On the category CatSet of usual set based categories, there is a "folk" model structure, as described on the first page of Model structures for homotopy of internal categories by T. Everaert, R.W. Kieboom and T. Van der Linden. Namely: in CatSet, ws are weak equivalences, cs are functors injective on objects, fs are functors with the lifting property for isomorphisms. wfs are then precisely the full faithful functors surjective on objects.

Is there's any nice sense in which this model category structure on CatSet is unique?

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    $\begingroup$ Well, it's not unique in the sense of being unique: there are model structures using different classes of weak equivalences (isomorphisms, htpy. equiv. on classifying spaces, ...). Maybe you're asking whether it's the unique one with the same class of weak equivalences? $\endgroup$ Mar 19, 2010 at 13:44
  • $\begingroup$ Could you explain what you mean by "usual set based categories", for those of us too lazy to click a link? $\endgroup$ Mar 19, 2010 at 14:52
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    $\begingroup$ Also: it might be perceived as impolite that you don't type all your words out in full (e.g. "cs" for "cofibrations"). It might seem to suggest that the time it takes you to type those extra letters is more valuable than the time it takes readers to decode your abbreviations. You can fix it by clicking the "edit" button. $\endgroup$ Mar 19, 2010 at 14:55
  • $\begingroup$ By the way, Joyal gave a really good reason why we should call this the "natural" or "canonical" model structure on Cat in a discussion on the nForum. That might be instructive. $\endgroup$ Nov 18, 2010 at 12:52

2 Answers 2


As has been pointed out above, there are many possible model structures on Cat with different weak equivalences. This is the only proper model structure on Cat for which the weak equivalences are the categorical equivalences.

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    $\begingroup$ Dear Steve, as the notion of properness of a model category depends only on the class of weak equivalences (this follows easily from Prop. 2.7 in this paper of Charles Rezk: arXiv:math/0003065), you are in fact claiming that this model structure is uniquely determined by the notion of equivalence of categories. $\endgroup$ Nov 18, 2010 at 14:26
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    $\begingroup$ G'day Denis-Charles! Thanks, that's very interesting, I hadn't known that. $\endgroup$
    – Steve Lack
    Nov 18, 2010 at 22:51
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    $\begingroup$ In fact it's easy to prove directly that there is a unique choice of fibrations and cofibrations for this choice of weak equivalences. You show that if $f:A\to B$ is an equivalence for which every pullback is also an equivalence then $f$ is surjective on objects. You also show that if every pushout of $f$ is an equivalence then $f$ is injective on objects. $\endgroup$
    – Steve Lack
    Nov 28, 2010 at 22:30
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    $\begingroup$ Here is a detailed proof of Steve's claim: sbseminar.wordpress.com/2012/11/16/… $\endgroup$ Nov 16, 2012 at 13:58
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    $\begingroup$ I find this fact that properness is a property of the weak equivalences to be quite surprising! The explicit statement: $M$ is right proper iff for every weak equivalence $w: X \to Y$ the induced functor $Ho(M/X) \to Ho(M/Y)$ is an equivalence. Some related links: the arxiv version of Rezk's paper, the published version (where it's Prop 2.3 instead), and some discussion at the n-category cafe. $\endgroup$
    – Tim Campion
    Jul 5, 2016 at 17:46

I don't know if this is what you are looking for, but, there are certainly many model structures on the category of small categories which do not describe the same homotopy theory. For example, there is the Thomason model structure which has the same homotopy category as topological spaces:



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