Pushout (category theory)

The pushout, also called cokartesisches square or fiber sum, is a concept from the mathematical branch of category theory. This is the dual to the pullback construction.

Pushout of modules

Let and two homomorphisms between modules over a ring. Substituting, the pushout is defined by and as

It can be shown that and that has the following universal property:

If any module with homomorphisms and so, so there is exactly one homomorphism with and.

Pushout in Categories

By the above example motivated, we define the pushout in any of the categories as follows.

Let and two morphisms of a category. A pair of morphisms of this category is called pushout of if the following applies:

• Is a pair of morphisms with, so there is exactly one morphism and.

Sometimes it is called only the object is a pushout, referring to that there are morphisms that satisfy the above definition. Also, the diagram

Is sometimes called pushout.

Examples

• Each pullback in a category is a pushout in the dual category, because apparently the pushout is exactly the pullback dual concept.
• In an abelian category, the pushout is to

• Includes above names the zero object of an additive category, the pushout is equal to the direct sum.
• The introductory example shows that there is always pushouts in the category of -modules.
• In the category of groups always exists a pushout. With above notation this is equal to the free product modulo the normal subgroup generated by the natural pictures This construction occurs during a block of Seifert -van Kampen.
• In the category of commutative rings with unit element is the pushout with above names equal to the tensor product.
• In the category of rings with identity, there is not always a pushout.