(X) is equivalent to E over C.

Proof. For Part 1. let X be a complete Segal object. Then the right ﬁbration Ext(X)× ։ C is represented by X0 by Proposition 4.10.1 and so Ext(X) is small. One can show that × the terminal object in Ext(X) is given by the identity 1X0 ∈ Ext(X)(X0). Towards local smallness, let x,y : C → X0 be vertices in Ext(X) and consider the pullback square

X1(x,y) / X1 (s,t) (32) C / X0 × X0. (x,y)

1 ∗ in C. We show that the right ﬁbration (δ ) ↓ (x,y) ։ C/C is represented by the arrow X1(x,y) → C. By deﬁnition, we have the pullback square

1 ∗ / 1 (δ ) ↓ (x,y) / Rect(∂∆ , Ext(X))/(x,y) ·y s (33) Rect(∆1, Ext(X)) / Rect(∂∆1, Ext(X)). (δ1)∗

The upper right corner of the square is equivalent to the representable C/C , the lower right corner is given by the pullback C/X0 ×C C/X0 ≃ C/X0×X0 by Remark 3.6 and smallness of Ext(X). The vertical ﬁbration on the right hand side is hence equivalent to the map

(x,y)∗ : C/C → C/X0×X0 . By Proposition 4.10.1, the lower left corner is equivalent to the representable C/X1 , and it follows that the square (33) is equivalent to the Yoneda embedding applied to the square (32). In particular, the comma ∞-category (δ1)∗ ↓ (x,y) is equivalent to the representable C/X1(x,y).

34 The converse direction basically holds by a generalization of the internal full subcategory construction in [13, B2.3]. Let t ∈ E× be terminal. The internal full subcategory X(t) of p generated by t is given as follows. 1 ∗ ∗ ∗ Let X(t)0 := p(t) and X(t)1 be the object representing (δ ) ↓ (π1t,π2t) over X(t)0 × X(t)0, which exists by local smallness. Thus, on the one hand, we have a trivial ﬁbration × ։∼ 0 0 × E/t C/X(t)0 which yields an equivalence Rect(∆ , E) ≃ C/X(t)0 since Rect(∆ , E) = E and t ∈E× is terminal. On the other hand, we note that we have equivalences

1 1 ∗ ∗ ∗ Rect(∆ , E) ≃ (δ ) ↓ (π1t,π2t) ≃C/X(t)1 again because t is terminal. For integers n ≥ 2, we have equivalences

Rect(∆n, E) ≃ Rect(Sn, E) 1 1 ≃ Rect(∆ , E) ×E× ···×E× Rect(∆ , E) ≃C × ···× C /X(t)1 C/X(t)0 C/X(t)0 /X(t)1 ≃C /(X(t)1×X(t)0 ···×X(t)0 X(t)1) over C again because Rect( , E) preserves limits and weak categorical equivalences. It hence follows by Proposition 4.10.2 that there is a complete Segal object X(t) in C whose externalization is equivalent to the cartesian fibration p: E ։ C. For Part 2. recall that a right fibration p: E ։ C is small if and only it is representable, and that representable fibrations are locally small if and only if C has equalizers by Re- mark 3.10. Thus every small right fibration C is locally small since C is left exact, and by Part 1. we obtain a complete Segal object X in C whose externalization is equivalent to p. By Remark 4.8, this implies that X is a complete Segal groupoid. The other direction follows directly from Part 1.

Corollary 4.12. Suppose C has pullbacks. Then a cartesian ﬁbration p: E ։ C has G-comprehension for all ﬁnite monomorphisms G: I → J if and only if it is the externalization of a complete Segal object X in C.

Proof. By Theorem 4.11.1 and Corollary 3.19.

Corollary 4.13. Suppose C has pullbacks and let ic(C) ⊂ IC(C) be the full ∞-subcategory of small and locally small indexed ∞-categories over C. Then the externalization construction Ext: CS(C) → ic(C) is an equivalence of ∞-categories.

Proof. The externalization construction is fully faithful as noted in Remark 4.7 via [21, Corollary 3.26].

Remark 4.14 (The universal cartesian fibration). We have seen in Example 4.9 that the op op ։ op universal cartesian fibration π : Dat∞ Cat∞ is the externalization of the complete • op Segal object ∆ in Cat∞. In Example 5.13 we will argue that this result can be understood as a consequence of the rather plain fact that the ∞-category of small ∞-categories is a “theory of (∞, 1)-categories” in the sense of Toën ([29]). Against the background of Theorem 4.11, we already know from Proposition 3.20 that the universal cartesian fibration is small and that it has (∅ → ∆1)-comprehension. In virtue of the factorization ∅→ ∂∆1 → ∆1, it also follows from Lemma 3.15 directly that it is locally small as well. Unfolding the definitions, local smallness of the fibration πop means that every ∞-category in Cat∞ can be freely extended by an edge between any two of its objects.

35 This internalizability (or “representability”) of the universal cartesian fibration recovers the fact that the universal left fibration S∗ ։ S is corepresented by the terminal object in S as well ([17, Proposition 3.3.2.6]). Indeed, the universal right fibration is the pullback of the universal cartesian fibration πop along a left adjoint (Example 3.26). The following corollary then shows that the complete Segal object I∆• ∈ CS(Sop) yields op • op n an equivalence (S∗) ≃ Ext(I∆ ) over S . But every I∆ is contractible in S, and so • op op Ext(I∆ ) ≃ S/∗ ≃ (S∗/) . We obtain the following computation of reindexed externalizations along left adjoints, directly generalizing [13, Lemma 2.3.6].

Corollary 4.15. Let F : D→C be a functor between ∞-categories with pullbacks. Sup- pose it has a right adjoint R. Then for every complete Segal object X in C, there is a homotopy cartesian square of the form

Ext(R∗X) / Ext(X)

D / C. F

Proof. Consider the pullback P / Ext(X) ·y D / C. F

In order to show that P is equivalent to Ext(R∗X) over D, by Proposition 4.10 it suﬃces n to show that the right ﬁbrations Rect(∆ , P ) ։ D are represented by R(Xn). But we have pullback squares of the form

Rect(∆n, P ) / Rect(∆n, Ext(X)) ·y (34) D / C F and by Proposition 4.10 we know that Rect(∆n, Ext(X)) ։ C is equivalent to the rep- ։ n resentable s: C/Xn C. Thus, the pullback square (34) exhibits Rect(∆ , P ) to be the comma ∞-category F ↓ Xn. By assumption, this is represented by R(Xn).

