Twin TQFTs and Frobenius Algebras

We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra (C,W,z,z*) consists of a commutative Frobenius algebra C, a symmetric Frobenius algebraW, and an algebra homomorphism z : C -> W with dual z* : W -> C, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological QuantumFieldTheory defined on singular 2-dimensional cobordisms and showthat it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.