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.



Small model categories?
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.


