Timeline for Adjoints of exact functors between semisimple abelian categories
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 31, 2021 at 15:21 | vote | accept | Tim Montegue | ||
| Oct 31, 2021 at 15:01 | comment | added | Tim Montegue | I have added some comments above to clear up the quastions addressed here. | |
| Oct 31, 2021 at 14:54 | history | edited | Tim Montegue | CC BY-SA 4.0 |
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| Oct 31, 2021 at 14:53 | answer | added | rvk | timeline score: 4 | |
| Oct 31, 2021 at 14:33 | comment | added | rvk | Apropos to Jeremy Rickard's comment (which is very relevant IMO). I am implicitly assuming in my earlier comment that the category is as "nice as can be" - enriched over vector spaces, Schur's lemma, etc. However, it is worthwhile pinning this down, because there are situations where subtleties arise. | |
| Oct 31, 2021 at 14:29 | comment | added | rvk | You are going to run into trouble with existence if the category consists of "finite dimensional"/finite length objects in any sense. Eg. semisimple Lie algebra, finite dim reps (over $\mathbb{C}$). Consider the forgetful functor to vector spaces. If you allow "infinite" stuff and have arbitrary limits and the like, then a general adjoint functor theorem should come into play. In any finite length situation, if the adjoint exists, then it is probably close to an equivalence or embedding (if it doesn't kill an object, the adjunction maps on simples are injective automatically). Induction. | |
| Oct 31, 2021 at 14:21 | comment | added | Jeremy Rickard | Could you say what you mean by a “semisimple abelian category”? I know of three (contradictory) definitions. | |
| Oct 31, 2021 at 14:05 | comment | added | Maxime Ramzi | To apply an adjoint functor theorem you'll need more hypotheses | |
| Oct 31, 2021 at 13:43 | comment | added | Tim Montegue | But I guess the existence should follow from some version of the adjoint functor theorem. | |
| Oct 31, 2021 at 13:02 | comment | added | Tim Montegue | I am asking about for existence of the adjoint as part of the question. | |
| Oct 31, 2021 at 12:56 | comment | added | rvk | Is your functor already part of an adjoint pair, or are you asking for existence of adjoint as part of the question? | |
| Oct 31, 2021 at 12:35 | history | asked | Tim Montegue | CC BY-SA 4.0 |