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>


[2] <https://ncatlab.org/nlab/show/model+structure+on+operads>



Is the category of topological operads left proper?
It is not left proper. You can find a counterexample (at least for operads in simplicial sets but you can just apply realization to this counterexample) in section 4 of <https://arxiv.org/abs/1411.4668>.


