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,



[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

  • 2
    $\begingroup$ 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 $\endgroup$ Apr 27 at 20:00

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.


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