nLab
property sup

Idea

The property (sup) is an optional property of an abelian category, introduced in (Gabriel).

Definition

An abelian category has property (sup) if:

Definition (sup)

For any ascending chain $\Omega$ of subobjects of a fixed object $M$, the supremum of $\Omega$ exists; and, for any subobject $L\hookrightarrow M$, the canonical morphism

$$\sup \{L\cap P\mid P\in \Omega \}\to (\sup \Omega )\cap L$$sup\{L\cap P | P\in \Omega\}\to (sup \Omega) \cap L

is an isomorphism.

Examples

Gabriel’s property (sup) is satisfied by any Grothendieck category (in some expositions it is listed as a part of the definition), e.g. the category of all modules over a fixed ring $R$, and the category of sheaves of abelian groups on a fixed topological space $X$.

According to the Appendix B of the Thomason–Trobaugh contribution to Grothendieck Festschrift, it is not known whether the category ${\mathrm{Qcoh}}_Y$ of quasicoherent sheaves over an arbitrary scheme $Y$ is a Grothendieck category (although this is known for a large class), but it is an elementary fact that it does satisfy Gabriel’s property (sup). If an abelian category is noetherian it clearly satisfies the property (sup).

References

• Pierre Gabriel, Des catégories abéliennes