# Is the category of topological operads left proper?

I just learned that there is a model structure on the category $$Op_{Top}$$ of topological operads, due to Berger-Moerdjik [1], obtained by right transfer of the Quillen model structure on $$Top$$.

Since $$Top$$ is a proper model category, I was wondering if $$Op_{Top}$$ is proper aswell. It should be right proper since all its objects are fibrant, but what about left properness?

Thank you for any help,

Tommaso

References:

[1] Axiomatic homotopy theory for operads, Clemens Berger and Ieke Moerdijk https://arxiv.org/pdf/math/0206094.pdf

• In addition to Geoffroy's answer, there is the possibility of partial repair: the property that, for any $\Sigma$-cofibrant operad $P$, the slice category of operads over $P$ is proper. This is means that a proper replacement of the model structure of operads is the category of operads equipped with a map to $E_\infty$, where $E_\infty$ is a fixed $\Sigma$-cofibrant replacement of $Comm$. See Theorem 8.7 and its corollaries in arXiv:1109.1004 Apr 27 at 20:00