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

and

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

In a quasitopos

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

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)