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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?

EDIT:

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.

Shlomi A
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  • I don't think so, model categories are (co)complete. Do you know many small (co)complete categories? – Fernando Muro Oct 03 '12 at 19:23
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    @Muro: all complete lattices are complete categories, so now you do. – Wouter Stekelenburg Oct 03 '12 at 19:28
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    In http://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. – Colin McQuillan Oct 03 '12 at 19:29
  • @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. – Fernando Muro Oct 03 '12 at 19:40
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    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! :-)

     – Shlomi A Oct 03 '12 at 21:46
  • Maybe you'd better edit your question then, Shlomi A. Of course, much of the literature these days uses all small limits and colimits, so you might consider whether Quillen's original definition is going to be the one you really want to work with. – Todd Trimble Oct 03 '12 at 21:54
  • Yes, I'm aware of it Todd. I've also just edited the question. Say, might there be some small sub-category of simplicial sets, for instance, that satisfies Quillen's original definition for a model category? – Shlomi A Oct 03 '12 at 22:08

2 Answers2

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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.

Zhen Lin
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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.

Todd Trimble
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