---
tag: XVG7FPG227
kind: concept
name: Well-powered category
msc: [18A32, 18A20]
specializes: [A2DE66937D]
sources:
  - url: https://en.wikipedia.org/wiki/Subobject
---

## Summary

A [[category|A2DE66937D]] is well-powered when every object has only a set of [[subobjects|FS1FKAE89H]], not a proper class of them.
This size condition holds in most categories of everyday algebra and topology.
It is a hypothesis of the special adjoint functor theorem and of several existence results for factorisations and limits.

## Explanations

### L1 human

In a large category, there can be a proper class of monomorphisms into an object, because there is a proper class of sets that could serve as domains.
After identifying monomorphisms that differ by an isomorphism, one usually gets back a set.
For example, the subobjects of a set $X$ correspond to the subsets of $X$, and these form the power set of $X$, which is a set.
A well-powered category is one where this collapse to a set happens for every object.