---
tag: TAVG968272
kind: definition
name: Direct limit
msc: [18A30]
defines: X9HEQ29Q49
uses: [D9RY4667K6, A2DE66937D, 859F0TFVA3, BJS1CGTKQ5]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_limit
---

## Definition

Let $(I, \le)$ be a [[directed set|D9RY4667K6]] and let $\mathcal{C}$ be a [[category|A2DE66937D]].
A *direct system* in $\mathcal{C}$ over $I$ is a family of objects $A_i$ for $i \in I$ with [[morphisms|859F0TFVA3]] $f_{ij} \colon A_i \to A_j$ for all $i \le j$, such that:

1. $f_{ii} = \id_{A_i}$ for all $i$.
2. $f_{jk} \circ f_{ij} = f_{ik}$ for all $i \le j \le k$.

A *direct limit* of this system is an object $A$ with morphisms $\phi_i \colon A_i \to A$ such that:

1. $\phi_j \circ f_{ij} = \phi_i$ for all $i \le j$.
2. For every object $B$ with morphisms $g_i \colon A_i \to B$ such that $g_j \circ f_{ij} = g_i$ for all $i \le j$, there is a unique morphism $u \colon A \to B$ with $u \circ \phi_i = g_i$ for all $i$.

## Remarks

The direct limit is written $\varinjlim A_i$.
It is the [[colimit|BJS1CGTKQ5]] of the diagram given by the system, and it is unique up to a unique isomorphism.
In sets, groups, rings and modules it always exists.
There it is the disjoint union of the $A_i$ divided by a relation: $a_i \in A_i$ and $a_j \in A_j$ are identified when $f_{ik}(a_i) = f_{jk}(a_j)$ for some $k \ge i, j$.

Example: let $A_n = \Z$ for $n \in \N$, and let $f_{n,n+1}$ be multiplication by $2$.
The maps $\phi_n \colon \Z \to \Z[1/2]$, $\phi_n(a) = a/2^n$, make $\Z[1/2]$ a direct limit.
Clause 1 holds because $\phi_{n+1}(2a) = 2a/2^{n+1} = \phi_n(a)$.

Non-example: the maps $\phi_n(a) = a/2^n$ into $\Q$ satisfy clause 1 but not clause 2.
For $B = \Z[1/2]$ with the maps above, there is no group homomorphism $u \colon \Q \to \Z[1/2]$ with $u(a/2^n) = a/2^n$ for all $a$ and $n$.
Indeed $u(1/3)$ would be an element $x$ with $3x = 1$, and $\Z[1/2]$ has no such element.