Approximate Fibrations in Higher Topos Theory

TL;DR

This paper integrates the concept of approximate fibrations into higher topos theory, providing new characterizations and generalizations that connect shape theory with topos-theoretic frameworks.

Contribution

It introduces a topos-theoretic definition of approximate fibrations, compares it with classical notions, and extends shape-theoretic characterizations to a broader topos context.

Findings

Defined approximate fibrations for geometric morphisms of ∞-topoi

Provided several shape-theoretic characterizations

Generalized Lurie's shape-theoretic results to topos theory

Abstract

The goal of this paper is to put the theory of approximate fibrations into the framework of higher topos theory. We define the notion of an approximate fibration for a general geometric morphism of $\infty$-topoi, give several characterizations in terms of shape theory and compare it to the original definition for maps of topological spaces of Coram and Duvall. Furthermore, we revisit the notion of cell-like maps between topoi, and generalize Lurie's shape-theoretic characterization by giving a purely topos-theoretical proof.