  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
algebra W, and an algebra homomorphism z: C - > W with dual z^*: W -> C,
satisfying some extra conditions. We also introduce a generalized 2-dimensional
Topological Quantum Field Theory defined on singular 2-dimensional cobordisms
and show that it is equivalent to a twin Frobenius algebra in a symmetric
monoidal category.
