On homotopy invariance for algebras over colored PROPs

TL;DR

This paper establishes a homotopy invariance result for algebras over cofibrant colored PROPs within monoidal model categories, with applications to homotopy topological conformal field theories.

Contribution

It introduces a model category structure on colored PROPs and their algebras and proves a homotopy invariance theorem for these algebras, extending previous results to a broader context.

Findings

Categories of colored PROPs and algebras can be equipped with model structures

Homotopy invariance holds for algebras over cofibrant colored PROPs

Homotopy topological conformal field theories are invariant under homotopy

Abstract

Over a monoidal model category, under some mild assumptions, we equip the categories of colored PROPs and their algebras with projective model category structures. A Boardman-Vogt style homotopy invariance result about algebras over cofibrant colored PROPs is proved. As an example, we define homotopy topological conformal field theories and observe that such structures are homotopy invariant.