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).
Abrams, Two-dimensional topological quantum field theories a nd Frobenius algebras, Journal of Knot Theory and Its Ramifications 5 (5), (1996), 569–587
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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).