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We construct a family of unoriented 2-dimensional cobordism theoriesparametrized by certain triples of sequences. We also prove that somespecializations of these sequences yield equivalences with an exterior productof Deligne categories. In particular, one of these specializations recovers thegeneralized Deligne category $\operatorname{Rep}(S_t\wr \mathbb Z_2)$.

Unoriented 2-dimensional TQFTs and the category $\operatorname{Rep}(S_t\wr \mathbb Z_2)$