Generators and relations for the 2-dimensional unoriented cobordism category

Question

It is very well known in the field of TQFT that the 2-dimensional oriented cobordism category is generated by the disk and the pair of pants (each going in both directions), subject to a finite set of relations. Those generators and relations are equivalent to the morphisms and axioms of Frobenius algebras.

What is the analogue statement in the unoriented case? It is easy to see that it suffices to add the Moebius strip to the generators, but is there a provably sufficient set of relations for them?

I feel like this must be worked out somewhere, but I'm having trouble to find anything. If someone could give a reference where generators and relations are listed, this would be helpful!

Answer 1

My initial answer was wrong, here's the correct version plus a reference: Turaev-Turner

New generating morphisms: The Mobius strip $\emptyset \rightarrow S^1$ and the "orientation reversing" diffemorphism of the circle $S^1 \rightarrow S^1$.

New relations: Orientation reversal is an involution. Orientation reversal plays well with all the other generators. Orientation reversal composed with a twice punctured unoriented surface is itself. The punctured Klein bottle is both the product of two Mobius strips, and a composition of copairing, orientation reversal, and pants.

Algebraically, you have a Frobenius algebra involution $\phi: A \rightarrow A$ and an element $\theta \in A$ such that:

1. $\phi(\theta v) = \theta v$
2. $(m \circ (\phi \otimes id) \circ \Delta)(1) = \theta^2$.

It actually follows from these definitions, see Lemma 2.8, that $\theta^3 = (m \circ \Delta)(\theta)$. That is the simplest relation you can state without making reference to the orientation reversing map, but you need the orientation reversing map in order for $\mathrm{Hom}(S^1,S^1)$ to be correct, which is why my initial guess was wrong.