Yoneda's lemma for internal higher categories

Abstract

We develop some basic concepts in the theory of higher categories internal to an arbitrary ∞topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal categories. Contents 1. Introduction Motivation Main results Related work Acknowledgment 2. Preliminaries 2.1. General conventions and notation 2.2. Set theoretical foundations 2.3. ∞-topoi 2.4. Universe enlargement 2.5. Factorisation systems 3. Categories in an ∞-topos 3.1. Simplicial objects in an ∞-topos 3.2. Categories in an ∞-topos 3.3. Functoriality and base change 3.4. The (∞, 2)-categorical structure of Cat(B) 3.5. Cat(S)-valued sheaves on an ∞-topos 3.6. Objects and morphisms 3.7. The universe for groupoids 3.8. Fully faithful and essentially surjective functors 3.9. Subcategories 4. Groupoidal fibrations and Yoneda's lemma 4.1. Left fibrations 4.2. Slice categories 4.3. Initial functors 4.4. Covariant equivalences 4.5. The Grothendieck construction 4.6. Yoneda's lemma References

Limits and colimits in internal higher category theory

“…General conventions and notation. We generally follow the conventions and notation from [Mar21], but there will be a few deviations. We therefore briefly recall our main setup below.…”

“…Recollection on B-categories. In this section we recall the basic framework of higher category theory internal to an ∞-topos from [Mar21]. We refer the reader to [Mar21] for proofs and a more detailed discussion.…”

“…This paper is the second in a series in which we intend to add to this language by developing categorical tools in the context of higher categories internal to an ∞-topos B. As a first step towards this goal, Yoneda's lemma for internal higher categories was proven in [Mar21] by the first-named author. In this article we continue this work with an extensive discussion of adjunctions, (co)limits, Kan extensions and free (co)completions in the internal setting.…”

“…A higher category C internal to an ∞-topos B is a simplicial object in B satisfying the Segal conditions and univalence (see [Mar21,Definition 3.2.4] for a precise definition). The study of internal categories in B is equivalent to the study of sheaves of ∞-categories on B.…”

“…Let us from now on fix an ∞-topos B. In [Mar21] the first-named author proved Yoneda's lemma for internal higher categories. More precisely, it was shown that for any B-category C there is a fully faithful functor of B-categories h : C → [C op , Ω] such that the functor…”