Pushout (category theory)

In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain. The pushout consists of an object P along with two morphisms X → P and Y → P that complete a commutative square with the two given morphisms f and g. In fact, the defining universal property of the pushout (given below) essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are

P

=

X

⊔

Z

Y

{\displaystyle P=X\sqcup _{Z}Y}

and

P

=

X

+

Z

Y

{\displaystyle P=X+_{Z}Y}

.

The pushout is the categorical dual of the pullback.

Universal property

Explicitly, the pushout of the morphisms f and g consists of an object P and two morphisms i1 : X → P and i2 : Y → P such that the diagram

commutes and such that (P, i1, i2) is universal with respect to this diagram. That is, for any other such triple (Q, j1, j2) for which the following diagram commutes, there must exist a unique u : P → Q also making the diagram commute:

As with all universal constructions, the pushout, if it exists, is unique up to a unique isomorphism.

Examples of pushouts

Here are some examples of pushouts in familiar categories. Note that in each case, we are only providing a construction of an object in the isomorphism class of pushouts; as mentioned above, though there may be other ways to construct it, they are all equivalent.

Suppose that X, Y, and Z as above are sets, and that f : Z → X and g : Z → Y are set functions.