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Yoneda's Lemma for Internal Higher Categories

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YONEDA'S LEMMA FOR INTERNAL HIGHER CATEGORIES LOUIS MARTINI Abstract. We develop some basic concepts in the theory of higher categories internal to an arbitrary 1- topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal categories. Contents 1. Introduction 2 Motivation 2 Main results 3 Related work 4 Acknowledgment 4 2. Preliminaries 4 2.1. General conventions and notation4 2.2. Set theoretical foundations5 2.3. 1-topoi 5 2.4. Universe enlargement 5 2.5. Factorisation systems 8 3. Categories in an 1-topos 10 3.1. Simplicial objects in an 1-topos 10 3.2. Categories in an 1-topos 12 3.3. Functoriality and base change 16 3.4. The (1; 2)-categorical structure of Cat(B) 18 3.5. Cat(S)-valued sheaves on an 1-topos 19 3.6. Objects and morphisms 21 3.7. The universe for groupoids 23 3.8. Fully faithful and essentially surjective functors 26 arXiv:2103.17141v2 [math.CT] 2 May 2021 3.9. Subcategories 31 4. Groupoidal fibrations and Yoneda's lemma 36 4.1. Left fibrations 36 4.2. Slice categories 38 4.3. Initial functors 42 4.4. Covariant equivalences 49 4.5. The Grothendieck construction 54 4.6. Yoneda's lemma 61 References 71 Date: May 4, 2021. 1 2 LOUIS MARTINI 1. Introduction Motivation. In various areas of geometry, one of the principal strategies is to study geometric objects by means of algebraic invariants such as cohomology, K-theory and (stable or unstable) homotopy groups. Usually these invariants are constructed in a functorial way, in the sense that they define (possibly higher) presheaves on a suitable (higher) category C of the geometric objects of interest. Usually, C is equipped with a Grothendieck topology that encodes the topological behaviour of the geometric objects contained in C, and one typically expects reasonable algebraic invariants to respect this topology in that they should define (higher) sheaves on C. In that way, one can study the global nature of these objects by means of their local behaviour. For example, if A is an abelian group and if n ≥ 0 is an integer, there is a higher sheaf Hn(−;A) for the ´etaletopology on the category Sch of schemes that takes values in the 1-category Grp(S) of group n objects in 1-groupoids, such that for any scheme X the group π0H (X; A) 2 Grp(Set) is the nth ´etale cohomology group of X with coefficients in A [Lur09a, Section 7]. In modern flavours of geometry, the study of categorified invariants has come more and more into focus. For example, one of the fundamental invariants that one can associate to a scheme X is its unbounded derived category D(X). In order to turn this construction into a sheaf on Sch, i.e. in order to be able to glue derived categories, one first passes to an 1-categorical enhancement: there is a sheaf X 7! QCoh(X) on Sch that sends X to the symmetric monoidal stable 1-category of quasi-coherent sheaves on X such that the homotopy category of QCoh(X) recovers the derived category D(X) of X. The properties of QCoh and their relations to the geometry of the scheme X have been investigated by numerous authors, for example by Lurie in [Lur18] and by Ben-Zvi, Francis and Nadler in [BZFN10]. Furthermore, the sheaf QCoh plays a fundamental role in geometric representation theory; in [Gai14] Gaitsgory studies section-wise module 1- categories over the sheaf QCoh, i.e. the sheaf of 1-categories that are acted on by QCoh. Another categorical invariant on schemes comes from motivic homotopy theory: the assignment that sends a scheme X to its unstable motivic homotopy 1-category H(X) defines a sheaf for the Nisnevich topology on Sch, and similarly the assignment X 7! SH(X) in which SH(X) is the stable motivic homotopy 1-category on X defines a sheaf for the Nisnevich topology on Sch [Hoy17]. The latter plays a prominent role in the formalisation of the six operations in motivic homotopy theory, see for example [CD19] for an overview. It has been long known that for many choices of an (a priori higher) category A, the datum of an A-valued sheaf on Sch is equivalent to that of an A-object internal to the 1-topos ShS(Sch), or the 1-topos ShSet(Sch) in the case that A is actually a 1-category. For example, a sheaf of abelian groups on Sch is simply an abelian group internal to ShSet(Sch), and the collection of structure sheaves OX for X 2 Sch are encoded by a single ring object in ShSet(Sch), the sheaf represented by the affine line. By making use of the internal logic of the 