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I added the remark that the canonical model structure on Cat is the model structure obtained by transferring the projective model structure on bisimplicial sets.
That statement sounded very confusing until I looked at the page and realized you were transferring it along Rezk’s classifying diagram functor!
Surely you’ll get the same thing if you transfer the injective/Reedy model structure?
There is a unique model structure on Cat with the weak equivalences the equivalences of categories (at least assuming AC). So if these other variants transfer then they give the same model structure.
Huh. That’s surprising – for instance, Beke has shown that there are different choices of cofibrations in $sSet$ for the usual weak homotopy equivalences in $sSet$. How does one prove uniqueness for $Cat$?
Another nice exercise along these lines is to show (again assuming AC) that the category of sets has precisely nine Quillen model structures, no more, no less.
Chris, that’s just great; thank you! I’ve now mentioned this at canonical model structure on Cat (just something quick for now, with links). I hope a fuller treatment will appear at some point within the nLab (and really this result deserves to be shouted from the rooftops; I had not heard it either).
Very nice result! I suppose this is the ultimate justification for the name ‘canonical’.
I went ahead and did a write-up of the uniqueness result, pretty much exactly Chris’s proof with only a few minor changes.
I added a reference to the earliest reference for the canonical model structure (although only for 1-groupoids) that I am aware of.
@Charles where did you add this? Neither canonical model structure on groupoids or canonical model structure on Cat show recent edits. Is this reference prior to
D.W. Anderson, Fibrations and Geometric Realizations, Bull. Am. Math Soc. 84, 765-786, (1978), 765-786.
as shown at canonical model structure on groupoids?
Oh, apparently I added it to canonical model structure, which seems not to be cross-linked with the pages you mention.
Thanks for the alert. I have added the missing cross-links.
Looking at the entry made me move all the references to the bottom of the entry and edit a bit to make it more standard. (new References section).
Also, I notice that the pointer to Anderson has been there are model structure on Grpd all along. Everbody who cares about the quality of the nLab: please take this incident as further incentive to be mindful about adding sufficient cross-links.
The link in
Along similar lines, one can prove (assuming AC) that there are nine – count ’em, nine – Quillen model structures on $Set$.
is broken.
That page is available via Wayback
but its math symbols are broken.
However Omar Antolín Camarena is now in Mexico and hosts the page there so I swapped in the link
I believe that’s a UK/US thing, Tim. They do insist on speaking differently from/than us.
There are occasions even in English where ’different’ does take ’than’, but the one I changed did not seem to be one of them. (I loved the announcement over the intercom on a US flight that ‘the plane will momentarily be landing’! Very confusing for an English speaker. Perfectly clear for an Unitedstatesian speaker. (I do not know the correct translation of the French term ’étatsunian’. ) There should be a good term for the language spoken in the US, and which is closely related to English. :-)
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