pushout

A **pushout** is an ubiquitous construction in category theory providing a a colimit for the diagram $\bullet\leftarrow\bullet\rightarrow\bullet$. It is dual to the notion of a pullback.

In the category Set a ‘pushout’ is a quotient of the disjoint union of two sets. Given a diagram of sets and functions like this:

$\array{
&&&&
C
&&&&
\\
&
&& f \swarrow
&
& \searrow g
&&
\\
&& A &&&& B
}$

the ‘pushout’ of this diagram is the set $X$ obtained by taking the disjoint union $A + B$ and identifying $a \in A$ with $b \in B$ if there exists $x \in C$ such that $f(x) = a$ and $g(x) = b$ (and all identifications that follow to keep equality an equivalence relation).

This construction comes up, for example, when $C$ is the intersection of the sets $A$ and $B$, and $f$ and $g$ are the obvious inclusions. Then the pushout is just the union of $A$ and $B$.

Note that there are maps $i_A : A \to X$, $i_B : B \to X$ such that $i_A(a) = [a]$ and $i_B(b) = [b]$ respectively. These maps make this square commute:

$\array{
&&&&
C
&&&&
\\
&
&& f \swarrow
&
& \searrow g
&&
\\
&& A &&&& B
\\
&
&& {}_{i_A}\searrow
&
& \swarrow_{i_B}
&&
\\
&&&&
X
&&&&
}$

In fact, the pushout is the universal solution to finding a commutative square like this. In other words, given *any* commutative square

$\array{
&&&&
C
&&&&
\\
&
&& f \swarrow
&
& \searrow g
&&
\\
&& A &&&& B
\\
&
&& {}_{j_A}\searrow
&
& \swarrow_{j_B}
&&
\\
&&&&
Y
&&&&
}$

there is a unique function $h: X \to Y$ such that

$h i_A = j_A$

and

$h i_B = j_B .$

Since this universal property expresses the concept of pushout purely arrow-theoretically, we can formulate it in any category. It is, in fact, a simple special case of a colimit.

A **pushout** is a colimit of a diagram like this:

$\array{
&&&&
c
&&&&
\\
&
&& f \swarrow
&
& \searrow g
&&
\\
&& a &&&& b
}$

Such a diagram is called a span. If the colimit exists, we obtain a commutative square

$\array{
&&&&
c
&&&&
\\
&
&& f \swarrow
&
& \searrow g
&&
\\
&& a &&&& b
\\
&
&& {}_{i_a}\searrow
&
& \swarrow_{i_b}
&&
\\
&&&&
x
&&&&
}$

and the object $x$ is also called the **pushout**. It has the universal property already described above in the special case of the category $Set$.

Other terms: $x$ is a **cofibred coproduct** of $a$ and $b$, or (especially in algebraic categories when $f$ and $g$ are monomorphisms) a free product of $a$ and $b$ with $c$ **amalgamated**, or more simply an **amalgamation** (or **amalgam**) of $a$ and $b$.

The concept of pushout is a special case of the notion of **wide pushout** (compare wide pullback), where one takes the colimit of a diagram which consists of a set of arrows $\{f_i: c \to a_i\}_{i \in I}$. Thus an ordinary pushout is the case where $I$ has cardinality $2$.

Note that the concept of pushout is dual to the concept of pullback: that is, a pushout in $C$ is the same as a pullback in $C^{op}$.

See pullback for more details.

…

Revised on August 10, 2016 05:22:26
by Thomas Holder
(176.7.88.1)