Timeline for Model category structure on Set without axiom of choice
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 1, 2010 at 23:00 | comment | added | Reid Barton | Oops, I got something backwards, and you're absolutely right. | |
| Jan 1, 2010 at 17:17 | comment | added | Mike Shulman | I would call the (mono, split epi) model structure on Set the injective one and this one requiring COSHEP the projective one, since the fibrant objects of the first are the injective objects in Set, while the cofibrant objects of the second are the projective objects in Set. Actually, on a general topos there can be three model structures: the "projective" one (relative-projective, epi), which exists iff there are enough projectives, the "injective" one (mono, relative-injective) which always exists, and another one (complemented-mono, split-epi) which also always exists. | |
| Jan 1, 2010 at 5:06 | comment | added | Mike Shulman | In Makkai's foundational paper on anafunctors ("Avoiding the axiom of choice in general category theory"), he considers the question of whether the category of anafunctors between two small categories is essentially small. He proves that it is if you assume a weak form of AC called the "small cardinality selection axiom," which in turn follows from Blass' axiom of "small violations of choice." | |
| Dec 31, 2009 at 23:19 | comment | added | Reid Barton | I agree with your conclusion, that we cannot expect model categories to be as effective in a world without any form of AC. I wonder whether there is some other machinery to prove that the category obtained by formally inverting weak equivalences is locally small, to compute its Hom-sets, etc. | |
| Dec 31, 2009 at 23:17 | comment | added | Reid Barton | Well, there is always the "projective" model structure on Set whose cofibrations are all monomorphisms and whose acyclic fibrations are split surjections. Under COSHEP, we can also form the "injective" model structure, the one we're talking about here, and the two are Quillen equivalent; under AC, they're identical. Without AC there is a "projective" folk model structure on Cat also, where the fibrations have an additional splitting requirement. But it seems to have nothing to do with anafunctors. | |
| Dec 31, 2009 at 23:06 | vote | accept | Reid Barton | ||
| Dec 31, 2009 at 19:01 | history | edited | Mike Shulman | CC BY-SA 2.5 |
added reference to Set
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| Dec 31, 2009 at 18:51 | history | answered | Mike Shulman | CC BY-SA 2.5 |