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 **un**oriented 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!



Generators and relations for the 2-dimensional unoriented cobordism category
My initial answer was wrong, here's the correct version plus a reference: [Turaev-Turner](https://arxiv.org/pdf/math/0506229.pdf)


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.