If C is complete, we obtain the following characterization of indexed ∞-categories which arise from complete Segal objects.

op Proposition 4.16. Suppose C is complete. Then a C-indexed ∞-category I : C → Cat∞ is equivalent to the externalization of a complete Segal object X in C if and only if it has a left adjoint.

op Proof. Suppose I : C → Cat∞ has a left adjoint L. We obtain a pullback square of the form / op Un(I) Dat∞ ·y πop C / Catop I ∞ where the bottom map has a right adjoint. The cartesian ﬁbration πop is the externalization of the complete Segal object ∆• by Example 4.14, and hence the pullback Un(I) is equivalent to the externalization of the complete Segal object Lop(∆•) by Corollary 4.15.

36 Vice versa, let X be a complete Segal object in C and consider its externalization

op [ ,X]C Ho∞(ph) C −−−−−→ CS(S) −−−−−→ Cat∞.

Since Ho∞(ph) has a left adjoint (being the underlying ∞-functor of the right Quillen ∗ functor ph), it suﬃces to show that [ , X]C = X ◦ y has a left adjoint as well. Therefore, we show that for all simplicial spaces Z ∈ sS, the comma ∞-category / Z ↓ [ , X]C / (sS)Z/ ·y t Cop / sS [ ,X]C

has an initial object (or equivalently, that it is corepresentable over Cop). First, if Z is a representable y([n]), we obtain a diagram as follows.

op \ C / Cop / N(∆) / S Xn/ y(Xn)/ y([n])/ •

d op / ˆop / \ / C / C ∗ / N(∆) / S y X ev([n])

The square on the right hand side is homotopy cartesian in the Joyal model structure 4 ∗ by Examples 2.4.4. Since ev([n]) ◦ X = ev(Xn), the right composite rectangle is a homotopy pullback as well for the same reason. This implies that the center square is a homotopy pullback. The square on the left hand side is a homotopy pullback, because the Yoneda embedding is fully faithful. Thus, eventually, the left composite rectangle is a homotopy pullback, which yields the equivalence Cop ≃ y([n]) ↓ [ , X] Xn/ C over Cop. \ For general simplicial spaces Z ∈ N(∆), one can now simply use that Z is the colimit \ of representables. Indeed, it follows that the comma ∞-category N(∆)Z/ is a homotopy limit of corepresentables, and hence so is the pullback along [ , X]C by what we just have shown, since homotopy pullbacks preserve homotopy limits. Since C is assumed to be complete, this limit of corepresentables over Cop is corepresented by the respective colimit in Cop. op op op It follows that [ , X]C : C → Fun((∆) , S) has a left adjoint, and so does [ , X]C : C → CS(S), since the inclusion CS(S) ⊂ Fun(N(∆)op, S) is fully faithful.

In particular, if C is a presentable ∞-category, a cartesian ﬁbration over C is the externalization of a complete Segal object if and only if it preserves limits by the Adjoint Functor Theorem. For right ﬁbrations this is exactly the basic fact that limit preserving presheaves are representable ([17, Proposition 5.5.2.2]).

Remark 4.17. If C is complete, it follows that every complete Segal object X in C yields a pullback square of the form / op Ext(X) Dat∞ ·y πop C / Catop Ext(X) ∞

4This also can be derived directly from Corollary 4.15 and the fact that the evaluation functor at [n] has a left adjoint.

37 where the bottom map has a right adjoint. Thus, by Lemma 3.24, it follows that all comprehension schemes satisﬁed by the universal cartesian ﬁbration are transferred to all internal ∞-categories in C whenever C is complete. We will study the externalization of complete Segal objects in the model categorical context in Section 5 and see there that, in analogy to Proposition 4.16, they exactly correspond to right Quillen functors into the Joyal model structure on simplicial sets.

4.2 Univalence over (∞, 1)-categories For the rest of this section suppose C is a left exact locally cartesian ∞-category. For every 1 map q : E → B in C we can associate the full comprehension ∞-category Fq ⊆ Fun(∆ , C) where Fq(C) ⊆C/C consists of those maps with codomain C which arise as pullback of q along some map C → B. By local cartesian closedness of C, there is a Segal object N (q) in C – the nerve of q – such that

Ext(N (q)) ≃ Fq (35)

∗ ∗ over C. Thenerve of q is determined by equivalences N (q) ≃ B and N (q)1 ≃ [π1q,π2q]B×B ∗ ∗ where the right hand side denotes the exponential of the maps π1q and π2q over B × B ([21, Remark 6.22]). Apart from notational diﬀerences, existence of the equivalence (35) is exactly the content of [21, Lemma 6.20]. This equivalence takes a map f : C → B ∗ in Ext(N (q))0 to the pullback f q in (Fp/C)0, and a map α: C → Ext(N (q))1 to the ∗ ∗ homotopically unique map between (d1α) q and (d0α) q that is represented by α as a ∗ ∗ map into the exponential [π1q,π2q]B×B. We follow Rasekh ([21, Deﬁnition 6.24]) and say a map q in C is univalent if the Segal object N (q) is complete.

Proposition 4.18. Suppose C is locally cartesian closed.

1. For every map q ∈C, the cartesian ﬁbration Fq ։ C is locally small. 2. A full and replete comprehension ∞-category J ⊆ Fun(∆1, C) is small if and only if it is the externalization Ext(N (q)) for some univalent map q ∈ C. That is, if and only if there is a univalent map q ∈C such that J ≃ Fq. In either case, it follows that the map q is a terminal object in J ×. 3. A map q : C → D in C is univalent if and only if the natural transformation × C( , D) → Ext(N (q)) of presheaves over C given by 1D 7→ 1D is a natural equivalence.

