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Gonçalo Tabuada has shown that there is a Quillen model category structure on the category of small dg-categories, i.e. the category of small categories enriched over chain complexes (for a fixed commutative ring k). Julia Bergner has shown that the category of simplicial categories, i.e. categories enriched over simplicial sets, also has a model category structure.

The above two results are connected by the fact that chain complexes and simplicial sets are both monoidal model categories (in the sense of Hovey Chapter 4). Toen in the introduction to his paper "Homotopy theory of dg-categories and derived Morita theory" mentions that work has been done by Tapia to put a model structure on categories enhanced over very general monoidal model categories. Unfortunately there seem to be no mentions of this work since.

I'm very interested in the following case of the above statement. Let $X$ be a nice scheme, say noetherian, separated, and finite dimensional. Even assuming regular would be ok. The category $\mathcal{C}$ of chain complexes of quasi-coherent $X$-modules is a monoidal model category by Gillespie; see Theorem 6.7 of the paper http://arxiv.org/abs/math/0607769.

Question: Does anyone know of a reference, or general machinery, that shows that there is a model structure on small categories enriched over $\mathcal{C}$ in which the weak equivalences are the functors that induce an equivalence after taking homology? I suppose one could go through and try to mimic Tabuada's proof, but that would not be ideal.

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In case you haven't seen it, Gillespie has a preprint on his webpage which gets a flat model category structure on $\mathcal{C}$. I don't know enough about the subject to know if this will help you, but I figured it couldn't hurt. Here's a link: phobos.ramapo.edu/~jgillesp/updated%20qcsheaf.pdf –  David White Jun 27 '11 at 20:18

2 Answers 2

up vote 3 down vote accepted

There were two preprints posted on the arXiv this week that seem to answer the question. They are Dwyer-Kan homotopy theory of enriched categories by Fernando Muro and On the homotopy theory of enriched categories by Clemens Berger and Ieke Moerdijk.

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I was delighted to see that both of these papers appeared on the arXiv in the same week. I was on my way to find this question to propose that one of them might just contain the answer! –  Clark Barwick Jan 12 '12 at 13:54

Lurie has a general construction (HTT A.3.2.4) of a model structure on $S$-enriched categories when $S$ is a combinatorial monoidal model category which is suitably nice (every object has to be cofibrant, and weak equivalences have to be stable under filtered colimits). This is a direct generalization of the Bergner model structure on simplicial categories and (I believe) the model structure on DG-categories when $S$ is the monoidal model category of either simplicial sets or chain complexes.

Namely, you can define a morphism of $S$-enriched categories to be a weak equivalence if when you pass to the homotopy category (so get a morphism of $ho(S)$-enriched categories), you get an equivalence of enriched categories. Unlike Bergner, though, Lurie seems to construct the model structure by building cofibrations and weak equivalences, and shows that in special cases of $S$ the fibrations coincide with what you want (basically, something that induces fibrations on the hom-spaces $\hom(x, y) \to \hom(Fx, Fy)$ and has some "quasi-fibration" categorical property).

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Thanks! This is exactly what I had hoped to find. –  Jesse Burke Sep 23 '11 at 7:32
    
In the model category structure I cited above, not every object is cofibrant. I also don't know of any other monoidal model category structures on chain complexes of qc sheaves. So unfortunately the result in HTT doesn't seem to apply and I don't see an obvious workaround. So for these reasons, I'm going to un-accept the answer (I should have checked closer before accepting it). Thank you again though, for pointing out this very nice result. –  Jesse Burke Sep 25 '11 at 11:18

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