Relative left properness of colored operads

TL;DR

This paper investigates the model category structure of colored symmetric operads, demonstrating a relative left properness condition and providing an example showing it is not fully left proper.

Contribution

It establishes a relative left properness property for the model structure on colored operads and clarifies the limitations of full left properness.

Findings

Weak equivalences between $ ext{Sigma}$-cofibrant operads are closed under cobase change.

The model structure on colored symmetric operads is not fully left proper.

Provides an example illustrating the failure of left properness.

Abstract

The category of $\mathfrak{C}$-colored symmetric operads admits a cofibrantly generated model category structure. In this paper, we show that this model structure satisfies a relative left properness condition, i.e., that the class of weak equivalences between $\Sigma$-cofibrant operads is closed under cobase change along cofibrations. We also provide an example of Dwyer which shows that the model structure on $\mathfrak{C}$-colored symmetric operads is not left proper.