Morphism

In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Although many examples of morphisms are structure-preserving maps, morphisms need not be maps, but they can be composed in a way that is similar to function composition.

Morphisms and objects are constituents of a category. Morphisms, also called maps or arrows, relate two objects called the source and the target of the morphism. There is a partial operation, called composition, on the morphisms of a category that is defined if the target of the first morphism equals the source of the second morphism. The composition of morphisms behaves like function composition (associativity of composition when it is defined, and existence of an identity morphism for every object).

Morphisms and categories recur in much of contemporary mathematics. Originally, they were introduced for homological algebra and algebraic topology. They belong to the foundational tools of Grothendieck's scheme theory, a generalization of algebraic geometry that applies also to algebraic number theory.

A category C consists of two classes, one of objects and the other of morphisms. There are two objects that are associated to every morphism, the source and the target. A morphism f from X to Y is a morphism with source X and target Y; it is commonly written as f : X → Y or X f→ Y the latter form being better suited for commutative diagrams.

For many common categories, an object is a set (often with some additional structure) and a morphism is a function from an object to another object. Therefore, the source and the target of a morphism are often called domain and codomain respectively.

Morphisms are equipped with a partial binary operation, called composition (partial because the composition is not necessarily defined over every pair of morphisms of a category). The composition of two morphisms f and g is defined precisely when the target of f is the source of g, and is denoted g ∘ f (or sometimes simply gf). The source of g ∘ f is the source of f, and the target of g ∘ f is the target of g. The composition satisfies two axioms:

For a concrete category (a category in which the objects are sets, possibly with additional structure, and the morphisms are structure-preserving functions), the identity morphism is just the identity function, and composition is just ordinary composition of functions.

The composition of morphisms is often represented by a commutative diagram. For example,