Here are two definitions as currently in the stacks project:<\/p>\n
Here the subquotients and extensions are taken in the bigger abelian category. The formal definitions can be found here<\/a>.<\/p>\n Yesterday I realized I had confused these two notions. In some situations the first is more appropriate (e.g., the kernel of an exact functor is a Serre subcategory) and in others the second is better (e.g., given a weak Serre subcategory B of A the derived category D_B(A) makes sense).<\/p>\n Nomenclature: I think the notion of a Serre subcategory is pretty standard, in the sense that all of the definitions of a Serre subcategory of an abelian category that I have seen are equivalent to the one above (single exception: nlab). Serre used the same definition (in the case that the ambient category is the category of abelian groups). On the other hand, the notion of a “weak Serre subcategory” is nonstandard. In some papers\/books the terminology “thick subcategory” is used for this, but unfortunately in many texts “thick subcategory” is synonymous with “Serre subcategory”. In fact, it seems that the notion of a “thick subcategory” is very malleable — there is no real agreement on what this term should mean, and, googling, I found at least one instance where this confusion led to a mathematical error. In the case of subcategories of a triangulated categories I decided to avoid using “thick” and I have used “saturated” just like Verdier does in his thesis. (Unfortunately, some authors use “saturated” to mean “closed under isomorphism”, but they seem in the minority.)<\/p>\n Is there a word, other than “thick”, we can use to describe weak Serre subcategories?<\/p>\n","protected":false},"excerpt":{"rendered":" Here are two definitions as currently in the stacks project: A Serre subcategory of an abelian category is a strictly full subcategory closed under taking subquotients and closed under taking extensions. A weak Serre subcategory of an abelian category is … Continue reading