nLab well-powered category

Definition

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 $C^{op}$ is well-powered, we say that $C$ 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 $f: A \to B$ 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.

Examples

• Every Grothendieck topos (indeed, every locally small elementary topos) is well-powered (by the existence of a subobject classifier and the smallness of hom sets).

• More generally, every locally presentable category is well-powered, since it is a full reflective subcategory of a presheaf topos, so its subobject lattices are subsets of those of the latter.

• Indeed, every category $C$ with a small dense subcategory $A$ is well-powered, since the restricted Yoneda embedding $C \to [A^{op},Set]$ is then fully faithful and preserves monomorphisms, so it embeds the subobject posets of $C$ as sub-posets of those of $[A^{op},Set]$. This includes in particular all accessible categories.

• On the other hand, a locally small category with a strong generator can fail to be well-powered; a counterexample is here.

• Every locally presentable category, indeed every accessible category with pushouts, is well-copowered. This is shown in Adamek-Rosicky, Proposition 1.58 and Theorem 2.49. Whether this is true for all accessible categories depends on what large cardinal properties hold: by Corollary 6.8 of Adamek-Rosicky, if Vopenka's principle holds then all accessible categories are well-copowered, while by Example A.19 of Adamek-Rosicky, if all accessible categories are well-copowered then there exist arbitrarily large measurable cardinals.

References

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge University Press, (1994)