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I work with the category of $\Delta$-generated spaces. I call reparametrization category a small strict semimonoidal topologically enriched category $(\mathcal{P},\otimes)$ such that $\mathcal{P}(\ell,\ell')$ is contractible for all objects $\ell,\ell'$ and such that for all maps $\phi:\ell\to \ell'$, and all $\ell'_1,\ell'_2$ such that $\ell'_1\otimes \ell'_2=\ell'$, there exist $\phi_1:\ell_1\to \ell'_1$ and $\phi_2:\ell_2\to \ell'_2$ such that $\phi=\phi_1\otimes \phi_2$.

I work with three reparametrization categories: the terminal category, and $\mathcal{G}$ and $\mathcal{M}$ defined as follows. The semigroup of objects is $(]0,+\infty[,+)$, $\mathcal{G}(\ell,\ell')$ is the space of nondecreasing homeomorphisms from $[0,\ell]$ to $[0,\ell']$ and $\mathcal{M}(\ell,\ell')$ is the space of nondecreasing surjective maps from $[0,\ell]$ to $[0,\ell']$.

I wonder to what extend $\mathcal{G}$ and $\mathcal{M}$ could be considered as cofibrant replacements of the terminal category. Hence the following question:

I wonder what kind of model structure could be put on the category of small strict semimonoidal topologically enriched categories. I am a bit lost in the literature about this kind of subject.

This notion is introduced in Homotopy theory of Moore flows (I) (Definition 4.3).

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  • $\begingroup$ Semi-monoidal means justs an associative tensor products and no units? $\endgroup$
    – Simon Henry
    Commented Sep 4, 2023 at 15:44
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    $\begingroup$ I suspect there are some interesting model structure (or maybe semi-model structure), notably at least one that capture the Dwyer-Kan equivalence, but I'm not aware of any literature on this, and I don't see any quick way to obtain it. Meaning some, probably not difficult but tedious, work need to be done to build it. The only kind of shortcut I can think of is to see if the category you are looking at is enriched over Space and if so see if it can be exploited through some flavor of Cisinski-Olschok theory... $\endgroup$
    – Simon Henry
    Commented Sep 4, 2023 at 20:25
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    $\begingroup$ If that doesn't pan out, my second approach would be to try to adapt the methods from arxiv.org/abs/1201.1575 which deals with the non-monoidal case. $\endgroup$
    – Simon Henry
    Commented Sep 4, 2023 at 20:27
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    $\begingroup$ I agree not every objects should be cofibrant . But that doesn't have to be the case. In arxiv.org/abs/1201.1575 they chose to take an "injective" model structure in the case the set of objects is fixed (that is cofibration are the cofibration in V) but I believe they could have taken instead a projective one (where the fibrations are fibrations in V) and then cofibrant objects are "free on graphs" in a cellular sense. And in any case in the monoidal case I expect cofibrant objects should have free monoids of objects. $\endgroup$
    – Simon Henry
    Commented Sep 5, 2023 at 12:42
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    $\begingroup$ In the case of Cisinski-Olschok theory, the requirement that every object is cofibrant fron Olschok paper can just be dropped if you are willing to work with semi-model categories (See section 6 of arxiv.org/pdf/2005.02360.pdf ) , so with this kind of methods you can chose pretty much the cofibrations you want. Though in this case I feel like this approach only works when the set of objects is fixed and one still needs to go through a similar argument to allow the set of objects to vary... $\endgroup$
    – Simon Henry
    Commented Sep 5, 2023 at 12:48

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