2D HQFTs and Frobenius $(\mathcal{G},\mathcal{V})$-categories

TL;DR

This paper introduces a new framework for 2D Homotopy Quantum Field Theories using Frobenius categories associated with pairs of spaces, providing a classification theorem that generalizes previous algebraic structures.

Contribution

It defines and classifies 2D HQFTs with target pairs of spaces using crossed loop Frobenius categories, extending the concept beyond traditional group-based models.

Findings

Classification of 1D $(X,Y)$-HQFTs via dualizable representations of $ ext{Pi}_1(X,Y)$.

Introduction of crossed loop Frobenius $( ext{G}, ext{V})$-categories for 2D HQFTs.

A general classification theorem for 2D $(X,Y)$-HQFTs using these new categorical structures.

Abstract

Homotopy Quantum Field Theories as variants of Topological Quantum Field Theories are described by functors from some cobordism category, enriched with homotopical data, to a symmetric monoidal category $\mathcal{V}$. A new notion of HQFTs is introduced using target pairs of spaces $(X,Y)$ acounting for basepoints being sent to points in $Y$. Such $(X,Y)$-HQFTs are classified in dimension 1 by dualizable representations of $\mathcal{G}:=\Pi_1(X,Y)$, the relative fundamental groupoid. For dimension 2, the notion of crossed loop Frobenius $(\mathcal{G},\mathcal{V})$-categories is introduced, generalizing crossed Frobenius $G$-algebras, where $G$ is only a group. After stating generalities of these multi-object generalizations, a classification theorem of 2-dimensional $(X,Y)$-HQFTs via crossed loop Frobenius $(\mathcal{G},\mathcal{V})$-categories is proven.