Props in model categories and homotopy invariance of structures

Abstract

Abstract We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object 𝑋 homotopy equivalent to an algebra 𝐴 over a cofibrant prop P inherits a P-algebra structure so that 𝑋 defines a model of 𝐴 in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of 𝐴∞-algebras.