On the PROP corresponding to bialgebras

A PROP is a symmetric monoidal category, whose set of objects is the set of natural numbers and on objects the monoidal structure is given by the addition. An algebra over a PROP is a symmetric strict monoidal functor to the tensor category of vector spaces. We give an explicite construction of the PROP whose category of algebras is equivalent to the category of bialgebras (= associative and coassociative bialgebras)

Andr\'e-Quillen homology via functor homology

We obtain Andr\'e-Quillen homology for commutative algebras using relative homological algebra in the category of functors on finite pointed set