Timeline for Cofibrant replacements of a given object in a combinatorial model category
Current License: CC BY-SA 3.0
9 events
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| Feb 4, 2020 at 20:25 | comment | added | Tim Campion | @PhilippeGaucher I don't believe the class of cofibrations, or even cofibrant objects, is always accessible -- or at least if it is, some set-theoretical hypothesis is needed. For, following Rosicky, there is a combinatorial model structure on $Ab$ with cofibrations generated by $0 \to \mathbb Z$ and weak equivalences the isomorphisms. The cofibrant objects are the free abelian groups. But under $V=L$, the free abelian groups are not closed under $\lambda$-filtered colimits for any $\lambda$, and in particular they aren't accessible. | |
| May 10, 2013 at 10:08 | answer | added | Philippe Gaucher | timeline score: 3 | |
| Apr 30, 2013 at 9:02 | history | edited | arsmath |
Add model categories tag.
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| Apr 30, 2013 at 4:10 | answer | added | Dylan Wilson | timeline score: 0 | |
| Apr 29, 2013 at 15:09 | comment | added | Dylan Wilson | I think the answer is yes: you should be able to describe the category of cofibrant replacements as a homotopy limit of categories you already know are accessible... If I have time I'll try to think of the exact limit you want. (Probably you need that cofibrant objects form an accessible subcategory, and the category of weak equivalences are accessible, and undercategories of accessible categories are accessible, etc.) | |
| Apr 29, 2013 at 10:39 | comment | added | Philippe Gaucher | I added the reason in my question. | |
| Apr 29, 2013 at 10:38 | history | edited | Philippe Gaucher | CC BY-SA 3.0 |
added 475 characters in body
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| Apr 29, 2013 at 9:12 | comment | added | Fernando Muro | I don't see why such a colimit of cofibrant objects should be cofibrant, although I have the feeling that it is true. | |
| Apr 29, 2013 at 9:00 | history | asked | Philippe Gaucher | CC BY-SA 3.0 |