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All the standard examples for model categories are large categories. Is it possible to have a small model category? Are there any interesting examples?

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Since a complete small category is a preorder (proposition V.2.3 in MacLane's Categories), I'd be glad to compromise the limit axioms to be as in Quillen's original definition, demanding only finite limits and colimits. In particular, I don't consider a trivial model structure to be interesting.

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  • $\begingroup$ I don't think so, model categories are (co)complete. Do you know many small (co)complete categories? $\endgroup$
    – Fernando Muro
    Commented Oct 3, 2012 at 19:23
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    $\begingroup$ @Muro: all complete lattices are complete categories, so now you do. $\endgroup$
    – Wouter Stekelenburg
    Commented Oct 3, 2012 at 19:28
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    $\begingroup$ In arxiv.org/abs/1209.2699 Section 3.2 it is argued that it is reasonable to allow model categories to be only finitely (co)complete, and indeed this was in Quillen's original definition. $\endgroup$
    – Colin McQuillan
    Commented Oct 3, 2012 at 19:29
  • $\begingroup$ @Wouter, nice point, I wonder whether there's any model category structure there. @Colin, indeed, as you say, Quillen's 1967 original definition of model categories only asks for finite (co)limits, not a new discovery. $\endgroup$
    – Fernando Muro
    Commented Oct 3, 2012 at 19:40
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    $\begingroup$ Alright. Fernando, by proposition V.2.3 (Freyd) in MacLane's Categories, a small complete category is a preorder. Therefore if I want to find a non-trivial example (or one which is not a preorder) then I better relax the axioms to Quillen's original definition, which required only finite (co)limits. Colin, the reference you've given looks neat. Thanks! :-) $\endgroup$
    – Shlomi A
    Commented Oct 3, 2012 at 21:46

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One of Quillen's original examples was the category of chain complexes of finitely-generated modules over a ring – this is obviously equivalent to a small category, and of course, one has to use Quillen's original definition which only required limits and colimits for finite diagrams, rather than the usual definition nowadays which demands limits and colimits for all small diagram.

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Any complete preorder where all isomorphisms are weak equivalences and all morphisms are both fibrations and cofibrations is an example. Reference.

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  • $\begingroup$ Well, perhaps I should have phrased my question differently. Is there any such non-trivial model structure? $\endgroup$
    – Shlomi A
    Commented Oct 3, 2012 at 21:28
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    $\begingroup$ there are preorders with a non-trivial model structure $\endgroup$
    – mmm
    Commented Jul 31, 2013 at 13:24

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