A pair of homotopy-theoretic version of TQFT's induced by a Brown functor

TL;DR

This paper develops a homotopy-theoretic framework for projective TQFTs using Brown functors, deriving obstruction classes and applying them to recover known TQFTs like Dijkgraaf-Witten and Turaev-Viro.

Contribution

It introduces a novel homotopy-theoretic approach to projective TQFTs via Brown functors and constructs explicit obstruction classes, connecting to established TQFTs.

Findings

Constructed pair of projective HTQFTs from Brown functors.

Derived formulae for obstruction classes in the homotopy-theoretic setting.

Reproduced Dijkgraaf-Witten and Turaev-Viro TQFTs from homology theory.

Abstract

The purpose of this paper is to study some obstruction classes induced by a construction of a homotopy-theoretic version of projective TQFT (projective HTQFT for short). A projective HTQFT is given by a symmetric monoidal projective functor whose domain is the cospan category of pointed finite CW-spaces instead of a cobordism category. We construct a pair of projective HTQFT's starting from a $\mathsf{Hopf}^\mathsf{bc}_k$-valued Brown functor where $\mathsf{Hopf}^\mathsf{bc}_k$ is the category of bicommutative Hopf algebras over a field $k$ : the cospanical path-integral and the spanical path-integral of the Brown functor. They induce obstruction classes by an analogue of the second cohomology class associated with projective representations. In this paper, we derive some formulae of those obstruction classes. We apply the formulae to prove that the dimension reduction of the cospanical and spanical path-integrals are lifted to HTQFT's. In another application, we reproduce the Dijkgraaf-Witten TQFT and the Turaev-Viro TQFT from an ordinary $\mathsf{Hopf}^\mathsf{bc}_k$-valued homology theory.