Pushout

Definition

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

be a span of morphisms .

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

such that

(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

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

Then the pushout is the quotient set

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

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

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