Pushout - Knowlpedia

Let $\mathcal{C}$ be a category and let

X \xleftarrow{f} Z \xrightarrow{g} Y

be a span of morphisms.

A pushout of $(f,g)$ is an object $P$ together with morphisms

i_X:X\to P,\qquad i_Y:Y\to P

such that

i_X\circ f = i_Y\circ g

(using composition), and satisfying the following universal property:

For any object $W$ and morphisms $u:X\to W$ , $v:Y\to W$ with $u\circ f = v\circ g$ , there exists a unique morphism $[u,v]:P\to W$ such that

[u,v]\circ i_X = u,\qquad [u,v]\circ i_Y = v.

When it exists, the pushout is unique up to unique isomorphism and is often denoted $X\amalg_Z Y$ .

Relationship to other constructions

A pushout is a special colimit: it is the colimit of the diagram $X \leftarrow Z \rightarrow Y$ .
It is dual to a pullback.
In many categories, it can be seen as a “coproduct with identifications,” relating it to the coproduct.

Examples

Example (Set)

In $\mathbf{Set}$ , form the disjoint union $X\amalg Y$ and impose the smallest equivalence relation $\sim$ such that

f(z)\sim g(z)\quad\text{for all }z\in Z.

Then the pushout is the quotient set

P = (X\amalg Y)/{\sim},

with $i_X,i_Y$ induced by the coproduct injections $X\to X\amalg Y$ , $Y\to X\amalg Y$ .

Example (Top)

In $\mathbf{Top}$ , the pushout of $X \xleftarrow{f} Z \xrightarrow{g} Y$ is the quotient space

P = (X\amalg Y)/{\sim},

where $\sim$ identifies $f(z)$ with $g(z)$ for each $z\in Z$ , and the topology is the quotient topology. This models “gluing spaces along a common subspace.”

Example (Grp)

In $\mathbf{Grp}$ , the pushout of $X \xleftarrow{f} Z \xrightarrow{g} Y$ is the amalgamated free product

X *_Z Y,

characterized by the universal property that homomorphisms out of $X$ and $Y$ that agree on $Z$ factor uniquely through $X *_Z Y$ .