Timeline for Magic behind idempotent-complete categories a.k.a. why (sometimes) be Karoubian is sexier than be Abelian
Current License: CC BY-SA 4.0
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| Jan 18, 2021 at 15:26 | comment | added | Todd Trimble | @PaulTaylor Yes, but it seems OP is working in a Ab-enriched context. It's additivity (existence of direct sums) plus splitting of idempotents which is operative here: the absolute colimit completion. | |
| Jan 18, 2021 at 13:59 | comment | added | Paul Taylor | Splitting idempotents has nothing at all to do with additivity or Abelianness. It's an easy construction that can be applied to any category at all. It's used in plenty of other subjects. For example it gives continuous lattices from algebraic ones and Scott-continuous maps. | |
| Jan 18, 2021 at 13:42 | answer | added | Jeroen van der Meer | timeline score: 2 | |
| Jan 6, 2021 at 13:05 | comment | added | Donu Arapura | @GhostinGrothendieckuniverse It's not a question of moral/philosophical principles, it's about what can be proved. If we stick to pure motives modulo homological equivalence, then Grothendieck conjectured this is abelian. However, I don't think anyone knows how to prove it. If we switch to motives mod numerical equivalence, then Jannsen did prove it's abelian. | |
| Jan 6, 2021 at 1:10 | history | became hot network question | |||
| Jan 6, 2021 at 0:53 | history | edited | user267839 | CC BY-SA 4.0 |
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| Jan 6, 2021 at 0:48 | comment | added | user267839 | @DonuArapura: So the philosophy here with Karoubian completion is indeed: "Be satisfied with what you get", right? In case of pure motives we obtain in a relatively easy way our Karoubian category after completion which already have a lot of nice properties which an Abelian category would have, but there is just no canonical way known to extend it to an Abelian category, so that's a 'stay happy with what we have' philosophy? Then that's the whole moral? | |
| Jan 5, 2021 at 21:22 | answer | added | Qiaochu Yuan | timeline score: 22 | |
| Jan 5, 2021 at 18:05 | comment | added | Donu Arapura | Karoubian categories are more abundant than abelian ones, but I don't know if that makes them more interesting or in any sense preferred. By the way, although the category of Grothendieck motives is contructed as a Karoubian completion, one really wants it to be abelian... | |
| Jan 5, 2021 at 17:38 | comment | added | Mohan | May be the basic example is the category of vector bundles which is Karoubian, but not abelian? | |
| Jan 5, 2021 at 17:13 | history | edited | David C | CC BY-SA 4.0 |
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| Jan 5, 2021 at 17:09 | history | edited | user267839 | CC BY-SA 4.0 |
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| Jan 5, 2021 at 16:51 | history | edited | user267839 | CC BY-SA 4.0 |
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| Jan 5, 2021 at 16:46 | history | asked | user267839 | CC BY-SA 4.0 |