---
tag: PYSBNWCHCF
kind: concept
name: Finitely complete category
msc: [18A35]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_category
---

## Summary

A finitely complete category is a [[category|A2DE66937D]] in which every finite diagram has a [[limit|CHS5AFJGWE]].
Equivalently, it has a terminal object and all [[pullbacks|Q05K04QJ71]].
Finite limits are enough for much of category theory, such as internal groups, regular categories and elementary toposes.
Every [[complete category|FWGPKXCPB7]] is finitely complete.

## Explanations

### L1 human

A finite diagram is a picture with finitely many objects and arrows.
A category is finitely complete when every such picture has a limit, such as a product of two objects or a solution set of an equation.
The category of finite sets is finitely complete, since finite products and subsets of finite sets are finite.
It is not complete, because infinite products of finite sets are not finite.