Virtual operad algebras and realization of homotopy types

We prove that the category of algebras over a cofibrant operad admits a closed model category structure. This leads to the notion of “virtual operad algebra” – the algebra over a cofibrant resolution of the given operad. In particular, virtual commutative algebras can serve to an algebraic description of homotopy p-types as in the recent preprint of Mandell (M. Mandell, E∞-algebras and p-adic homotopy theory, Hopf preprint server, October, 1998). Our main result allows one to simplify the proof of Mandellʹs theorem.