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When can I localise a model category by a set(or class) of morphisms, and how do I describe the localised model category ?

By 'localise' I mean to find a localisation functor $q_S : M \rightarrow M_S$ with the usual universality property: that is, $q_S(s)$ is a weak equivalence for any $s\in S$, and any functor $q:M \rightarrow M'$ with this property factors through $q_S$. Importantly, $M_S$ and $M$ may have different underlying categories and so it is not necessarily a Bousfield localisation.

The model category (defined here) I am interested in, is very degenerate: its underlying category is a partially ordered set (or class).

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  • $\begingroup$ If I'm not mistaken, it's essentially a sort of set-theoretic condition on the class of morphisms we want to localize at? That is, we basically take (homotopy) colimits over acyclics w.r.t. to our class W of morphisms, and if that class is generated under homotopy colimits by a set, then our localization exists. $\endgroup$
    – Jonathan Beardsley
    Commented Feb 10, 2012 at 21:40
  • $\begingroup$ I believe your universal property will force the underlying categories to be equal. I mean, you could take $M'$ to be the model structure on $M$ where the weak equivalences are the isomorphisms and all maps are both fibrations and cofibrations. $\endgroup$
    – David White
    Commented Nov 10, 2012 at 0:22
  • $\begingroup$ If my comment above is correct, then the Bousfield localization (if it exists) will be the localization in your sense, by its well-known universal property. However, your type should always exist (because the class of $M'$ is nonempty, so you can just take the one "closest" to $M$) even if Bousfield localization does not. But because you haven't required the cofibrations in $M$ and $M_S$ to match up, I'm not sure if yours is good for anything. Still, +1 since it got me thinking. $\endgroup$
    – David White
    Commented Nov 10, 2012 at 0:28
  • $\begingroup$ Quite the opposite, I want cofibrations to change. The point is, there is a sequence of related model category structures (with more and more cofibrations), and I wanted to see whether it is a sequence of localisations $\endgroup$
    – mmm
    Commented Jul 31, 2013 at 10:35
  • $\begingroup$ thanks for your responses. I just saw them now, (and do not get email notifications), and not sure if you'll ever see my response... $\endgroup$
    – mmm
    Commented Jul 31, 2013 at 10:36

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