Classifying spaces of algebras over a prop

TL;DR

This paper proves that weak equivalences between cofibrant props induce homotopy equivalences between their classifying spaces of algebras, extending known results from operads to props with new methods.

Contribution

It generalizes homotopy invariance of classifying spaces from operads to props, overcoming the lack of model structure on algebras over props with novel techniques.

Findings

Weak equivalences induce weak homotopy equivalences of classifying spaces

Extension to colored props in symmetric monoidal model categories

New methods to handle absence of model structure on algebras

Abstract

We prove that a weak equivalence between cofibrant props induces a weak equivalence between the associated classifying spaces of algebras. This statement generalizes to the prop setting a homotopy invariance result which is well known in the case of algebras over operads. The absence of model category structure on algebras over a prop leads us to introduce new methods to overcome this difficulty. We also explain how our result can be extended to algebras over colored props in any symmetric monoidal model category tensored over chain complexes.