Proof. For Part 1. just note that the square

1 ∗ 1 (δ ) 1 Rect(∆ , Fq) / Rect(∂∆ , Fq) _ _

(δ1)∗ Rect(∆1, Fun(∆1, C)) / Rect(∂∆1, Fun(∆1, C))

1 is a pullback square since the inclusion ι: Fq ֒→ Fun(∆ , C) is full. It hence follows that for all q1,q2 ∈ Fq(C), C ∈C, the square

1 ∗ / / 1 (δ ) ↓ (q1,q2) / / Rect(∂∆ , Fq)/(q ,q ) _ _ 1 2

1 ∗ / / 1 1 (δ ) ↓ (ι(q1), ι(q2)) Rect(∂∆ , Fun(∆ , C))/(ι(q1),ι(q2)) is a pullback as well. The vertical arrow on the right hand side is a categorical equivalence since the inclusion ι is replete (using [3, Lemma 4.5] as referred to in the beginning of

38 Section 3.2). It thus follows that the vertical arrow on the left hand side is a categorical equivalence. The bottom left corner has a terminal object, since t: Fun(∆1, C) ։ C is locally small by Example 3.14. Hence, the top left corner has a terminal object as well, which shows Part 1. For Part 2. if J is small, then its core J × has a terminal object q : C → D and it follows 1 that J = Fq since J ⊆ Fun(∆ , C) is full and replete. By Part 1. and Theorem 4.11, we obtain a complete Segal object X such that Ext(X) ≃ Fq. Furthermore, by the proof of Theorem 4.11 we see that X0 ≃ D and X1 is the object in C representing the right fibration 1 ∗ ∗ ∗ ։ (δ ) ↓ (π1(q),π2(q)) C associated to the cartesian fibration Ext(N (q)) ։ C. By the proof of Part 1. X1 is also 1 ∗ ∗ ∗ ։ the object representing the right fibration (δ ) ↓ (ι(π1 (q)), ι((π2 (q))) C associated to the target fibration t: Fun(∆1, C) ։ C. By Proposition 3.14, this is the exponential of ∗ ∗ π1q and π2q over D × D. But by [21, Remark 6.22], this is exactly N (q)1. Since both X and N (q) are Segal objects, we obtain a pointwise equivalence X ≃ N (q) in sC. Hence, the Segal object N (q) is complete, and the map q is univalent. The other direction follows immediately from Theorem 4.11.1. × For Part 3. if q is univalent, then the identity 1D is terminal in Ext(N (q)) by the proof of Theorem 4.11.1 and the statement follows. Vice versa, if the object 1D is terminal in Ext(N (q))×, then, via the equivalence (35), the object q is terminal in the core of the small, full and replete comprehension ∞-category Fq. By Part 2. we obtain a univalent ′ ′ map q which is terminal in the core of Fq′ = Fq. It follows that q and q are equivalent, and hence that q is univalent as well.

The statement of Proposition 4.18.3 is equivalent to [21, Theorem 6.28]. In the case that C is a presentable ∞-category, Gepner and Kock have characterized univalence of maps q in C in terms of a sheaf property of the indexed ∞-category Fq ([9, Proposition 3.8]). This can be directly generalized as follows.

Corollary 4.19. Suppose C is locally cartesian closed and complete and q is a map in op C. Then q is univalent in C if and only if the indexed ∞-category Fq : C → Cat∞ has op a left adjoint. In particular, the functor Fq : C → Cat∞ preserves limits whenever q is univalent.

Proof. If q is univalent, then Fq has a left adjoint by Proposition 4.16 and the equivalence in (35). Vice versa, if Fq has a left adjoint, it is small in virtue of smallness of the universal cartesian ﬁbration and Lemma 3.24. By Proposition 4.18.2, it follows that there is a ′ univalent map q in C such that Fq ≃ Fq′ . As in the proof of Proposition 4.18.2, it follows that N (q) ≃N (q′) pointwise and so q is univalent.

Proposition 4.18.2 shows that a map q in C is univalent if and only if it is terminal in the core of the full comprehension ∞-category Fq ։ C. That means, a map q in C is univalent if and only if it is a (−1)-truncated object in the core Fun(∆1, C)×, in the sense 1 × that for all arrows f ∈ Fun(∆ , C), the mapping space Fq (f,q) is (−1)-truncated.

F ×(f,q) ≃ ∗ if f ∈ F , Fun(∆1, C)×(f,q) ≃ q q (∅ otherwise.

This motivates to define an object x ∈ E to be univalent if it is (−1)-truncated in the core E×. Accordingly, one can define the univalent completion of an object x ∈ E to be its (−1)-truncation in E× whenever it exists. This generalizes the notion of univalent completion as defined in [5] and studied in more generality in [5, 5].

39 Remark 4.20. According to this definition of univalent objects in a fibered ∞-category × p: E ։ C, the full ∞-subcategory Euniv ⊂ E of univalent objects in E is the ∞-poset of “propositions” in E× (borrowing homotopy type theoretic notation) by construction. Since p× : E× ։ C is a right fibration, a map in E× is a monomorphism if and only if its image under the fibration p is a monomorphism in C. This follows directly from the characterization of monomorphisms in an ∞-category D as those arrows f : C → D such that D/f → D/D is fully faithful ([17, 5.5.6]). Now, every map between (−1)- truncated objects is a monomorphism and the domain of a monomorphism into a (−1)- truncated object is (−1)-truncated itself ([17, Lemma 5.5.6.14]). It thereby follows that the ∞-category Euniv is fibred over the wide ∞-subcategory C−1 of C generated by the monomorphisms in C. More precisely, if p−1 : E−1 ։ C−1 denotes the pullback of p along the inclusion C−1 ⊂ C, it follows that Euniv ։ C−1 is a full and replete subfibration of × × ։ p−1 : E−1 C−1 by [3, Lemma 4.5]. Remark 4.21. In Remark 3.27 we discussed criteria towards a potential classification of elementary ∞-toposes via comprehension schemes. Against the background of the results in this section, let us review the discussion and once again assume for the moment being that C is presentable. Then C is an ∞-topos if and only if colimits in C are universal and its canonical indexing op C/ : C → Cat∞ preserves limits (via [17, Theorem 6.1.3.9]). If this canonical indexing had a left adjoint, we would obtain a homotopy cartesian square of the form

