---
tag: 975PMY4FG8
kind: concept
name: Filtered category
msc: [18A30]
sources:
  - url: https://en.wikipedia.org/wiki/Filtered_category
same_as:
  wikidata: Q5449247
  wikipedia: Filtered category
---

## Summary

A filtered category is a [[category|A2DE66937D]] in which every finite diagram has a cocone.
It generalizes a [[directed set|D9RY4667K6]]: any two objects map to a common object.
Also, any two parallel arrows become equal after composing with a further arrow.
Colimits over filtered categories behave like unions, and in sets they commute with finite limits.
They are the basis of the theory of locally presentable categories.

## Explanations

### L1 human

Think of a process that runs forward without end, where any two stages are followed by a later stage that contains both.
A filtered category is such a process, with the extra rule that two different ways of going forward can always be made to agree later on.
A directed set, such as the natural numbers with their usual order, is the simplest example.
A category with a terminal object is also filtered, since everything maps to the terminal object.