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If $H\to\mathrm O$ is a tangential structure (e.g. orientation, spin), let $\mathsf{Bord}_2^H$ denote the category whose objects are 1-dimensional manifolds with $H$-structure and whose morphisms $M_1\to M_2$ are diffeomorphism classes of bordisms from $M_1$ to $M_2$ with $H$-structure. This category is symmetric monoidal under disjoint union.

The structure of the oriented bordism category $\mathsf{Bord}_2^{\mathrm{SO}}$ is well-known, as part of the proof that a 2D oriented TQFT is equivalent to a Frobenius algebra. (Joachim Kock has a good set of notes on the subject.)

The structure of the spin bordism category was worked out by Moore and Segal in "$D$-branes and $K$-theory in 2D topological field theory," and further discussed in a paper of Gunningham.

My question is: have descriptions of the categories $\mathsf{Bord}_2^{\mathrm{Pin}^+}$ and $\mathsf{Bord}_2^{\mathrm{Pin}^-}$ been written down?

I'm interested in particular in 2D extended TQFT, hence in the relevant bordism 2-categories, but a nonextended description would still be helpful.

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    $\begingroup$ As far as I know this has not appeared in the literature in either the extended or nonextended cases. $\endgroup$ – Chris Schommer-Pries Mar 29 '18 at 14:59

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