2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/153680We define closed model category structures on different categories connected to the world of operad algebras over the category C(k) of (unbounded) complexes of k-modules: on the category of operads, on the category of algebras over a fixed operad, on the category of modules over a fixed operad algebra. In Sections 2 - 6 we define the necessary structures and provide some standard comparison results. In Section 7 we define cotangent complex. In Section 8 we define a canonical structure of homotopy Lie algebra on the tangent complex.AMS-LaTeX v1.2 (Compatibility mode), 39 pagesQuantum Algebratext