Is the empty subcategory thick, localizing, topologizing, etc

Let $A$ be an abelian category. There are various types of full subcategories. I often wonder if it is assumed that these are nonempty, since in most proofs this is used implicitely, but also the empty case then works as a special case. To be sure, I ask you:

• $S \subseteq A$ is called thick iff $S$ is closed under subquotients and extensions. Note that $\emptyset$ is thick according to this definition. Is this allowed in the literature you know? Remark that $S$ is nonempty iff $0 \in S$.

• $S \subseteq A$ is called topologizing iff $S$ is closed under subquotients and direct sums. The same as above.

• $S \subseteq A$ is called Serre iff $S = S^{-}$. Here $S^-$ consists of those objects, such that every nonzero subquotient of it contains a nonzero subobject isomorphic to an object of $S$. Since $0 \in S^{-}$, this always implies that $S$ is nonempty, ok.

• $S \subseteq A$ is called localizing iff $S$ is thick and $A \to A/S$ admits a right adjoint. I think that the construction $A/S$ makes only sense if $S$ is nonempty. Thus we should add that $S$ is nonempty (i.e. $0 \in S$)?

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1 Answer

Normally subcategories of Abelian Categories are assumed nonempty for the simple reason that they need contain the zero object.

If you want your subcats to be abelian, then they need this zero. For this reason, I would say no.

This is at least the viewpoint in NCAG.

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Your first sentence does not mean what you meant. – Mariano Suárez-Alvarez Oct 28 '10 at 17:32
Thanks Mariano. – BBischof Oct 29 '10 at 7:28