Frobenius algebras, categorification and 2d TQFTs

Commutative Frobenius algebras can be regarded essentially in two ways. On one hand, they are finite dimensional algebras equipped with a pairing enjoying some special properties;
on the other hand, they can be seen as two dimensional oriented Topological Quantum Field theories, namely representations of the oriented cobordism category in two dimensions.
Both notions can be categorified, providing apparently different results. In this talk, I will present the main results from a work in progress with Domenico Fiorenza, which show how
the different notions of categorified commutative Frobenius algebras can be explained in terms of (higher) representations of two dimensional cobordism categories.