---
tag: FWGPKXCPB7
kind: concept
name: Complete category
msc: [18A35]
specializes: [PYSBNWCHCF]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_category
same_as:
  wikidata: Q4370335
  wikipedia: Complete category
---

## Summary

A complete category is a [[category|A2DE66937D]] in which every small diagram has a [[limit|CHS5AFJGWE]].
Products, equalizers, pullbacks and inverse limits then all exist.
The categories of sets, groups, rings, modules and topological spaces are complete.
Completeness is a main hypothesis of the adjoint functor theorems.

## Explanations

### L1 human

A limit is a best way to combine a family of objects and maps into one object, like a product or a set of solutions of equations.
A category is complete when you can always do this, as long as the family is indexed by a set.
The category of sets is complete: the limit is a set of compatible families of elements.
The category of finite sets is not complete, because an infinite product of two-element sets is infinite.