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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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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? –  Charles Rezk Mar 19 '10 at 13:44
    
Could you explain what you mean by "usual set based categories", for those of us too lazy to click a link? –  Tom Leinster Mar 19 '10 at 14:52
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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. –  Tom Leinster Mar 19 '10 at 14:55
    
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. –  Harry Gindi Nov 18 '10 at 12:52

2 Answers 2

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:

http://ncatlab.org/nlab/show/Thomason+model+structure

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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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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. –  Denis-Charles Cisinski Nov 18 '10 at 14:26
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G'day Denis-Charles! Thanks, that's very interesting, I hadn't known that. –  Steve Lack Nov 18 '10 at 22:51
    
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. –  Steve Lack Nov 28 '10 at 22:30
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Here is a detailed proof of Steve's claim: sbseminar.wordpress.com/2012/11/16/… –  Chris Schommer-Pries Nov 16 '12 at 13:58

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