On a pair of extensions of Mayer-Vietoris functors

TL;DR

This paper introduces two canonical extensions of Mayer-Vietoris functors to cospans of spaces, connecting classical homology theories with models in topological quantum field theory through a homotopy-theoretic framework.

Contribution

It provides a novel pair of extensions of Mayer-Vietoris functors to cospans, linking algebraic topology with quantum field theory models via a universal cospan category.

Findings

Derived extensions relate to classical homology theories.

Connected extensions to topological quantum field theory models.

Revealed partial universality of cospan categories of CW-spaces.

Abstract

The purpose of this paper is to give a pair of canonical extensions of Mayer-Vietoris functors (e.g. homology or cohomology theories of spaces) to cospans of spaces. One of the pair, called cospanical extension in this paper, is closely related with the classical theory part of abelian Dijkgraaf-Witten-Freed-Quinn model and bicommutative Turaev-Viro-Barrett-Westbury model. We derive those extensions by revealing a partial universality of a cospan category of CW-spaces which is a homotopy-theoretic analogue of cobordism categories.