Stratified spaces, Directed Algebraic Topology, and State-Sum TQFTs

TL;DR

This paper explores how directed topology can be applied to stratified spaces to construct state-sum TQFTs with defects, extending existing theories to include more complex defect orientations and structures.

Contribution

It introduces a novel application of directed topology to stratified spaces, enabling the construction of new state-sum TQFTs with oriented defects and extending Dijkgraaf-Witten theories.

Findings

Directed topology provides a framework for stratified space analysis.

New state-sum TQFTs with defect orientations are constructed.

Extension of Dijkgraaf-Witten TQFTs to complex defect configurations.

Abstract

We apply the theory of directed topology developed by Grandis [9, 10] to the study of stratified spaces by describing several ways in which a stratification or a stratification with orientations on the strata can be used to produce a related directed space structure. This description provides a setting for the constructions of state-sum TQFTs with defects of [5, 8], which we extend to a similar construction of a Dijkgraaf- Witten type TQFT in the case where the defects (lower dimensional strata) are not sources or targets, but sources on one side and targets on the other, according to an orientation convention.