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Suppose we have a functor between two model categories $F\colon \mathcal{C}\to \mathcal{D}$. Suppose we don't know that it is a part of a Quillen pair. Nevertheless, it can still happen that the derived functor $\mathbb{L}F\colon \mathrm{Ho}(\mathcal{C})\to \mathrm{Ho}(\mathcal{D})$ exists. Then we apriori don't know that it can be computed by the formula $$\mathbb{L}F(A)=\gamma F(R)\phantom{aaaa}(*)$$ for some cofibrant replacement $R\to A$ of $A$ (here $\gamma$ denotes the localization funtor).

Can we somehow extend (or maybe we should say "complete") the categories $\mathcal{C}$ and $\mathcal{D}$ to $\hat{\mathcal{C}}$ and $\hat{\mathcal{D}}$, and extend the functor $F$ to $\hat{F}$ so that there will exist $\mathbb{L}\hat{F}$ which will restrict to $\mathbb{L}F$, and such that $\hat{F}$ can be computed by the formula $(*)$?

It feels like there should exist something analogous to the case of abelian categories. There, if we don't have enough resolutions, we can always complete our categories and extend the functor in the desired way (like extending category of coherent sheaves to quasi-coherent sheaves in order to have injective resolutions).

Or maybe we can extend $\mathcal{C},\mathcal{D},F$ in such a way that $\hat{F}$ becomes a part of a Quillen pair? Is there any result like that?

Thank you very much for your help!

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Are you assuming at least that $F$ has a right adjoint $G$? –  David White Aug 12 at 3:03
    
@DavidWhite No, I don't think so... At least for the first question. For the second question I guess we need to assume something about $F$. For example, it should commute with colimits or limits (if we want it to be left or right Quillen). Maybe we can assume it already has an adjoint if it is necessary. –  Sasha Patotski Aug 12 at 3:09
    
Adding new resolutions seems like a terrible idea. As far as I know, the only reason why model structures do actually compute the classical derived functors of homological algebra is because there are enough resolutions already. –  Zhen Lin Aug 12 at 8:02
    
@ZhenLin I am not asking for minimal such extensions. Maybe one can embed the model categories $\mathcal{C}$ and $\mathcal{D}$ into something much bigger... –  Sasha Patotski Aug 12 at 21:24
    
Well, one can embed model categories satisfying certain smallness conditions into combinatorial model categories, but this procedure can't really be used to turn an arbitrary functor into a left Quillen functor. At minimum the original functor must preserve finite colimits, cofibrations, and trivial cofibrations. –  Zhen Lin Aug 12 at 22:46

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