Relative left properness of colored operads

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, ie 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.