Admissibility and rectification of colored symmetric operads

TL;DR

This paper provides broad conditions for the admissibility and rectification of colored symmetric operads in symmetric monoidal model categories, enabling transfer and equivalence of algebraic structures across different contexts.

Contribution

It introduces a flexible criterion for admissibility of colored symmetric operads and a necessary and sufficient condition for rectification of weak equivalences, extending the applicability of operad theory.

Findings

All colored symmetric operads in certain model categories are admissible.

Weak equivalences of admissible operads can be rectified under specific conditions.

Quillen equivalences of base categories induce equivalences of algebra categories.

Abstract

We establish a highly flexible condition that guarantees that all colored symmetric operads in a symmetric monoidal model category are admissible, i.e., the category of algebras over any operad admits a model structure transferred from the original model category. We also give a necessary and sufficient criterion that ensures that a given weak equivalence of admissible operads admits rectification, i.e., the corresponding Quillen adjunction between the categories of algebras is a Quillen equivalence. In addition, we show that Quillen equivalences of underlying symmetric monoidal model categories yield Quillen equivalences of model categories of algebras over operads. Applications of these results include enriched categories, colored operads, prefactorization algebras, and commutative symmetric ring spectra.