nLab locally small object

Definition

An object $A$ in a category $C$ is a locally small object if the full subcategory of the slice category $C/A$ on the monomorphisms is essentially small (has a small skeleton). In other words, isomorphism classes of monomorphisms with target $A$ (=subobjects of $A$) form a set.

An object $B$ in a category $C$ is colocally small if it is locally small in the dual category. In other words, (isomorphism classes of) quotient objects form a set.

A category is well-powered if its every object is locally small. In older literature and in the subject of abelian categories one sometimes says locally small category for a well-powered category, clashing with the weaker standard notion of a locally small category (all Hom-classes are sets).

Literature

• (1.2 in) N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs 3, Academic Press 1973, xii + 470 pp
• Bo Stenström, Rings of quotients