What is "pushout"

n. (context category theory English) Given a pair of morphisms $f:Zrightarrow\; X$ and $g:\; Zrightarrow\; Y$ with a common domain, ''Z'', their '''pushout''' is a pair of morphisms $i\_1:Xrightarrow\; P$ and $i\_2:Yrightarrow\; P$ as well as their common codomain, ''P'', such that the equation $i\_1\; circ\; f\; =\; i\_2\; circ\; g$ is satisfied, and for which there is the ''universal property'' that for any other object ''Q'' for which there are also morphisms $j\_1:\; Xrightarrow\; Q$ and $j\_2:\; Yrightarrow\; Q$; there is a unique morphism $u:\; Prightarrow\; Q$ such that $u\; circ\; i\_1\; =\; j\_1$ and $u\; circ\; i\_2\; =\; j\_2$.

A pushout is a student that leaves their school before graduation, through the encouragement of the school. A student who leaves of their own accord (e.g., to work or care for a child), rather than through the action of the school, is considered a school dropout. In typical use, the category of pushouts excludes students who have been formally expelled from school for violating rules (e.g., for being violent).

Students may be pushed out of school because their presence in the school creates difficulty in meeting some goal of the school. For example, in the case where funding for the school is dependent upon scholastic achievement of the students, if the school can get rid of low-performing students, average test scores on academic performance tests will go up, thus increasing funding. Schools may pushout truant students, who formally enroll in classes, but then refuse to attend.

In some low-performing schools in Chicago combined dropout/pushout rates have exceeded 25% in one year.

Children are also pushed from schools because they present discipline problems. Within youth advocacy and activist communities, pushout is a term that recognizes the intersecting forces of oppression most commonly responsible for high school "drop outs" within marginalized communities of color, allowing for the responsibility to be placed on those forces, rather than the youth impacted by unequal education, economics, disciplinary actions, and racism.

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 which 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. A common notation for the pushout is

P = X ⊔ Y
.

The pushout is the categorical dual of the pullback.