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I have given Grothendieck construction for model categories its own entry, in order to have a place for recording references. In particular I added pointer to the original references (Roig 94, Stanculescu 12)
(There used to be two places in the entry Grothendieck construction where an attempt was made to list the literature on the model category version, but they didn’t coincide and were both inclomplete. So I have replaced them with pointers to the new entry.)
The following ought to be true, I have added a first version of a proof here (labeled “tentative”, still need to polish and double-check):
Let $\mathcal{C}$ be a combinatorial simplicial model category in which all objects are cofibrant. Then then pseudofunctor on the model category of simplicial groupoids which sends a simplicial groupoid $\mathcal{X}$ to the projective model structure on simplicial functors from $\mathcal{X}$ to $\mathcal{C}$
$\array{ sGrpd &\longrightarrow& ModCat \\ \mathcal{X} &\mapsto& sFunc(\mathcal{X},\,\mathcal{C})_{proj} \\ \Big\downarrow\mathrlap{^f} && \Big\downarrow\mathrlap{^f_!} \\ \mathcal{Y} &\mapsto& sFunc(\mathcal{Y},\,\mathcal{C})_{proj} }$is relative and proper, hence the integral model structure on the category of parameterized $\mathcal{C}$ objects
$\mathcal{C}_{sGrpd} \;\;\coloneqq\;\; \underset{\mathcal{X} \in sGrpd}{\int} sFunc(\mathcal{X},\,\mathcal{C})_{proj}$exists.
I have fine-tuned the proof further (here), and generalized the setup and statement.
Even though the cofibrancy assumption is pretty strong, the statement does apply (as far as its proof is correct) to three pleasant examples (now listed here).
I will later polish the write-up further, but need to go offline now for a while.
I had scratched the previous text under “Examples” after all, and am starting afresh now (here), for the moment just with two elementary examples (here):
(1.) model structures on indexed sets of objects in a given model category, and
(2.) the specialization of this example to the case of skeletal simplicial groupoids.
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