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Added the definitions of Karoubian category and Karoubi envelope that appear in (an exercise in) SGA 4.
A stupid question: why do they call that difference kernel the image of p? In what sense is it the image?
The equaliser of an idempotent and the identity is (as an object) also the coequaliser of the same pair. So you can think of it either as a quotient or as a subobject!
But doesn’t this article duplicate what is at Cauchy complete category and Karoubi envelope?
The equaliser of an idempotent and the identity is (as an object) also the coequaliser of the same pair.
This I do understand; if one can think of Ker(id, p) as the kernel of p then one can also see it as the image, since it is canonically isomorphic to Coker(id, p). What i don't understand is in what sense Ker(id, p) is the kernel of p? (Why is a Karoubi category one where all idempotents "have a kernel"?) I thought the kernel is the object represented by the functor Y -> Ker(Hom(Y,X) -> Hom(Y,X)), or the equaliser Ker(p, 0) when C admits zero morphisms.
But doesn’t this article duplicate what is at Cauchy complete category and Karoubi envelope?
Hm... it's strange that there weren't even any links between these two pages. I suppose we will have to do some reorganization.
This I do understand; if one can think of Ker(id, p) as the kernel of p then one can also see it as the image, since it is canonically isomorphic to Coker(id, p). What i don’t understand is in what sense Ker(id, p) is the kernel of p? (Why is a Karoubi category one where all idempotents “have a kernel”?) I thought the kernel is the object represented by the functor Y -> Ker(Hom(Y,X) -> Hom(Y,X)), or the equaliser Ker(p, 0) when C admits zero morphisms.
Oh, I think that’s just a French-ism. It seems to me people close to algebraic geometry like saying ‘kernel’ instead of ‘equaliser’.
Hm… it’s strange that there weren’t even any links between these two pages.
??? Karoubi envelope links directly to Cauchy complete category, in more than one place. And I see that Cauchy complete category links back to Karoubi envelope as well.
Oh, I think that’s just a French-ism. It seems to me people close to algebraic geometry like saying ‘kernel’ instead of ‘equaliser’.
Ah, I see.
Hm... it's strange that there weren't even any links between these two pages.
??? Karoubi envelope links directly to Cauchy complete category, in more than one place. And I see that Cauchy complete category links back to Karoubi envelope as well.
Sorry, I meant between the pages Karoubian category and Karoubi envelope.
Of course, linking between pages is the easiest thing in the world. If you see pages that ought to be linked, just put them in (as I just did, cross-linking between Karoubian category and Karoubi envelope).
While I was at it, I added some more material to Karoubian category (e.g., answering a question about a converse statement).
I added some more material to Karoubian category (e.g., answering a question about a converse statement).
Thanks!
Since $p:X\to X$ is idempotent iff $\id_X - p$ is idempotent, this is the same as saying every idempotent has a kernel.
Ah, this was precisely the point I was missing.
I moved a bunch of redirects pointing to various synonyms of “idempotent completion” from Karoubian category to idempotent completion. Okay?
I also made some slight cosmetic edits in the former entry.
The entries Karoubian category, , Karoubi envelope, Cauchy completion confuse me quote a lot in the case of $Ab$-enriched categories (what is the default case for Karoubian envelope in the practical mathematics). If understood the terminology, the $n$Lab says that the Cauchy completion is the general case of enriched categories and Karoubi envelope is the default for usual categories. This seems to say that in the case of Ab-enriched categories that the underlying Set-category of the Karoubian completion is the Karoubian completion of the underlying Set-category. But this seems not to be so. Borceux’s monograph’s proof of the existence of completion takes Yoneda embedding, notes that it is Karoubi complete, takes the representables and their split subobjects where one particular splitting is chosen for each such object. So, in the case of a ring viewed as a $Ab$-enriched category with one object, one would look just at projective ideals (+with splitting within the ring), while the true Karoubian envelope is the category of finitely generated projectives. Thus the Karoubi completion of $Ab$-enriched categories is completion under idempotents and direct sums. $Ab$-enriched category does not need to have the direct sums so it is not obvious how the general discussion implies we have to take retracts of finite coproducts ?
Zoran, I expect you’re right that the situation is a bit of a mess, but let me try to summarize what I think you are saying and then propose a remedy:
The nLab says that the Karoubi envelope is the Cauchy completion for categories considered qua $Set$-enriched categories, but
Usually in mathematics, when the term “Karoubi envelope” comes up, it is understood as pertaining to $Ab$-enriched categories, where Cauchy completion means something else (closure under absolute colimits, which includes direct sums and splitting of idempotents).
I think that sounds right to me, particularly the second point. I think merging idempotent completion into Karoubi envelope is therefore a bad idea; we should have them as separate entries. (I can check once this comment is posted, but I prefer the term idempotent-splitting completion and hope that at least there’s a redirect for that. Edit: yes there is, but the redirect is to Karoubi envelope.)
Of course Cauchy completion, for which we actually have multiple entries, is the most general (pertaining to all forms of enrichment), and I think deserves a stand-alone entry.
I expected the answer from you – and was lucky to get it quickly!
let me try to summarize what I think you are saying
Yes, you summarized what I said that I suspected that it was true, but was not sure. Otherwise I would not ask (I am trying to compare many external references, but it is hard to quickly find explicit all what I need in a due comparison).
closure under absolute colimits, which includes direct sums and splitting of idempotents
Not only finite direct sums ?
No, I meant only finite direct sums for the $Ab$-enriched case.
After googling a little bit, it seems that all the hits I got pointed to Karoubi envelope as a synonym of idempotent-splitting completion (even for $Ab$-enriched categories). So what can I say? It would be nice if Karoubi envelope in the literature on $Ab$-enriched categories meant the “true Karoubi envelope” as Zoran put it, but for whatever reason people don’t seem to adopt the “true” concept under that name. Does Borceux treat the general enriched case?
Just in case anyone is wondering, the “Anonymous Coward” who performed the last edits at Karoubi envelope was me (it was an accident; I’m on vacation and working from a computer different from my own).
Added:
Thanks. We should also add citation for why, when and by whom these structures were named after Karoubi.
Incidentally, the same problem persists for the entry Karoubi envelope and also with the Wikipedia entry of the same name (here): These entries never say what Karoubi had to do with their content.
added pointer to:
also added two reference that actually say “Karoubian category”:
Bruno Kahn, Appendix A.1 of: Zeta and L-Functions of Varieties and Motives, Cambridge University Press (2020) [doi:10.1017/9781108691536]
Stacks Project, Karoubian categories[tag:09SF]
(for bystanders: see also the parallel discussion in another thread here)
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