(∞, 1)-Categorical Comprehension Schemes Raffael Stenzel October 20, 2020 Abstract In this paper we define and study a notion of comprehension schemes for cartesian fibrations over (∞, 1)-categories, generalizing Johnstone’s respective notion for ordinary fibered categories. This includes natural generalizations of smallness, local smallness and the notion of definability in the sense of B´enabou, as well as of Jacob’s comprehension cat- egories. Thereby, among others, we will characterize numerous categorical properties of ∞-toposes, the notion of univalence and the externalization of internal (∞, 1)-categories via comprehension schemes. An example of particular interest will be the universal carte- sian fibration given by externalization of the “freely walking chain” in the (∞, 1)-category of small (∞, 1)-categories. In the end, we take a look at the externalization construc- tion of internal (∞, 1)-categories from a model categorical perspective and review various examples from the literature in this light. Contents 1 Introduction 1 2 Preliminaries on indexed (∞, 1)-category theory 5 3 (∞, 1)-Comprehension schemes 9 3.1 Definitionsandexamples. 9 3.2 Definability and comprehension (∞, 1)-categories . 27 4 Externalization of internal (∞, 1)-categories 29 4.1 Definition and characterization via (∞, 1)-comprehension . 30 4.2 Univalence over (∞, 1)-categories . 38 arXiv:2010.09663v1 [math.CT] 19 Oct 2020 5 Indexed quasi-categories over model structures 41 1 Introduction What is a comprehension scheme? Comprehension schemes arose as crucial notions in the early work on the foundations of set theory, and hence found expression in a large variety of foundational settings for mathematics. Particularly, they have been introduced to the context of categorical logic first by Lawvere and then by B´enabou in the 1970s. Hence, they have been studied in different forms and have been applied to many examples throughout the literature of category theory. The notion of a comprehension scheme as used in this paper is ultimately The author acknowledges the support of the Grant Agency of the Czech Republic under the grant 19-00902S. 1 rooted in the Axiom scheme of Restricted Comprehension (often referred to as the Axiom scheme of Separation) which is part of the Zermelo-Fraenkel axiomatization of set theory. 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], B´enabou 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 for- mal/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 elemen- tary 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 hyper- doctrines ([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 ex- amples 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).