nLab pushout

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 sum (Gabriel & Zisman (1967), p. 1) 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

• (monomorphism,topos), or
• (strong monomorphism,quasitopos)

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

Example

A pushout of injections of Sets is called the union of the sets.

Example

A pushout of groups in Grps is called their amalgamated free product

Example

In topology, space attachments are pushouts in Top.

Related entries

• amalgamation property

• wide pushout

• coequalizer

• colimit

• pullback

• pushout-product

• homotopy pushout

References

Early use of the terminology “pushout”:

• Barry Mitchell, Section I.7 of: Theory of categories, Pure and Applied Mathematics 17, Academic Press (1965) [ISBN:978-0-12-499250-4]

Early use of the terminology “amalgamated sums”:

• Pierre Gabriel, Michel Zisman, p 1 of: Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer (1967) [doi:10.1007/978-3-642-85844-4, pdf]

Textbook accounts:

• Saunders MacLane, p. 65-66 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, Section 2.5 in Vol. 1: Basic Category Theory of: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)