1 / op Fun(∆ , C) Dat∞

t πop / op C Cat∞ C/ where the bottom map has a right adjoint R. This implied that the canonical ﬁbration of C over itself is equivalent to Ext(R∆•) by Corollary 4.15 and Example 4.9, and in particular that it is small and locally small. Vice versa, every such C is an ∞-topos. op Brought back down to earth, we see that although the canonical indexing C/ : C → Cat∞ may be limit preserving, it basically cannot ever have a left adjoint in virtue of size issues (for instance the adjoint functor theorems fail, because the opposite of a presentable ∞-category is not presentable). The underlying slogan

“An ∞-topos is a presentable ∞-category which is small over itself.” or, in more animated terms,

“An ∞-topos is a presentable ∞-category which comprehends itself.” still holds up to local approximations, by which we mean the following. 1 1 Given a regular cardinal κ, let Fun(∆ , C)κ ⊂ Fun(∆ , C) be the full (and replete) ∞-subcategory generated by the relative κ-compact maps in C and consider its associated 1 comprehension ∞-category Fun(∆ , C)κ ։ C.

Corollary 4.22. A presentable ∞-category C is an ∞-topos if and only if the following two conditions hold. 1. The canonical ﬁbration Fun(∆1, C) ։ C is locally small. 1 2. For all suﬃciently large regular cardinals κ, the comprehension ∞-category Fun(∆ , C)κ over C is small.

Proof. Immediate by Proposition 4.18.2 and [17, Theorem 6.1.6.8].

40 For general ∞-categories C, for any pullback stable class S of morphisms in C we may consider its associated full comprehension ∞-category S ⊆ Fun(∆1, C). Then the deﬁnition of an elementary ∞-topos C as proposed by Shulman and used by Rasekh ([22, 3.2]) is equivalent to the following.

Corollary 4.23. Let C be an ∞-category with finite limits and colimits. Then C is an “elementary higher topos” ([22, Definition 3.5]) if and only if the following two conditions hold. 1. The canonical fibration Fun(∆1, C) ։ C is well-powered.

2. There is a collection A = {Si | i ∈ I} of “closed” ([22, Deﬁnition 3.4]) subclasses 1 1 Si ⊂ Fun(∆ , C) such that i∈I Si = Fun(∆ , C) and such that the associated comprehension ∞-categories S are small over C for every i ∈ I. iS

Thus, we obtain the following slogan.

“An elementary higher topos is a well-powered ∞-category C which can be covered by an atlas A of small neighbourhoods in C.”

It follows that elementary higher toposes are locally small over themselves via [22, The- orem 3.11]. Thus, they are “vertically small” over themselves and can be covered by “horizontally small” neighbourhoods.

5 Indexed quasi-categories over model structures

In this section we take a look at indexed ∞-categories over an ∞-category C from a homotopical algebraic perspective. We therefore will assume that C is the homotopy ∞-category of a model category M and study functors of type M → (S, QCat). To provide a general theory of indexed quasi-categories over a model category M, one is exposed to various choices in which class of functors of the form M → (S, QCat) to consider. One may for instance take the class of weak equivalence preserving functors, or the class of Quillen pairs, or anything inbetween. Each one of the Examples 2.2 can be constructed in one such fashion or the other, in certain cases requiring additional properties on M. The primary aim of this section is to relate the externalization construction studied in Section 4 to homotopical algebraic notions considered at various places in the literature. Therefore, we will restrict our attention to the class of right Quillen functors Mop → (S, QCat). The reason is the following. Given any model category M, recall that its derived mapping spaces M(A, B)h may be calculated in terms of coframes on its cofibrant objects A or frames on its fibrant objects B (see [10, 5]). Here, a frame on a fibrant object B is a homotopically constant Reedy ∼ fibrant simplicial object X in M such that X0 = B. In [10, Corollary 5.4.4], Hovey shows that for every fibrant object B in M and every frame X on B, the left Kan extension of X along the Yoneda embedding y : ∆ → S yields a Quillen adjunction

/ Mop M Lany(X): (S, Kan) o ⊢ : ( , X(·))

M M h such that (A, X(·)) ≃ (A, B) for all coﬁbrant objects A. Thus, we can assign to every simplicial object X in M a right adjoint Mop → S which is a right Quillen functor whenever X is a frame. Since frames always exist on all ﬁbrant objects in M in a homotopically unique way ([10, Theorem 5.2.8]), we may think of these right Quillen functors

M Mop ( , X(·)): → (S, Kan)

41 as the representable M-indexed Kan complexes. In the following we will note that in fact op every right Quillen functor of type M → (S, Kan) is exactly of the form M( , X(·)) for a frame X in M. So the M-indexed Kan complexes which are right Quillen functors are exactly the representable ones. If we define Ext(X) := M( , X(·)) for every simplicial object X in M, then we can op show that Ho∞(Ext(X)): C → S is the externalization construction of the complete Segal groupoid Ho∞(X) in C as defined in Section 4 whenever X is a frame. More generally, in this section we will characterize the right Quillen functors Mop → (S, QCat) exactly as those which are the externalization of a complete Segal object in M. Thus, in the sense of Theorem 4.11 and Proposition 4.16, we may think of the right Quillen functors into the Joyal model structure as exactly the small and locally small M-indexed quasi-categories. Whenever M is a logical model category (so M can be assigned an internal type theory that is able to express univalence among others), we obtain as special case that a fibration p: E ։ B in M with fibrant base is univalent if and only if the externalization of its associated internal ∞-category is a right Quillen functor Mop → (S, QCat). In this context we will see that the fibration extension property ([24, (3′)]) and the weak equivalence extension property ([24, (4′)]) can be similarly expressed as right Quillen conditions for a slice 2-functor associated to the fibration p. We consequently obtain an “externalized” explanation of the well known relationship between these two properties and univalence of p. Therefore, we first recall the definition of complete Segal objects in a model category M and define their externalization. For a simplicial object X ∈ sM, consider the left Kan extension X / Mop ∆ = ④ y ④ ④ \X:=LanyX ④ S, of X along the Yoneda embedding. By definition, for every X ∈ sM, we have isomorphisms n ∼ n n n X = Xn. The functor \ X takes the n-th boundary inclusion δ : ∂∆ ֒→ ∆ to \ ∆ n the n-th matching object MnX → Xn in M, and the n-th spine inclusion Sn ֒→ ∆ to the n-th Segal map Xn → X1 ×X0 ···×X0 X1.

