Bousfield Localization and Algebras over Colored Operads

TL;DR

This paper develops a general framework for establishing model structures on algebras over colored operads, with minimal assumptions on the underlying model category, and studies how Bousfield localization affects these algebraic structures.

Contribution

It introduces a unified approach to constructing (semi-)model structures for colored operads under various cofibrancy conditions, expanding the applicability of algebraic localization techniques.

Findings

Established conditions for model structures on colored operad algebras.

Provided criteria for Bousfield localization to preserve these algebras.

Unified framework applicable to multiple classes of colored operads.

Abstract

We provide a very general approach to placing model structures and semi-model structures on algebras over symmetric colored operads. Our results require minimal hypotheses on the underlying model category $\mathcal{M}$, and these hypotheses vary depending on what is known about the colored operads in question. We obtain results for the classes of colored operad which are cofibrant as a symmetric collection, entrywise cofibrant, or arbitrary. As the hypothesis on the operad is weakened, the hypotheses on $\mathcal{M}$ must be strengthened. Via a careful development of the categorical algebra of colored operads we provide a unified framework which allows us to build (semi-)model structures for all three of these classes of colored operads. We then apply these results to provide conditions on $\mathcal{M}$, on the colored operad $\mathcal{O}$, and on a class $\mathcal{C}$ of morphisms in $\mathcal{M}$ so that the left Bousfield localization of $\mathcal{M}$ with respect to $\mathcal{C}$ preserves $\mathcal{O}$-algebras.