A Synthetic Perspective on (∞,1)-Category Theory: Fibrational and Semantic Aspects

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.

Section: Lari Familiesmentioning

confidence: 85%

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confidence: 99%

“…The work presented here appears as [73,Chapter 5]. This PhD thesis was written at TU Darmstadt under the supervision of Thomas Streicher.…”

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confidence: 99%

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confidence: 99%

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

2022

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“…Indeed, continuing the program of [RS17], some parts of synthetic (∞, 1)-category theory have recently in simplicial HoTT been developed in [CRS18,BW21,Wei22,Mar22], and in [WL20] within a (bi-)cubical variant of the theory. Many of these developments are crucially inspired by results from Riehl-Verity's ∞-cosmos theory, a very general and powerful account to model-independent (∞, n)-category theory itself not formulated within type theory.…”

Section: Introductionmentioning

confidence: 99%

Strict stability of extension types

2022

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

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.