nLab
well-powered category

Contents

Defnition

A category CC 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 XX, the (generally large) preordered set of monomorphisms with codomain XX 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 opC^{op} is well-powered, we say that CC is well-copowered (although “cowell-powered” is also common).

Properties

Relation to local smallness

A well-powered category with binary products is always locally small, since morphisms f:ABf: A \to B can be identified with particular subobjects of A×BA \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.

Examples

Revised on July 10, 2014 20:09:19 by Urs Schreiber (89.204.130.36)