---
tag: PZHDNJ2J73
kind: concept
name: Locally presentable category
msc: [18C35]
specializes: [FC8C0V4TFZ]
sources:
  - url: https://en.wikipedia.org/wiki/Accessible_category
---

## Summary

A locally presentable category is an [[accessible category|FC8C0V4TFZ]] that has all small colimits.
Every object is a highly filtered colimit of objects from a fixed small set of presentable objects.
Sets, groups, rings, modules, small categories and presheaf categories are locally presentable, while topological spaces are not.
These categories are also complete, and the adjoint functor theorem takes a simple form for them.

## Explanations

### L1 human

A locally presentable category is a large category built from a small set of building blocks in a controlled way.
In the category of groups, every group is a filtered colimit of finitely presented groups.
Because of this, questions about the whole category can often be reduced to questions about the small building blocks.