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Recently, there was a discussion in the page about Serre subcategories in The Stacks project. I usually follow both nLab and The Stack project for these things but it seems that the definitions of Serre subcategories are different here and there. I just want to clarify if both definitions are correct or there is an error on any of them. Am I missing something? The discussion is about the use of “if” or “iff” conditions under the arrows involving $M,M',M''$ (or $A,B,C$) in the pages Serre subcategory and https://stacks.math.columbia.edu/tag/02MO (there is even more discussion (going on) in https://stacks.math.columbia.edu/tag/02MN).
Please note that the $n$Lab page Serre subcategory is at best a stub that is hardly meant to be authorative but is instead waiting for a kind expert soul to take care of it!
The page history shows that its content was jotted down 12 years ago (rev 1) by a user no longer active here, and not substantially touched by any expert author. Related problems with related entries had recently been raised in comment 94319 and I have tried to patch it up then (comment 94328), but, as I disclaimed there, I am neither expert on this nor am I investing serious thoughts into it (being busy elsewhere).
So if there is a contradiction between the StacksProject and these pages, it’s certainly these pages that need attention. You would do the $n$Lab community a great service if you took care of this! (To start with, this may require minimal work, just replacing wrong or non-existent definition with correct ones, or even just with pointers to correct ones). Editing pages here is pretty straightforward, and I’d be happy to help if any issues arise.
If M=M’, the map M→M’ is identity, and the map M’→M” is the zero map, then the sequence M→M’→M” is exact.
Thus, the “only if” part of “if and only if” implies that any object M” belongs to any Serre subcategory.
So it seems like the “only if” part should be removed.
The “if and only if” version is meant for talking about short exact sequences $0 \to M \to M' \to M'' \to 0$. E.g. as implied by lemma 12.10.2 of the stacks project page.
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