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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
The changes from v9 to v10 left this page with a second paragraph that jumps between topics mid-sentence. The part starting with “where the colimit is over all” comes out of nowhere, and all of the material about quotient categories is gone.
I only remember that so much has to be done to make this entry good, but what I was to do in that particular attempt I do not recall. I will resume on this some time soon.
Serre’s “original article” is just the case in the category of Abelian groups and contains the quotient construction. Most of the work later has been within the category of (all) modules over a fixed ring, or slightly more generally within a Grothendieck category. If one goes beyond the Grothendieck categories, then the role which Serre categories have in fundamental theorems usually require an additional condition, thus a more restrictive definition may be appropriate for that generalization. This is a bit difficult to cover as there is no terminological consensus in the literature.
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