Two-dimensional homotopy quantum field theories with target pairs (X,Y) are classified by crossed loop Frobenius categories over the relative fundamental groupoid of (X,Y).
Rodrigues, Homotopy Quantum Field Theories and the Homotop y Cobordism Category in Dimension 1 + 1, J
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2D HQFTs and Frobenius $(\mathcal{G},\mathcal{V})$-categories
Two-dimensional homotopy quantum field theories with target pairs (X,Y) are classified by crossed loop Frobenius categories over the relative fundamental groupoid of (X,Y).