Deﬁnition 5.1. A simplicial object X in M is

1. Reedy fibrant if the matching object MnX → Xn is a Kan fibration for all n ≥ 0. 2. a Segal object if it is Reedy fibrant and the Segal maps

Xn → X1 ×X0 ···×X0 X1

are homotopy equivalences for all n ≥ 2.

Definition 5.2. A Segal object X in M is complete if the functor \ X : Sop → M takes the endpoint inclusion ∆0 → I∆1 to an acyclic fibration. This definition of completeness is a straight forward generalization of Rezk’s original definition of completeness for Segal spaces. It is used in [14] to define a Quillen equivalence op from Rezk’s model structure (S∆ , CS) for complete Segal spaces to the Joyal model structure (S, QCat). A comparison to the “internal” definitions of completeness as used in [18] and [21] for example, is given in [27]. Definition 5.1 is chosen in such a way that we immediately obtain the following characterization.

Lemma 5.3. The left Kan extension \ X : Sop → M takes • boundary inclusions (and hence all monomorphisms) in S to ﬁbrations in M if and only if X is Reedy ﬁbrant;

42 • furthermore inner horn inclusions (and hence all mid anodyne morphisms) in S to acyclic ﬁbrations if and only if X is a Segal object; • furthermore the endpoint inclusion ∆0 → I∆1 to an acyclic ﬁbration if and only if X is a complete Segal object.

Proof. By definition, a simplicial object X in M is a Segal object if and only if the functor \ X : Sop → M takes the boundary inclusions to fibrations, and the spine inclusions to acyclic fibrations in M. The latter holds if and only if the functor \ X takes the class of inner horn inclusions to acyclic fibrations by [14, Lemma 1.21, Lemma 3.5].

The functor \ X : S → Mop naturally comes equipped with a right adjoint X/ , and we obtain an equivalence

op r.a. Lany : sM → Fun(M , S) (36) where the codomain denotes the full subcategory of right adjoints in Fun(Mop, S). This is noted as well already in [10, Proposition 3.1.5]. op Let CS(M) ⊆ M∆ denote the full subcategory of complete Segal objects in M and consider the full subcategory Fun(Mop, (S, QCat))r.Q. ⊆ Fun(Mop, S) of right Quillen functors. op Similarly, let Fr(M) ⊆ M∆ be the full subcategory of frames in M, and let Fun(Mop, (S, Kan))r.Q. ⊆ Fun(Mop, S) be the full subcategory of right Quillen functors.

Proposition 5.4. The assignment X 7→ Ext(X) := X/ yields equivalences of categories 1. Ext: CS(M) → Fun(Mop, (S, QCat))r.Q., 2. Ext: Fr(M) → Fun(Mop, (S, Kan))r.Q., which in both cases map a complete Segal object X to the functor M 7→ M(M, X(·)). Proof. Since the map (36) is an equivalence, for Part 1. we only have to show that a simplicial object X in M is a complete Segal object if and only if its associated functor Ext(X)= X/ is a right Quillen functor with respect to the Joyal model structure. If X is a complete Segal object, by Lemma 5.3 we know that the left adjoint \X : (S, Qat) → Mop preserves cofibrations and takes both inner horn inclusions and the endpoint inclusion I∆1 to acyclic cofibrations. But since a map between quasi-categories is a fibration →֒ 0∆ in (S, QCat) if and only if it has the right lifting property against the inner horn inclusions and ∆0 → I∆1, this implies that the right adjoint X/ preserves fibrations between fibrant objects. This implies that X/ is a right Quillen functor e.g. by [14, Proposition 7.15]. The converse direction follows directly from Lemma 5.3. For Part 2. we similarly need to show that a simplicial object X in M is a frame if and only if its associated functor Ext(X) = X/ is a right Quillen functor with respect to the Quillen model structure for Kan complexes. But a Reedy fibrant X is homotopically i constant if and only if all boundary maps d \ X : Xn+1 → Xn are acyclic fibrations (since the degeneracies are sections thereof). By an application of [14, Lemma 3.7] this in turn holds if and only if the functor \X takes all anodyne maps in S to acyclic cofibrations in Mop. Thus, it follows that X is a frame if and only if the functor \ X : (S, Kan) → Mop is a left Quillen functor.

Given a simplicial object X in M, there are various ways to think about the functor ⊢ Ext(X)= \ X = M( , X(·)). It is part of the adjoint pair ( \ X) (X/ ) deﬁned in [14] and the weighted limit of X. In the given context we think of it as the externalization of the internal ∞-category X whenever X is a complete Segal object.

Remark 5.5. For a frame X in M, the functor Ext(X) is just the derived mapping space h functor M( , X0) , since X is a simplicial frame on X0.

43 Remark 5.6. If M is a model category and C is a strict category object in M, we can construct the internal nerve N(C) ∈ sM in M. If by h we denote the left adjoint to the nerve functor N : Cat → S, then the push forward h ◦ Ext(N(C)): Mop → S → Cat is exactly the externalization of the category object C in the 1-categorical sense of [12, 7.3] and [13, B2.3].

Remark 5.7. In [14, 1], the authors construct a right Quillen functor k! : (S, QCat) → (S, Kan) which comes with a natural weak equivalence to the core functor (·)≃ : QCat → Kan when restricted to the full subcategory of quasi-categories. If I∆n denotes the nerve ! n of the free groupoid generated by [n], then it is given by the formula k (S)n = S(I∆ ,S). Thus, postcomposition of a right Quillen functor Ext(X): Mop → (S, QCat) with k! yields a right Quillen functor k! ◦ Ext(X): Mop → (S, Kan). Hence, by Proposition 5.4, there is a frame X≃ such that Ext(X≃) =∼ k! ◦ Ext(X) which we deﬁne to be the core of the complete Segal object X.

Lemma 5.8. For every complete Segal object X in M, the functor

op Ho∞(Ext(X)): C → Cat∞ is equivalent to the externalization Ext(Ho∞(X)) as deﬁned in Section 4.

