Timeline for Is $G\mapsto \operatorname{Hol}(G)$ the object component of any functor on the category of groups?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Oct 28, 2020 at 21:02 | vote | accept | Ali Taghavi | ||
| Oct 28, 2020 at 6:02 | history | edited | YCor | CC BY-SA 4.0 |
formatting
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| Oct 28, 2020 at 5:37 | answer | added | Qiaochu Yuan | timeline score: 9 | |
| Oct 28, 2020 at 4:55 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
edited title
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| Oct 28, 2020 at 4:54 | comment | added | Ali Taghavi | @DavidRoberts based on what you suggest, I revise the title of my question.thank you! | |
| Oct 28, 2020 at 4:50 | comment | added | Ali Taghavi | @DavidRoberts Thanks! BTW i should say Z(G) is not a component of a functor I was mistaken I wrote G' | |
| Oct 28, 2020 at 4:37 | comment | added | David Roberts♦ | "Usually this is done via canonical maps(in a natural manner)" Hmm, I don't think so. "Canonical" is an informal word with no good definition that I've seen. Naturality here is a red herring, I agree. I'll leave it to the group theorists to answer your question. It is clear what you want, I just suspect the answer is no. | |
| Oct 28, 2020 at 4:35 | comment | added | Ali Taghavi | @DavidRoberts every functor should maps a morphism to a morphism. Usually this is done via canonical maps(in a natural manner). But on our qurstion we do not require any naturality. | |
| Oct 28, 2020 at 4:29 | comment | added | David Roberts♦ | Then I don't understand what this means: "In the question we do not require that the morphism between holomorphs would be canonically assigne to a morphisms from G to H". What is (the arrow component of) a functor but some such assignment? | |
| Oct 28, 2020 at 4:27 | comment | added | Ali Taghavi | @DavidRoberts but is not the phrase of my question clear, equivalent to the same terminology you said? | |
| Oct 28, 2020 at 4:26 | comment | added | Ali Taghavi | @DavidRoberts morphisms are homomorphism . As you said we ask if "holomorph" the object component of a functor. Apart from your example G \to G' has the same problem,i think. Thanks for giving a better statement of my question,that is object component of a functor. | |
| Oct 28, 2020 at 4:19 | comment | added | David Roberts♦ | If you don't require that, what do you mean by a functor? What are the morphisms in your category of groups if not homomorphisms? I think more details are needed if you aren't taking the standard definitions (though I assume you really do want them). But my comment is not on firm ground, it just poses one potential problem: it is definitely true that $G\mapsto Aut(G)$ is not the object component of a functor out of the (standard) category of groups. I'm just cautious about a claim the holomorph construction will fix the issues with it. | |
| Oct 28, 2020 at 4:16 | comment | added | Ali Taghavi | @DavidRoberts In the question we do not require that the morphism between holomorphs would be canonically assigne to a morphisms from G to H. May you more explain on your comment? | |
| Oct 28, 2020 at 4:09 | comment | added | David Roberts♦ | This would induce homomorphisms $Aut(G) \to Aut(H)$ from homomorphisms $G\to H$, no? I don't think this is possible. | |
| Oct 28, 2020 at 4:05 | history | asked | Ali Taghavi | CC BY-SA 4.0 |