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Internal sums for synthetic fibered $(\infty,1)$-categories (2205.00386v3)

Published 1 May 2022 in math.CT, cs.LO, math.AT, and math.LO

Abstract: We give structural results about bifibrations of (internal) $(\infty,1)$-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 `{a} la B\'{e}nabou, 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 $(\infty,1)$-categories, interpreted as Rezk objects in any given Grothendieck--Rezk--Lurie $(\infty,1)$-topos.

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