---
tag: Z3MQ1D5Q38
kind: definition
name: Pushout
msc: [18A30]
defines: Q699KKMVJD
uses: [A2DE66937D, 859F0TFVA3, BJS1CGTKQ5]
sources:
  - url: https://en.wikipedia.org/wiki/Pushout_(category_theory)
---

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]], and let $f \colon Z \to X$ and $g \colon Z \to Y$ be [[morphisms|859F0TFVA3]] of $\mathcal{C}$.
A *pushout* of $f$ and $g$ is an object $P$ with morphisms $i_1 \colon X \to P$ and $i_2 \colon Y \to P$ such that:

1. $i_1 \circ f = i_2 \circ g$.
2. For every object $Q$ and morphisms $j_1 \colon X \to Q$ and $j_2 \colon Y \to Q$ with $j_1 \circ f = j_2 \circ g$, there is a unique morphism $u \colon P \to Q$ with $u \circ i_1 = j_1$ and $u \circ i_2 = j_2$.

## Remarks

Equivalently, $(P, i_1, i_2)$ is a [[colimit|BJS1CGTKQ5]] of the diagram $X \xleftarrow{f} Z \xrightarrow{g} Y$.
A pushout is unique up to a unique isomorphism, and one writes $P = X \sqcup_Z Y$ or $X +_Z Y$.

Example: in the category of sets, let $\sim$ be the smallest equivalence relation on $X \sqcup Y$ with $f(z) \sim g(z)$ for all $z \in Z$.
Let $P = (X \sqcup Y)/{\sim}$, and let $i_1, i_2$ send an element to its class.
Clause 1 holds by the choice of the relation.
For clause 2, the map $u$ must send the class of $x$ to $j_1(x)$ and the class of $y$ to $j_2(y)$; this is well defined because $j_1 \circ f = j_2 \circ g$.

Non-example: take $P = X \sqcup Y$ with the two inclusions and no identification.
If $Z$ is not empty, then $i_1(f(z))$ and $i_2(g(z))$ lie in different parts, so clause 1 fails.

In the category of groups the pushout is the amalgamated free product $X *_Z Y$.