Proof. Given a complete Segal object X in M, we obtain a C-indexed ∞-category

op Ho∞(Ext(X)): C → Cat∞ (37) by Proposition 5.4, as well as a complete Segal object Ho∞(X) in C with Ho∞(X)n = Xn for all n ≥ 0. In order to show that Ho∞(Ext(X)) ≃ Ext(Ho∞X)), by Proposition 3.23 and Proposition 4.10 it suﬃces to show that the composition

∆n ≃ op Ho∞(Ext(X)) (·) (·) C / Cat∞ / Cat∞ / S (38) is equivalent to the representable C( , Xn) for all n ≥ 0. Each component in the composition (38) is the underlying ∞-functor of the following respective right Quillen functor.

n Ext(X) (·)∆ ! Mop / (S, QCat) / (S, QCat) k / (S, Kan) (39)

By Proposition 5.4, for each n ≥ 0 we obtain a frame X(n) in M such that Ext(X(n)) is given by the composition (39). For every M ∈ M, we obtain a sequence of natural isomorphisms as follows.

M (n) ∼ (n) ∼ ! ∆n ∼ ∆n ∼ ∼ M (M, X0 ) = Ext(X )(M)0 = (k (Ext(X) ))0 = (Ext(X)) )0 = Ext(X)n = (M, Xn)

(n) ∼ M (n) so that X0 = Xn in and the simplicial object X is a frame of Xn. It follows that (n) Ext(X ) represents the derived mapping space functor M( , Xn)h, and we see that the presheaf (n) op Ho∞(Ext(X )): C → S is represented by Xn ∈C.

Remark 5.9. If the model category M satisﬁes certain additional properties (e.g. if it is simplicial, combinatorial and right proper, and its coﬁbrations are exactly the monomorphisms), one can construct the maps in (26) on the level of model categories such that the resulting functor Ext: CS(M) → Fun(Mop, (S, QCat))r.Q. yields the functor Ext: CS(C) → IC(C) from Section 4 on underlying ∞-categories.

44 Remark 5.10. Dugger showed in [7] that whenever M is combinatorial there is a model structure (sM, Fr) whose ﬁbrant objects are exactly the frames, such that the inclusion

∆: M → (sM, Fr) is the left part of a Quillen equivalence. Thus the composition

M −→∆ sM −−→Ext Fun(Mop, S)r.a. ≃ yields a Quillen equivalence to a model structure on the functor category Fun(Mop, S) whose fibrant objects are the right Quillen functors into (S, Kan). More generally, whenever the complete Segal objects arise as the fibrant objects of a model structure on sM (see [2]), then we obtain a Quillen equivalent model structure on the functor category Fun(Mop, S)r.a. whose fibrant objects are exactly the right Quillen functors into the Joyal model structure (S, QCat). The model categorical externalization construction of internal (∞, 1)-categories as given by Proposition 5.4 can be dualized to a nerve construction associated to internal (∞, 1)-co-categories. This comprises many examples throughout the literature.

Example 5.11. If the category Cat of small categories is equipped with the canonical model structure, the ordinary nerve

N : Cat → (S, QCat) is a right Quillen functor. That means it is a “small and locally small” indexed quasi- category over Catop and as such it is given by the externalization of the complete Segal object ∆• in Catop. The simplicial object ∆• itself is the internal nerve of the internal category F given by “free co-composition” on the cograph [0] ⊔ [0] → [1] in Cat. In that sense the internal co-category [0]⊔[0] → [1] in Cat is “co-univalent”, witnessed by the fact that the endpoint inclusion into the freely walking isomorphism is an acyclic cofibration in Cat. The internal category F in Catop can be externalized in the ordinary categorical sense, giving an indexing Ext(F): Cat → Cat which is isomorphic to the identity on Cat. Its Grothendieck construction Gr(Ext(F)) is given by pointed categories and lax pointed functors between them. Its associated opfibration pF : Gr(Ext(F)) ։ Cat is the universal opfibration in the sense that every opfibration with fibres in Cat is equivalent to the pullback of pF along its associated 1-categorical indexing.

Example 5.12. The homotopy coherent nerve N∆ with its left adjoint C yields a Quillen equivalence N∆ : S-Cat → (S, QCat) where the domain is the category of simplicially enriched categories equipped with the Bergner model structure. In particular, N∆ is the small and locally small indexed quasi- op category over S-Cat given by the externalization of the complete Segal object C|∆op in S-Catop. When considered as a cosimplicial object in S-Cat, this is known to be the Reedy coﬁbrant replacement (in the following denoted R) of the functor ∆• ∈ S-Cat∆. Or, that is, the Reedy ﬁbrant replacement of the same functor when considered as a simplicial object in S-Catop. Since the co-Segal maps of ∆• ∈ S-Cat∆ are in fact isomorphims, we again have that ∆• is the “co-nerve” of the internal co-category [0] ⊔ [0] → [1] in S-Cat, yielding an internal category F in the opposite category. Thus, we have C ≃ Ext(R(N(F)). In this sense, it also is presented by a “co-univalent” co-category object in S-Cat. Analogously one can show that Dwyer and Kan’s W¯ -construction for simplicial groupoids op ([8, 3]) is equivalent to Ext(R(N(F)) for F = ((d1, d0): I[1] → [0] × [0]) in S-Gpd .

45 Example 5.13. More generally, in [29], Toën considered criteria on a model category M for the existence of an “indexed complete Segal space” (Mop)op → (sS, CS) which is the right part of a Quillen equivalence. Therefore, he gives a general construction of functors SX : M → sS associated to cosimplicial objects X in M, which is part of a Quillen equivalence if and only if the pair (M, X) is a “theory of (∞, 1)-categories” ([29, Definition 4.2]). One essential part of this definition is that X is a complete Segal object in Mop with contractible base X0, there called interval ([29, Definition 3.4]). In the proof of [29, Theorem 5.1] he constructs the Quillen equivalence

op op SX : (M ) → (sS, CS) exactly such that its postcomposition with the projection ph : (sS, CS) → (S, QCat) yields op the externalization of the complete Segal object X in M . Since ph is a Quillen equivalence itself by [14], Toën’s theorem can be rephrased stating that a pair (M, X) is a theory of (∞, 1)-categories if and only if the externalization Ext(Xop): M → (S, QCat) is a Quillen equivalence. Suppose M is a logical model category ([1, Definition 1]), so that it comes equipped with an underlying type theory which is able to express univalence of type families. By definition, this just means that the pullback along fibrations in M, first, preserves acyclic cofibrations and, second, has a right adjoint. In this case we have the following result.

