nLab
thick subcategory

Definition

A full subcategory of a triangulated category is thick (or épaisse) if it is closed under extensions.

Sometimes the same definition is used in abelian categories as well. However, for many authors, including Pierre Gabriel, in abelian categories, this term denotes the stronger notion of a topologizing subcategory closed under extensions; in other words, a nonempty full subcategory $T$ of an abelian category $A$ is thick (in strong sense) iff with all objects contains all its subquotients and all extensions, i.e. for every exact sequence

$$0⟶M⟶M″⟶M\prime ⟶0$$0 \longrightarrow M\longrightarrow M''\longrightarrow M'\longrightarrow 0

in $A$, the object $M″$ is in $T$ iff $M$ and $M\prime$ are in $T$.

For some authors the thick subcategory (strong version) is called a Serre subcategory (in a weak sense), the term which we reserve for (generally) a stronger notion.

For any subcategory of an abelian category $A$ one denotes by $\overline{T}$ the full subcategory of $A$ generated by all objects $N$ for which any (nonzero) subquotient of $N$ in $T$ has a (nonzero) subobject from $T$. This becomes an idempotent operation on the class of subcategories of $A$ where $T\subset \overline{T}$ iff $T$ is topologizing. Moreover $\overline{T}$ is always thick in the stronger sense. Serre subcategories in the strong sense are those (nonempty) subcategories which are stable under the operation $T\mapsto \overline{T}$.

References

• Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France, 90 (1962), p. 323-448 (numdam)

• A. L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, MIA 330, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9

• Springer eom: localization of categories