A concept of $\frac{2}{3}$PROP and deformation theory of (co)associative bialgebras

We introduce a concept of $\frac{2}{3}$PROP generalizing the Kontsevich concept of $\frac{1}{2}$PROP. We prove that some Stasheff-type compactification of the Kontsevich spaces $K(m,n)$ defines a topological $\frac{2}{3}$PROP structure. The corresponding chain complex is a minimal model for its cohomology (both are considered as $\frac{2}{3}$PROPs). We construct a $\frac{2}{3}$PROP $\text{End}(V)$ 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 $\frac{2}{3}$PROPs.