# Small model categories?

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.

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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 3 '12 at 19:23
@Muro: all complete lattices are complete categories, so now you do. –  Wouter Stekelenburg Oct 3 '12 at 19:28
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. –  Colin McQuillan Oct 3 '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 3 '12 at 19:40
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 3 '12 at 21:46