Props in model categories and homotopy invariance of structures

TL;DR

This paper establishes a model structure for props in various symmetric monoidal model categories and demonstrates that homotopy equivalent objects can inherit algebraic structures, generalizing minimal models of A-infinity algebras.

Contribution

It proves that categories of props inherit model structures in broad settings and shows homotopy invariance of algebraic structures over props, extending known results to new contexts.

Findings

Props in symmetric monoidal model categories have inherited model structures.

Homotopy equivalent objects can inherit algebra structures over props.

Generalization of Kadeishvili's minimal model for A-infinity algebras.

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-infinity algebras.