---
tag: FC8C0V4TFZ
kind: concept
name: Accessible category
msc: [18C35]
sources:
  - url: https://en.wikipedia.org/wiki/Accessible_category
same_as:
  wikidata: Q4672496
  wikipedia: Accessible category
---

## Summary

An accessible category is a large category that is generated, under highly [[filtered colimits|975PMY4FG8]], by a small set of small objects.
Here small objects are the $\lambda$-presentable ones, those whose hom functor preserves $\lambda$-filtered colimits for a [[regular cardinal|13ZV2F8JAR]] $\lambda$.
Accessible categories are exactly the categories of models of sketches, and they include the categories of models of many first-order theories.
The cocomplete ones are the locally presentable categories.

## Explanations

### L1 human

Many large categories are determined by a small part of them.
Every set is the union of its finite subsets, and every vector space is the union of its finite-dimensional subspaces.
An accessible category is one where every object is glued in this way from a fixed small stock of small objects.
This lets one handle a huge category with set-sized data.