2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78841We 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-tipes as in the recent preprint of Mandell (electronic Hopf topology archive, october '98, see http://www.hopf.math.purdue.edu. Our main result allows one to significally simplify the proof of Mandell's theorem.12 pagesQuantum AlgebraAlgebraic Topology18G55; 55P99text