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

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 $\mathcal{G}:={\mathrm{\Pi }}_1(X,Y)$, 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.