2D HQFTs and Frobenius $(\mathcal{G},\mathcal{V})$-categories
Paul Großkopf
TL;DR
The paper extends HQFTs to target pairs $(X,Y)$ and develops a complete 1D classification in terms of dualizable representations of the relative fundamental groupoid $\mathcal{G}=\Pi_1(X,Y)$. For 2D HQFTs, it introduces crossed loop Frobenius $(\mathcal{G},\mathcal{V})$-categories as the governing algebraic structure and proves a classification theorem: 2D $(X,Y)$-HQFTs are equivalent to crossed loop Frobenius $(\mathcal{G},\mathcal{V})$-categories, generalizing the known group case to a groupoid setting. The approach mirrors Turaev's classical results while leveraging the relative homotopy data of $(X,Y)$, thereby providing a unified, functorial framework for HQFTs with basepoints and their 2D realizations. The outlook sketches open/closed HQFTs and related categorical refinements, indicating future work toward a crossed knowledgable Frobenius theory to capture boundary phenomena.
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 $\mathcal{G}:=Π_1(X,Y)$, 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.