Limits and colimits in internal higher category theory

Martini, Louis; Wolf, Sebastián

doi:10.48550/arxiv.2111.14495

Cited by 4 publications

(9 citation statements)

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“…General conventions and notation. We generally follow the conventions and notation from [Mar21] and [MW21]. For the convenience of the reader, we will briefly recall the main setup.…”

Section: Preliminariesmentioning

confidence: 99%

“…Recollection on B-categories. In this section we recall the basic framework of higher category theory internal to an ∞-topos from [Mar21] and [MW21].…”

Section: Factorisation Systemsmentioning

confidence: 99%

“…We used this result to prove Yoneda's lemma for B-categories. In joint work with Sebastian Wolf, we continued our study of internal higher categories in [MW21], where we developed a few basic tools such as the theory of adjunctions, limits and colimits as well as Kan extensions for B-categories. We also constructed the (large) B-category Cat B of small B-categories.…”

Section: Introductionmentioning

confidence: 99%

“…Note that by[MW21, Corollary 6.3.7] the assumptions on E 0 , E and D make sure that the functors of left Kan extension exist. Using the triangle identities, one can construct a commutative diagram…”

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confidence: 99%

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Cocartesian fibrations and straightening internal to an $\infty$-topos

Martini¹

2022

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We define and study cartesian and cocartesian fibrations between categories internal to an ∞-topos and prove a straightening equivalence in this context. Contents 1. Introduction 1 2. Preliminaries 4 3. Cocartesian fibrations 9 4. The marked model for cocartesian fibrations 15 5. The B-category of cocartesian fibrations 30 6. Straightening and unstraightening 35 7. Applications 49 Appendix A. The proof of Lemma 4.1.1 51 References 53

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“…General conventions and notation. We generally follow the conventions and notation from [Mar21] and [MW21]. For the convenience of the reader, we will briefly recall the main setup.…”

Section: Preliminariesmentioning

confidence: 99%

“…Recollection on B-categories. In this section we recall the basic framework of higher category theory internal to an ∞-topos from [Mar21] and [MW21].…”

Section: Factorisation Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Cocartesian fibrations and straightening internal to an $\infty$-topos

Martini¹

2022

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“…Rezk objects. Those have been investigated in various works and semantic settings, notably [51,50,43,10,60,37,38].…”

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confidence: 99%

Two-sided cartesian fibrations of synthetic $(\infty,1)$-categories

Weinberger¹

2022

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Within the framework of Riehl-Shulman's synthetic (∞, 1)-category theory, we present a theory of two-sided cartesian fibrations. Central results are several characterizations of the two-sidedness condition à la Chevalley, Gray, Street, and Riehl-Verity, a two-sided Yoneda Lemma, as well as the proof of several closure properties.Along the way, we also define and investigate a notion of fibered or sliced fibration which is used later to develop the two-sided case in a modular fashion. We also briefly discuss discrete two-sided cartesian fibrations in this setting, corresponding to (∞, 1)-distributors.The systematics of our definitions and results closely follows Riehl-Verity's ∞-cosmos theory, but formulated internally to Riehl-Shulman's simplicial extension of homotopy type theory. All the constructions and proofs in this framework are by design invariant under homotopy equivalence. Semantically, the synthetic (∞, 1)-categories correspond to internal (∞, 1)-categories implemented as Rezk objects in an arbitrary given (∞, 1)-topos.

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Internal sums for synthetic fibered $(\infty,1)$-categories

Weinberger¹

2022

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We give structural results about bifibrations of (internal) (∞, 1)categories with internal sums. This includes a higher version of Moens' Theorem, characterizing cartesian bifibrations with extensive aka stable and disjoint internal sums over lex bases as Artin gluings of lex functors.We also treat a generalized version of Moens' Theorem due to Streicher which does not require the Beck-Chevalley condition.Furthermore, we show that also in this setting the Moens fibrations can be characterized via a condition due to Zawadowski.Our account overall follows Streicher's presentation of fibered category theory à la Bénabou, generalizing the results to the internal, higher-categorical case, formulated in a synthetic setting.Namely, we work inside simplicial homotopy type theory, which has been introduced by Riehl and Shulman as a logical system to reason about internal (∞, 1)-categories, interpreted as Rezk objects in any given Grothendieck-Rezk-Lurie (∞, 1)-topos.

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Limits and colimits in internal higher category theory

Cited by 4 publications

References 11 publications

Cocartesian fibrations and straightening internal to an $\infty$-topos

Cocartesian fibrations and straightening internal to an $\infty$-topos

Two-sided cartesian fibrations of synthetic $(\infty,1)$-categories

Internal sums for synthetic fibered $(\infty,1)$-categories

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