Theorem 5.14 ([27, Corollary 4.5]). Let M be a logical model category in which all fibrant objects are cofibrant, and let p: E ։ B be a fibration in M with fibrant base. Then the following are equivalent. 1. p is a univalent fibration in the type theoretic fibration category underlying M ([25]). M ∗ ∗ 2. The nerve N(Fun(p)) ∈ s of the internal category Fun(p) := [π1E,π2 E]B×B over B × B is univalent in the sense of [27] (i.e. complete in the sense of [21] and [18]).

3. For any Reedy ﬁbrant replacement Xp of N(Fun(p)), the Segal object Xp is complete in the sense of Deﬁnition 5.2.

Thus, in virtue of Proposition 5.4, we obtain the following characterization of univalence.

Corollary 5.15. Let M be a logical model category in which all fibrant objects are cofibrant, p: E ։ B be a fibration in M with fibrant base. Then the following three conditions are equivalent. 1. p is a univalent fibration. op 2. For any Reedy fibrant replacement Xp of N(Fun(p)), the externalization Ext(Xp): M → (S, QCat) is a right Quillen functor. ≃ op 3. For any Reedy fibrant replacement Xp of N(Fun(p)), the externalization Ext((Xp) ): M → (S, Kan) is a right Quillen functor.

Proof. The only non-trivial direction is (3) ⇒ (2). Therefore, first note that, by Lemma 5.3 and an orthogonality argument, for every simplicial object X in M the functor Ext(X): Mop → (S, QCat) preserves acyclic fibrations if and only if X is Reedy fibrant. It takes fibrations to inner fibrations if and only if X is furthermore a Segal object, and it takes fibrations to fibrations in (S, QCat) if and only if X is furthermore complete. Since Xp is always a Reedy fibrant Segal object, for (2) to hold we hence only need to show that the functor Ext(Fun(p)) takes every fibration between fibrant objects in Mop to a map in S with the right lifting property against the endpoint inclusion ∆0 → I∆1. But by [14, Proposition

46 1.20] for every such ﬁbration B → A in Mop, we have commutative squares

! ∼ ≃ k Ext(Xp)(B) / / Ext(Xp)(B) / Ext(Xp)(B)

! ∼ ≃ k Ext(Xp)(A) / / Ext(Xp)(A) / Ext(Xp)(A)

1 ≃ in S. Every map from I∆ to Ext(Xp)(A) factors through the core Ext(Xp)(A) and thus ! through k Ext(Xp)(A). Hence, (3) implies the desired lifting property, and hence implies (2).

In the light of Corollary 5.15, we finish this section by giving an “external” explanation of two properties – the fibration extension property and the weak equivalence extension property – that are often used to verify whether a given fibration in a model category M as given in Theorem 5.14 is univalent or not. Therefore, for the rest of this section let M be a logical model category in which all objects are cofibrant and let p: E ։ B be a fibration in M with not necessarily fibrant base B. The underlying graph Fun(p) → B × B of the associated internal category Fun(p) over B × B is a fibration. Hence, via the usual recursion, one can construct a simplicial object Xp together with a Reedy acyclic cofibration N(Fun(p)) → Xp such that (Xp)0 = B, (Xp)1 = Fun(p), and Xp is Reedy fibrant if and only if B is fibrant in M. By the adjoint gymnastics exercised in the proof of Corollary 5.15, we see that the following three conditions are equivalent. (1) B is fibrant in M.

(2) Xp is Reedy fibrant (and hence a Segal object). op (3) Ext(Xp): M → S takes acyclic fibrations to maps with the right lifting property to ∅→ ∆0. If either of these conditions is satisfied, the following three are equivalent as well. (i) p: E ։ B is univalent.

(ii) Xp is a complete Segal object. op (iii) Ext(Xp): M → S takes ﬁbrations to maps with the right lifting property with respect to the endpoint-inclusion ∆0 → I∆1. To further simplify these conditions, we may consider the associated class

∼ ∗ Fp = {q : F ։ C |∃f : C → B : q = f p} of “reindexed” families of p, and quotient out all higher categorical data that is irrelevant for the two lifting properties (3) and (iii). Indeed, although the simplicial set I∆1 has non-degenerate simplices in all dimensions, it may without loss of generality replaced by (2) (2) the 2-dimensional simplicial set K = J ∪∆1 J introduced in the proof of Proposi- tion 3.17.2.(c). Therefore, let 2-Cat be the category of 2-categories and strict 2-functors. In analogy to Quillen’s original homotopy category of the model category M, one can define a homotopy 2-category Ho2(M) whose objects are the bifibrant objects in M, whose arrows are exactly the arrows between bifibrant objects in M, and whose 2-cells between a pair of arrows f, g : A → B are equivalence classes of homotopies between f and g, where we consider two homotopies H, J : A → PB with the same boundary to be equivalent if and only if there is a homotopy A → PB×B (PB) from H to J. Then a map f in M between bifibrant objects is a homotopy equivalence if and only if it is an equivalence in the 2-category Ho2(M). Note that all 2-cells in Ho2(M) are invertible. Most of the necessary calculations are contained in [10, 1.2] and [25, 3].

47 If LModCat denotes the category of logical model categories in which all objects are coﬁbrant, together with right Quillen functors between them, we obtain a functor

Ho2 : LModCat → 2-Cat. Here, note that in virtue of the cofibrancy condition we do not have to derive right Quillen functors between such model categories to obtain a 2-functor between the quo- tients Ho2(M) as defined above. This enables us to define a 2-categorical slice functor on M as follows. For any object C ∈ M the slice M/C is a logical model category again in which all objects are cofibrant. We thus we may define (Fp)/C to be the full 2-subcategory of Ho2(M/C ) generated by the class of objects contained in Fp. Given an arrow f : A → C M ∗ M M in , the pullback functor on model categories f : /C → /A is a right Quillen functor and hence yields an associated functor on homotopy 2-categories which we may restrict to the full subcategory (Fp)/C . We hence obtain a composite pseudo-functor as follows. M op / Ho2 (Fp)/ : M −−−−→ LModCat −−→ 2-Cat Lemma 5.16. If j : A → C is an acyclic cofibration in M, then the transition map

(Fp)/j : (Fp)/C → (Fp)/A is locally an acyclic ﬁbration in Cat. In other words, it is a 2-functor which is locally full faithful and locally surjective on objects.

