On a general notion of a polynomial identity and codimensions

Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category C as well as their codimensions in the case when C is linear over some field. The new cases include coalgebras, bialgebras, Hopf algebras, braided vector spaces, Yetter-Drinfel'd modules, etc. We find bases for polynomial identities and calculate codimensions in some important particular cases.