Timeline for left integration of functor in the category of groups
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Sep 10, 2023 at 18:52 | comment | added | Yemon Choi | @MarkGrant Is it clear that "left derived functors" and "half-exact functors" still work in this setting? BTW I think this is more like Bourne et al ("protomodular" etc etc) than Barr-Beck but I am very rusty so I may be confused. | |
| Sep 10, 2023 at 18:50 | comment | added | Yemon Choi | If you want an answer to this question, could you clarify what you mean by "left derived functor" and "half-exact functor" for a category that is not additive? There might be versions of these concepts that work for more general categories, but I think the person asking the question should make it clear what versions they have in mind, to reduce confusion for readers. | |
| Sep 1, 2023 at 19:10 | comment | added | Ali Taghavi | @MarkGrant so I realize that maybe the abelian case is not known | |
| Sep 1, 2023 at 19:08 | comment | added | Ali Taghavi | @MarkGrant I meant $G\mapsto G'=[G,G]$ yes it does not vanish on free group. Now I see why did you pointed out to abelian group. thanks again for your attention to my question. | |
| Sep 1, 2023 at 17:28 | comment | added | Mark Grant | @AliTaghavi: I see. I'm not sure what you mean by the commutator functor. If you mean the abelianisation, or the derived (commutator) subgroup, then neither of these vanish on free groups. I don't know the answer to your question for abelian groups, that's why I asked if you did. | |
| Sep 1, 2023 at 13:24 | comment | added | Ali Taghavi | @MarkGrant I did not see your two previous comment. the reason I considered the category of non abelian group is thst I was initially interested in commutator functor and wss curiious if it is derived functor. so i considered non abelian category. Any way what is the answer for abelian group? | |
| Sep 1, 2023 at 7:13 | comment | added | Bugs Bunny | "I would appreciate if you read my previous two comments." I did and see what happened. I still have not understood your question -- the right strategy is to give an answer then :-)) | |
| Aug 31, 2023 at 22:16 | comment | added | Ali Taghavi | @MarkGrant thank you very much for your very helpful comment | |
| Aug 31, 2023 at 19:44 | comment | added | Mark Grant | By the way, there exists a subject of non-abelian homological algebra (see work of Barr-Beck, Inassaridze and others). The projective objects in the category of groups are precisely the free groups. | |
| Aug 31, 2023 at 19:43 | comment | added | Mark Grant | @AliTaghavi: My comment was really about the motivation for the question. If you don't know this to be true in abelian categories, why would you suspect it to be true in the (non-abelian) category of groups? | |
| Aug 31, 2023 at 19:17 | comment | added | Ali Taghavi | @BugsBunny I would appreciate if you read my previous two comments. | |
| Aug 31, 2023 at 19:16 | comment | added | Ali Taghavi | @YCor what is the difference between endofunctor and a functor from a category to itself? You are right. I was mistaken touse the terminilogy for non abelian category. But what are obstructions to have the derived functors in non abelian categories? In what step of construction of derived functor we need "abelian"-ness? | |
| Aug 31, 2023 at 19:12 | comment | added | Ali Taghavi | @MarkGrant I was not aware of this in abelian category. Is the proof obvious in this case? | |
| Aug 31, 2023 at 15:47 | review | Close votes | |||
| Sep 15, 2023 at 3:08 | |||||
| Aug 31, 2023 at 15:28 | comment | added | Bugs Bunny | What is the model structure on the category of groups? Or is it just abelian groups?? | |
| Aug 31, 2023 at 9:15 | comment | added | YCor | do you mean an endofunctor? or you mean "to itself"? In any case, are there derived functors in such a non-additive setting? | |
| Aug 31, 2023 at 9:11 | comment | added | Mark Grant | Is this true for functors out of abelian categories? I wasn't aware. | |
| Aug 31, 2023 at 5:43 | history | asked | Ali Taghavi | CC BY-SA 4.0 |