# User Contributed Dictionary

### Noun

#### Translations

- Croatian: morfizam
- French: morphisme
- German: Morphismus
- Swedish: morfism

### See also

# Extensive Definition

In mathematics, a morphism is
an abstraction derived from structure-preserving mappings
between two mathematical
structures.

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 functions preserving this
structure. Nevertheless, morphisms are not necessarily functions,
and objects over which morphisms are defined are not necessarily
sets. Instead, a morphism is
often thought of as an arrow
linking an object called the domain to
another object called the codomain.
Hence morphisms do not so much map sets into sets, as embody a
relationship between some posited domain and codomain.

The notion of morphism recurs in much of
contemporary mathematics. In set theory,
morphisms are functions;
in topology, continuous
functions; in universal
algebra, homomorphisms; in group
theory, group
homomorphisms.

## Definition

There are two operations defined on every
morphism, the domain
(or source) and the codomain (or target).

If a morphism f has domain X and codomain Y, we
write f : X → Y. Thus a morphism is an arrow from its
domain to its codomain. The set 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. (Some authors write MorC(X,Y) or Mor(X, Y)).

For every three objects X, Y, and Z, there exists
a binary
operation hom(X, Y) × hom(Y, Z) → hom(X, Z)
called composition.
The composite of is written g o f, gf, or even fg. The composition
of morphisms is often represented by a commutative
diagram. For example,

Morphisms satisfy two axioms:

- Identity: for every object X, there exists a morphism idX : X → X called the identity morphism on X, such that for every morphism we have idB o f = f = f o idA.
- Associativity: h o (g o f) = (h o g) o f whenever the operations are defined.

When C is a concrete category, the identity
morphism is just the identity
function, and composition is just the ordinary composition
of functions. Associativity then follows, because the
composition of functions is associative.

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 specific morphisms

Monomorphism:
f : X → Y is called a monomorphism if f o g1 = f
o g2 implies g1 = g2 for all morphisms g1, g2 : Z → X.

It is also called a mono or a monic. The morphism
f has a left inverse if there is a morphism g:Y → X such that g o f
= idX. The left inverse g is also called a retraction of f.
Morphisms with left inverses are always monomorphisms, but the
converse is not always true in every category; a monomorphism may
fail to have a left-inverse.

A split monomorphism h : X → Y is a monomorphism
having a left inverse g : Y → X, so that g o h = idX. Thus h o g :
Y → Y is idempotent, so that (h o g)2
= h o g.

In concrete
categories, a function which has 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.

Epimorphism:
Dually, f : X → Y is called an epimorphism if g1 o f = g2 o
f implies g1 = g2 for all morphisms g1, g2 : Y → Z. It is also
called an epi or an epic. The morphism f has a right-inverse if
there is a morphism g : Y → X such that f o g = idY. The right
inverse g is also called a section of f. Morphisms having a right
inverse are always epimorphisms, but the converse is not always
true in every category, as an epimorphism may fail to have a right
inverse.

A split epimorphism is an epimorphism having a
right inverse.

In concrete
categories, a function which 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, every surjection has a section, a result equivalent to
the axiom of
choice.

Note that if a split monomorphism f has a
left-inverse g, then g is a split epimorphism and has right-inverse
f.

A bimorphism is a morphism that is both an
epimorphism and a monomorphism.

Isomorphism: f
: X → Y is called an isomorphism if there exists
a morphism g : Y → X such that f o g = idY and g o f = idX.

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.

Note that 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 which 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.

Endomorphism:
f : X → X is an endomorphism of X.

A split endomorphism is an idempotent
endomorphism f if f admits a decomposition f = h o g with g o 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.

## Examples

- In the concrete categories studied in universal algebra (groups, rings, modules, etc.), morphisms are called homomorphisms. Likewise, the notions of automorphism, endomorphism, epimorphism, homeomorphism, isomorphism, and monomorphism all find use in universal algebra.

- In the category of topological spaces, morphisms are continuous functions and isomorphisms are called homeomorphisms.

- In the category of smooth manifolds, morphisms are smooth functions and isomorphisms are called diffeomorphisms.

- In the category of small categories, functors can be thought of as morphisms.

- In a functor category, the morphisms are natural transformations.

For more examples, see the entry category
theory.

## See also

## External links

morphism in Catalan: Morfisme

morphism in German: Morphismus

morphism in Spanish: Morfismo

morphism in French: Morphisme

morphism in Korean: 사상 (범주론)

morphism in Italian: Morfismo

morphism in Japanese: 射 (圏論)

morphism in Portuguese: Morfismo (teoria das
categorias)

morphism in Swedish: Morfism

morphism in Chinese: 态射