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.
Morphisms are equipped with a partial binary operation, called composition.
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 collection of all morphisms from X to Y is denoted HomC(X, Y) or simply Hom(X, Y) and called the hom-set between X and Y.
The domain and codomain are in fact part of the information determining a morphism.
In practice, this is not a problem because if this disjointness does not hold, it can be assured by appending the domain and codomain to the morphisms (say, as the second and third components of an ordered triple).
An epimorphism can be called an epi for short, and we can use epic as an adjective.
In the category of sets, the statement that every surjection has a section is equivalent to the axiom of choice.
For example, in the category of commutative rings the inclusion Z → Q is a bimorphism that is not an isomorphism.
In particular, the Karoubi envelope of a category splits every idempotent morphism.