Approximate Fibrations in Higher Topos Theory

2025/10/28 by Kremer, Christian, Volpe, Marco
#Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2510.24629

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 ∞-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.

Citations

• Presentability and topoi in internal higher category theory
• Verdier duality on conically smooth stratified spaces
• Colimits and cocompletions in internal higher category theory
• The six operations in topology
• Yoneda's lemma for internal higher categories
• Approximately fibering a manifold over an aspherical one
• Higher Galois theory
• Higher Topos Theory
• Shrinking Cell-Like Decompositions of Manifolds. Codimension Three

Related

• Cocartesian fibrations and straightening internal to an ∞-topos
• A Theory of Elementary Higher Toposes
• On fibred products of toposes
• Local fibrations and morphisms of relative toposes
• Internal higher topos theory