Homotopy theory of non-symmetric operads II: change of base category and left properness

TL;DR

This paper establishes that Quillen equivalences between monoidal model categories induce Quillen equivalences between their operad and algebra categories, and demonstrates left properness and homotopy invariance results.

Contribution

It proves that Quillen equivalences transfer to operad and algebra categories and shows left properness, advancing homotopy theory of operads.

Findings

Quillen equivalences induce equivalences between operad categories

Left properness results for operads and their algebras

Homotopy invariance for associative operads

Abstract

We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of algebras over operads. We also show left properness results on model categories of operads and algebras over operads. As an application, we prove homotopy invariance for (unital) associative operads.