Monoidal categories graded by crossed modules and 3-dimensional HQFTs

TL;DR

This paper introduces $ ext{chi}$-graded monoidal and fusion categories based on crossed modules and constructs 3D Homotopy Quantum Field Theories using these categories, linking algebraic structures to topological quantum field theories.

Contribution

It develops the theory of $ ext{chi}$-graded monoidal categories and uses spherical $ ext{chi}$-fusion categories to build 3D HQFTs via state sum methods.

Findings

Defined $ ext{chi}$-graded monoidal and fusion categories.

Constructed 3D HQFTs with target space $B ext{chi}$.

Established a new link between algebraic categories and topological quantum field theories.

Abstract

Given a crossed module $\chi$, we introduce $\chi$-graded monoidal categories and $\chi$-fusion categories. We use spherical $\chi$-fusion categories to construct (via the state sum method) 3-dimensional Homotopy Quantum Field Theories with target the classifying space $B\chi$ of the crossed module $\chi$ (which is a homotopy 2-type).