Proof. Let j : A → C be an acyclic coﬁbration in M, let qi : Ei ։ C for i = 0, 1 be ﬁbrations in Fp and consider their pullbacks along j.

∗ j F1 / F1 ·y ∗ / ∗ j F2 F2 j q1 ·y q1 ∗ q2 j q2 A ∼ / C j

∗ Since j is an acyclic cofibration and the qi are fibrations, the maps j Fi → Fi are acyclic ∗ ∗ cofibrations as well. Hence, given a map f : j F1 → j F2 over A, we obtain a square in M of the form q∗j◦f j∗F 2 / F _1 2 ∗ q j q2 1 ∼ F / C. 1 q1

Maps of type F1 → F2 over B which pullback to f along j correspond exactly to lifts in this square. Such a lift always exists, which means that the functor ∗ ∗ ((Fp)j)1 : (Fp)/C (q1,q2) → (Fp)/C (j q1, j q2) is surjective on objects. Fully faithfulness can be shown in a similar way from the homotopy extension property of acyclic coﬁbrations. Indeed, given maps f, g : F1 → F2 over C in M and a homotopy H between j∗f and j∗g, then the existence of a lift in the square

j∗F H / P (j∗F ) / P (F ) _1 A 2 C 2 ∼ F1 / F2 ×C F2 (f,g) proves fullness of (Fp)/ , while homotopy uniqueness of such lifts proves faithfulness.

48 Deﬁnition 5.17. Let p: E ։ B be a ﬁbration in M. We say that Fp has 1. the weak equivalence extension property if every solid diagram of the form ∗ j F1 ❴❴❴❴❴❴❴❴ / F1 ❋❋ ✮ ❆ ❋❋w ❆ v ❋❋ ✰ ❆ ❋❋# ✲ ❆ ∗ j F2 ✵ / F2 q¯1 ✸ ·y q1 q¯2 ✻ q2 ✾ A / C j

where w is a weak equivalence between fibrationsq ¯1, q¯2 ∈ Fp, j is a cofibration and q2 is another fibration in Fp, has a dashed extension as above such that v is a weak equivalence between fibrations q1 ∈ Fp and q2 ∈ Fp and the back square is cartesian, too. 2. the fibration extension property if every solid span

F¯ ❴❴❴ / F y ✤ q · ✤ q¯ ✤ A ∼ / C j

where q ∈ Fp and j is an acyclic coﬁbration can be complemented to a cartesian square such thatq ¯ ∈ Fp is a ﬁbration, too.

3. the stratiﬁcation property if for every pair of ﬁbrations q1, q2 in Fp, every solid diagram of the form j∗F / E ❈ > ❈❈·y ⑦ ❈❈ ⑦ ·y ❈❈ ⑦ ❈! ⑦ q1 F p

q2 A a / B. ❈ > ❈❈ ⑥ ❈❈ ⑥ j ❈❈ ⑥ c ❈! ⑥ C with j : A → C a coﬁbration, there is a dotted extension such that the front square is cartesian as well, and such that the diagram commutes.

Proposition 5.18. Let p: E ։ B be a fibration in M. Let 2-Cat be equipped with the Lack model structure ([15]). op 1. The class Fp has the weak equivalence property if and only if (Fp)/ : M → 2-Cat preserves fibrations. op 2. The class Fp has the fibration extension property if and only if (Fp)/ : M → 2-Cat preserves acyclic fibrations.

Proof. The class Fp has the weak equivalence extension property if and only if for every cofibration j in M, the functor (Fp)/j has the right lifting property against the inclusion [0] → I[1]. That is, if and only if the functor (Fp)/j is a fibration in 2-Cat. The class Fp has the fibration extension property if and only if for every acyclic cofibration j in M, the functor (Fp)/j has the right lifting property against the endpoint inclusion ∅ → [0]. That is, if and only if the functor (Fp)/j is surjective on objects. By Lemma 5.16, the functor (Fp/ )h(j) is always a local acyclic fibration, but the local acyclic fibrations which are surjective on objects are exactly the acyclic fibrations in the Lack model structure on 2-Cat.

49 Thus, the class Fp has the weak equivalence extension property and the ﬁbration ex- op tension property if and only if the M-indexed 2-category (Fp)/ : M → 2-Cat satisﬁes the right Quillen conditions of a Quillen pair.

op Similarly, one can quotient out the higher data in the externalization Ext(Xp): M → S to obtain a functor 2 op Ext (Xp): M → 2-Cat. Since fibrancy of B and univalence of p are characterized by the right lifting property of the transition maps Ext(Xp)(j) for (acyclic) cofibrations j against the 2-dimensional arrows ∅→ ∆0 and ∆0 → I∆1, it follows that both can be verified by the same right lifting 2 properties of Ext (Xp)(j) in 2-Cat. Furthermore, we obtain a pseudo-natural transformation 2 Ext (Xp) → (Fp)/ 2 by choosing a pullback of p along every object C → B in Ext (Xp)(C). Then p: E ։ B satisfies the stratification property if and only if this pseudo-natural transformation takes cofibrations j : A → C in M to squares in 2-Cat such that the natural map

2 2 Ext (Xp)(C) → Ext (Xp)(A) ×(Fp)/A (Fp)/C is surjective on objects. One can thereby construct an “external” proof of the following fact used throughout the literature.

Proposition 5.19. Suppose p satisﬁes the stratiﬁcation property.

1. Then B is ﬁbrant if and only if Fp has the ﬁbration extension property.

2. If B is ﬁbrant, then p is univalent if and only if Fp has the weak equivalence extension property.

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Department of Mathematics and Statistics, Masaryk University, Kotla´rskˇ a´ 2, Brno 61137, Czech Republic E-mail address: [email protected]