23PROP and deformation theory of (co)associative bialgebras

We introduce a concept of 2 3 PROP generalizing the Kontsevich concept of 1 2 PROP. We prove that some Stasheff-type compactification of the Kontsevich spaces K(m,n) defines a topological 2 3 PROP structure. The corresponding chain complex is a minimal model for its cohomology (both are considered as 2 3 PROPs). We construct a 2 3 PROP End(V ) for a vector space V . Finally, we construct a dg Lie algebra controlling the deformations of a (co)associative bialgebra. Philosophically, this construction is a version of the Markl’s operadic construction from [M1] applied to minimal models of 2 3 PROPs.