1-topos of sheaves on Sch, one can therefore study such invariants in the same way as one studies their non-parametrized counterpart in the 1-category of 1-groupoids, see for example the PhD thesis of Blechschmidt [Ble17] for an application of these ideas to algebraic geometry in the 1-categorical case. The study of higher invariants, i.e. sheaves on Sch that take values in a reasonable 1-category A, thus naturally leads to the emerging field of homotopy type theory [Uni13]. In fact, it is now known [Shu19] that much of homotopy type theory has a model in an arbitrary 1-topos. By extending the same line of thought to the case of categorical invariants on schemes, the datum of a sheaf of 1-categories on Sch is equivalent to that of a category internal to the 1-topos ShS(Sch). One should therefore be able to formulate and study the properties of such sheaves by means of studying their categorical properties when viewed internally in the 1-topos ShS(Sch). Such an internal perspective on category theory is not only useful for the study of categorical invariants on schemes, but in other situations as well. For example, work of Barkwick, Glasman and Haines [BGH20, BH19] shows that the pro´etale 1- topos Spro´et of any coherent scheme S can be naturally regarded as a pyknotic 1-category, i.e. as a category YONEDA'S LEMMA FOR INTERNAL HIGHER CATEGORIES 3 internal to the 1-topos Pyk(S) of pyknotic 1-groupoids. Moreover, Sebastian Wolf shows in [Wol20] that internally in Pyk(S) the 1-topos Spro´et turns out to simply be given by a presheaf category, which amounts to a significant simplification of the structure of Spro´et. Although bare homotopy type theory is insufficient to argue synthetically about such structures (owing to the presence of non-invertible arrows), Riehl and Shulman [RS17] have proposed an additional layer to homotopy type theory that rectifies this problem and that is powerful enough to support a synthetic formulation of basic higher categorical constructions. In the present paper, our goal is to study sheaves of 1-categories from a semantic point of view: as categories internal to an 1-topos B. On the one hand, this is meant to provide tools for studying categorical invariants in geometry, such as those that are mentioned above. On the other hand, the study of categories in an 1-topos B should provide categorical semantics for the type theory developed by Riehl and Shulman. We do not aim to turn this into a formal statement, nevertheless we believe that developing the semantical side of the story might serve as a bridge to foster future development of the syntactic theory as well. Main results. The present paper constitutes the first in a series of papers aimed at developing the theory of categories internal to an arbitrary 1-topos B. Arguably, Yoneda's lemma is at the very core of any flavour of category theory, so in this paper we will focus on a proof of this result in the context of internal higher categories. We define the 1-category Cat(B) of categories in B as the full subcategory of the 1-category of simplicial objects in B that satisfy the Segal condition and univalence, mimicking the definition of complete Segal spaces given by Rezk [Rez01] as previously done by Lurie [Lur09b] and Rasekh [Ras18]. Since complete Segal spaces provide a model for the 1-category of 1-categories, the latter can be identified with Cat(S), and one obtains an equivalence Cat(B) ' ShCat(S)(B) between the 1-category of categories in B and the 1-category of Cat(S)-valued sheaves on B. Consequently, the sheaf B=− that sends an object A 2 B to the slice 1-category B=A defines a (large) category Ω in B that we will refer to as the internal universe of B and which represents the reflection of B within itself. We furthermore show that Cat(B) is cartesian closed, hence that for any two categories C and D in B there is a category [C; D] 2 Cat(B) of functors between C and D. The first of the two main results in this paper is an internal version of the Grothendieck construction: For any category C in B, we define a large category LFibC in B whose objects are given by left fibrations over C. We then show: Theorem 4.5.2. There is a canonical equivalence [C; Ω] ' LFibC that is natural in C. By making use of the Grothendieck construction, we derive our second main result, an internal version op of Yoneda's lemma. For any category C in B, we construct a mapping bifunctor mapC(−; −): C × C ! Ω whose transpose gives rise to the Yoneda embedding h: C ! [Cop; Ω].
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