nLab
well-powered category

Defnition

A category $C$ is well-powered if every object has a small poset of subobjects.

Assuming that by ‘subobject’ we mean (an equivalence class of) monomorphisms, this means that for every object $X$, the (generally large) preordered set of monomorphisms with codomain $X$ is equivalent to a small poset, or equivalently that this preordered set is essentially small. Variations exist that use notions of subobject other than monomorphisms.

If $C^{\mathrm{op}}$ is well-powered, we say that $C$ is well-copowered (although “cowell-powered” is also common).

Relation to local smallness

A well-powered category with binary products is always locally small, since morphisms $f:A\to B$ can be identified with particular subobjects of $A\times B$ (their graphs).

Conversely, any locally small category with a subobject classifier must obviously be well-powered. In particular, a topos is locally small if and only if it is well-powered.

There are interesting conditions and applications of the preorder on the sets of subobjects in well-powered categories, cf. e.g. property sup.