Approximate Fibrations in Higher Topos Theory

TL;DR

The paper develops a higher topos-theoretic framework for approximate fibrations by defining them for proper geometric morphisms $f:\mathscr{X}\to\mathscr{Y}$ and establishing shape-theoretic characterizations; it also recasts cell-like maps in purely topos-theoretic terms and supplies a topos-theoretic proof akin to Lurie’s results. For locally contractible topoi, approximate fibrations coincide with shape fibrations and shape quasi-fibrations, and for maps of locally compact ANRs the framework recovers classical equivalences between topological and shape-theoretic conditions. The main contributions include a purely topos-theoretic generalization of Lacher’s characterization of cell-like maps, relative shape and hereditary shape notions, and a precise corepresentation framework linking $F_*F^*$ to $f_\sharp(1_{\mathscr{X}})$. Overall, the work provides a bridge between geometric topology and higher topos theory, enabling obstruction-theoretic and manifold-topology techniques to be imported into an ∞-topos setting with potential new insights and methods.

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.

Theorems & Definitions (81)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 2.1
• Theorem 2.2
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Definition 2.6