Timeline for How should one call and use categories that are not locally small?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 2 at 20:05 | comment | added | Martin Brandenburg | The statement by Marc is not correct. See math.stackexchange.com/questions/3654695/… | |
| Nov 3, 2016 at 21:07 | vote | accept | Mikhail Bondarko | ||
| Oct 29, 2016 at 16:20 | answer | added | ACL | timeline score: 4 | |
| Oct 29, 2016 at 10:25 | answer | added | user337830 | timeline score: 3 | |
| Oct 29, 2016 at 9:41 | comment | added | Mikhail Bondarko | Yes, my "main" results concern locally small categories only. However, for a (locally small) subcategory $A$ of the category of functors from $C$ into $Ab$ it is rather convenient to say that $A$ is an exact subcategory of the "big abelian" category $AddFun(C,Ab)$. | |
| Oct 29, 2016 at 9:33 | history | edited | Mikhail Bondarko | CC BY-SA 3.0 |
added 175 characters in body; edited tags
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| Oct 29, 2016 at 9:25 | comment | added | HeinrichD | In my experience the category of all functors $\mathsf{Ab} \to \mathsf{Ab}$ is not really useful and not well-behaved. This is because many results in category theory, when applied to specific examples, require locally small categories. Instead, we should look at subcategories of functors $\mathsf{Ab} \to \mathsf{Ab}$ which are locally small, for example the accessible functors. Actually, one can do a lot of category theory only with locally small categories, which makes me wonder why we should need anything else. And many authors define categories as locally small per definition. | |
| Oct 28, 2016 at 16:10 | comment | added | Philippe Gaucher | @MikhailBondarko A category does not have to be locally small. The only precaution you have to take is not asserting that a proper class is a set if it is not a set. | |
| Oct 28, 2016 at 15:41 | comment | added | Mikhail Bondarko | Cannot $Fun(C.D)$ have "too many" objects? | |
| Oct 28, 2016 at 13:10 | comment | added | Marc Hoyois | Being locally small is invariant under equivalences of categories. In particular if C is essentially small and D is locally small then Fun(C,D) is locally small. | |
| Oct 28, 2016 at 10:07 | history | asked | Mikhail Bondarko | CC BY-SA 3.0 |