Morphism

In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.
In category theory, "morphism" is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism.
The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the "objects" are simply "sets with some additional structure", and "morphisms" are "structure-preserving functions". In category theory, morphisms are sometimes also called arrows.
Definition.
A category "C" consists of two classes, one of and the other of . There are two objects that are associated to every morphism, the and the . A morphism "f" with source "X" and target "Y" is written "f" : "X" → "Y", and is represented diagrammatically by an from "X" to "Y".
For many common categories, objects are sets (often with some additional structure) and morphisms are functions from an object to another object. Therefore, the source and the target of a morphism are often called and respectively.
Morphisms are equipped with a partial binary operation, called . 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,
The collection of all morphisms from "X" to "Y" is denoted Hom"C"("X","Y") or simply Hom("X", "Y") and called the hom-set between "X" and "Y". Some authors write Mor"C"("X","Y"), Mor("X", "Y") or C("X", "Y"). Note that the term hom-set is something of a misnomer, as the collection of morphisms is not required to be a set. A category where Hom("X", "Y") is a set for all objects "X" and "Y" is called locally small.
Note that the domain and codomain are in fact part of the information determining a morphism. For example, in the category of sets, where morphisms are functions, two functions may be identical as sets of ordered pairs (may have the same range), while having different codomains. The two functions are distinct from the viewpoint of category theory. Thus many authors require that the hom-classes Hom("X", "Y") be disjoint. 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).
Some special morphisms.
Monomorphisms and epimorphisms.
A morphism "f": "X" → "Y" is called a monomorphism if "f" ∘ "g"1 = "f" ∘ "g"2 implies "g"1 = "g"2 for all morphisms "g"1, "g"2: "Z" → "X". A monomorphism can be called a "mono" for short, and we can use "monic" as an adjective. A morphism "f" has a left inverse or is a split monomorphism if there is a morphism "g": "Y" → "X" such that "g" ∘ "f" id"X". Thus "f" ∘ "g": "Y" → "Y" is idempotent; that is, ("f" ∘ "g")2 "f" ∘ ("g" ∘ "f") ∘ "g" "f" ∘ "g". The left inverse "g" is also called a retraction of "f".
Morphisms with left inverses are always monomorphisms, but the converse is not true in general; a monomorphism may fail to have a left inverse. In concrete categories, a function that has a left inverse is injective. Thus in concrete categories, monomorphisms are often, but not always, injective. The condition of being an injection is stronger than that of being a monomorphism, but weaker than that of being a split monomorphism.
Dually to monomorphisms, a morphism "f": "X" → "Y" is called an epimorphism if "g"1 ∘ "f" = "g"2 ∘ "f" implies "g"1 = "g"2 for all morphisms "g"1, "g"2: "Y" → "Z". An epimorphism can be called an "epi" for short, and we can use "epic" as an adjective. A morphism "f" has a right inverse or is a split epimorphism if there is a morphism "g": "Y" → "X" such that "f" ∘ "g" id"Y". The right inverse "g" is also called a section of "f". Morphisms having a right inverse are always epimorphisms, but the converse is not true in general, as an epimorphism may fail to have a right inverse.
If a monomorphism "f" splits with left inverse "g", then "g" is a split epimorphism with right inverse "f". In concrete categories, a function that has a right inverse is surjective. Thus in concrete categories, epimorphisms are often, but not always, surjective. The condition of being a surjection is stronger than that of being an epimorphism, but weaker than that of being a split epimorphism. In the category of sets, the statement that every surjection has a section is equivalent to the axiom of choice.
A morphism that is both an epimorphism and a monomorphism is called a bimorphism.
Isomorphisms.
A morphism "f": "X" → "Y" is called an isomorphism if there exists a morphism "g": "Y" → "X" such that "f" ∘ "g" = id"Y" and "g" ∘ "f" = id"X". If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so "f" is an isomorphism, and "g" is called simply the inverse of "f". Inverse morphisms, if they exist, are unique. The inverse "g" is also an isomorphism, with inverse "f". Two objects with an isomorphism between them are said to be isomorphic or equivalent. 
While every isomorphism is a bimorphism, a bimorphism is not necessarily an isomorphism. For example, in the category of commutative rings the inclusion Z → Q is a bimorphism that is not an isomorphism. However, any morphism that is both an epimorphism and a "split" monomorphism, or both a monomorphism and a "split" epimorphism, must be an isomorphism. A category, such as Set, in which every bimorphism is an isomorphism is known as a balanced category.
Endomorphisms and automorphisms.
A morphism "f": "X" → "X" (that is, a morphism with identical source and target) is an endomorphism of "X". A split endomorphism is an idempotent endomorphism "f" if "f" admits a decomposition "f" = "h" ∘ "g" with "g" ∘ "h" = id. In particular, the Karoubi envelope of a category splits every idempotent morphism.
An automorphism is a morphism that is both an endomorphism and an isomorphism. In every category, the automorphisms of an object always form a group, called the automorphism group of the object.
Examples.
For more examples, see the entry category theory.