Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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


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.

share|improve this question
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
show 2 more comments

2 Answers

up vote 5 down vote accepted

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.

share|improve this answer
add comment

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

share|improve this answer
Well, perhaps I should have phrased my question differently. Is there any such non-trivial model structure? –  Shlomi A Oct 3 '12 at 21:28
there are preorders with a non-trivial model structure –  mmm Jul 31 '13 at 13:24
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.