# nLab pushout

Contents

category theory

## Applications

#### Infinity-limits

limits and colimits

# Contents

## Idea

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

## Pushouts in $Set$

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:

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:

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

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.

## Definition

A pushout is a colimit of a diagram like this:

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

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.

## Properties

### In any category

###### Proposition

(pushouts as coequalizers)

If coproducts exist in some category, then the pushout

is equivalently the coequalizer

of the two morphisms induced by $f$ and $g$ into the coproduct of $b$ with $c$.

###### Proposition

(pushouts preserves epimorphisms and isomorphisms)

Pushouts preserve epimorphisms and isomorphisms:

If

is a pushout square in some category then:

1. if $g$ is a epimorphism then $f_\ast g$ is an epimorphism;

2. if $g$ is an isomorphism then $f_\ast g$ is an isomorphism.

###### Proposition

(pasting law for pushouts)

Consider a commuting diagram of the following shape in any category:

If the left square is a pushout, then the total rectangle is a pushout if and only if the right square is a pushout.

###### Proof

See the proof of the dual property for pullbacks.

###### Proposition

The converse implication does not hold: it may happen that the outer and the right square are pushouts, but not the left square.

###### Proof

See the proof of the dual proposition for pullbacks.

### In a quasitopos

###### Proposition

pushout of strong monomorphism in quasitopos

Suppose that $(\mathrm{T},\mathcal{C})$ is either

Suppose that

is a commutative diagram in $\mathcal{C}$ such that

• $m$ is $\mathrm{T}$ in $\mathcal{C}$
• the diagram is a pushout in $\mathcal{C}$

Then

• $h$ is $\mathrm{T}$ in $\mathcal{C}$
• the diagram is a pullback in $\mathcal{C}$

See at quasitopos this lemma. Note that the result for quasitoposes immediately implies the result for toposes, since all monomorphisms $i: A \to B$ in a topos are regular ($i$ being the equalizer of the arrows $\chi_i, t \circ !: B \to \Omega$ in

where $\chi_i$ is the classifying map of $i$) and therefore strong.

## Examples

Textbook accounts: