Pushout (category theory)

In , 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 consisting of two morphisms f : Z → X and g : Z → Y with a common domain. The pushout consists of an 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 and . The pushout is the of the . 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. Here are some examples of pushouts in familiar . 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. The pushout of f and g is the disjoint union of X and Y, where elements sharing a common (in Z) are identified, together with the morphisms i1, i2 from X and Y, i.e. where ~ is the finest equivalence relation (cf. also this) such that f(z) ~ g(z) for all z in Z. In particular, if X and Y are subsets of some larger set W and Z is their intersection, with f and g the inclusion maps of Z into X and Y, then the pushout can be canonically identified with the union . A specific case of this is the cograph of a function. If is a function, then the cograph of a function is the pushout of f along the identity function of Y.

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In , a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function from a fixed index set to the class of sets. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a functor from a fixed index category to some category.

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