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

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 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 $(\infty ,1)$-categories, interpreted as Rezk objects in any given Grothendieck--Rezk--Lurie $(\infty ,1)$-topos.

arXiv comment59 pages. This text is based on Section 3.4 and Chapter 5 from author's PhD thesis arXiv:2202.13132, with varioius additions and improvements. Revised version, accepted for publication at the Journal of Pure and Applied Algebra