# 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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