---
tag: X9HEQ29Q49
kind: concept
name: Direct limit
msc: [18A30]
specializes: [BJS1CGTKQ5]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_limit
same_as:
  wikidata: Q906015
  wikipedia: Direct limit
---

## Summary

A direct limit, or inductive limit, is the [[colimit|BJS1CGTKQ5]] of a system of objects indexed by a [[directed set|D9RY4667K6]].
Its maps go from earlier objects to later ones.
It builds a large object as the union of a growing family of smaller ones.
Many objects of algebra are built this way, such as the [[Prüfer group|F0XAMM2PPS]] and stalks of sheaves.
The dual notion is the [[inverse limit|DZB378CNCK]].

## Explanations

### L1 human

Picture a chain of boxes, each one placed inside the next.
The direct limit is the big box that holds all of them at once.
Two elements of the boxes count as the same once they become equal in some later box.
For example, the groups $\frac{1}{2^n}\Z$ for $n = 0, 1, 2, \dots$ sit inside each other.
Their direct limit is the group of all fractions whose denominator is a power of $2$.