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In
mathematics, a '''morphism''' is an abstraction of a
function or
mapping between two spaces. The word can mean different things depending on the type of space in question. In
set theory, for example, morphisms are just functions, in
group theory they are
group homomorphisms, while in
topology they are
continuous functions. In the context of
universal algebra morphisms are generically known as
homomorphisms.
The abstract study of morphisms and the spaces (or objects) on which they are defined forms a branch of mathematics called
category theory. In category theory, morphisms need not be functions at all and are usually thought as ''arrows'' between two different objects (which need not be sets). Rather than mapping elements of one set to another they simply represent some sort of relationship between the domain and codomain.
Despite the abstract nature of morphisms, most people's intution about them (and indeed much of the terminology) comes from the case of the so-called
concrete categories where the objects are simply sets with some additional structure and morphisms are functions preserving this structure.
== Definition ==
A
category ''C'' is given by two pieces of data: a
class of ''objects'' and, for any two objects ''X'' and ''Y'', a
set of ''morphisms'' from ''X'' to ''Y''. Morphisms are often depicted as arrows between those objects, e.g. a morphism ''f'' from ''X'' to ''Y'' is denoted ''f'' : ''X'' → ''Y''. The set of all morphisms from ''X'' to ''Y'' is denoted Mor
''C''(''X'',''Y'') or sometimes Hom
''C''(''X'',''Y'').
For every three objects ''X'', ''Y'', and ''Z'' there exists a
binary operation Mor(''X'',''Y'') × Mor(''Y'',''Z'') → Mor(''X'',''Z'') called '''composition'''. The composition of ''f'' : ''X'' → ''Y'' and ''g'' : ''Y'' → ''Z'' is written as ''g''
O ''f'' or ''gf'' (Some authors write it as ''fg''). Composition of morphisms is often denoted by means of a
commutative diagram. For example,
Morphisms must satisfy two
axioms:
* I
DENTITY: for every object ''X'' there exists a morphism id
''X'' : ''X'' → ''X'' called the '''identity morphism''' on ''X'', such that for every morphism ''f'' : ''A'' → ''B'' we have id
''B'' O ''f'' = ''f'' = ''f''
O id
''A''
* A
SSOCIATIVITY: ''h''
O (''g''
O ''f'') = (''h''
O ''g'')
O ''f'' whenever the operations are defined.
When ''C'' is a concrete category, composition is just ordinary
composition of functions, the identity morphism is just the
identity function, and associativity is automatic (functional composition is associative by definition).
There are two operations defined on every morphism, the '''
domain''' (or '''source''') which assigns the morphism ''f'' : ''X'' → ''Y'' the object ''X'':
:dom(''f'') = ''X'',
and the '''
codomain''' (or '''target''') which assigns the morphism ''f'' : ''X'' → ''Y'' the object ''Y'':
:cod(''f'') = ''Y''.
== Types of morphisms ==
* Let ''f'' : ''X'' → ''Y'' be a morphism. If there exists a morphism ''g'' : ''Y'' → ''X'' such that ''f''
O ''g'' = id
''Y'' and ''g''
O ''f'' = id
''X'' then ''f'' is called an '''
isomorphism''' and ''g'' is said to be an '''inverse morphism''' of ''f''. Inverse morphisms, if they exist, are unique. It is easy to see that ''g'' is also an isomorphism with inverse ''f''. Two objects with an isomorphism between them are said to be
isomorphic or equivalent. Isomorphisms are the most important kinds of morphisms in category theory.
* A morphism ''f'' : ''X'' → ''Y'' is called an '''
epimorphism''' if ''g''
1 O ''f'' = ''g''
2 O ''f'' implies ''g''
1 = ''g''
2 for all morphisms ''g''
1, ''g''
2 : ''Y'' → ''Z''. It is also called an ''epi'' or an ''epic''. Epimorphisms in concrete categories are typically
surjective morphisms.
* A morphism ''f'' : ''X'' → ''Y'' is called an '''
monomorphism''' if ''f''
O ''g''
1 = ''f''
O ''g''
2 implies ''g''
1 = ''g''
2 for all morphisms ''g''
1, ''g''
2 : ''Z'' → ''X''. It is also called a ''mono'' or a ''monic''. Monomorphisms in concrete categories are typically
injective morphisms.
* If ''f'' is both an epimorphism and a monomorphism then ''f'' is called a '''bimorphism'''. Note that every isomorphism is bimorphism but, in general, ''not'' every bimorphism is an isomorphism.
* Any morphism ''f'' : ''X'' → ''X'' is called an '''
endomorphism''' of ''X''.
* An endomorphism that is also an isomorphism is called an '''
automorphism'''.
* If ''f'' : ''X'' → ''Y'' and ''g'' : ''Y'' → ''X'' satisfy ''f''
O ''g'' = id
''Y'' one can show that ''f'' is epic and ''g'' is monic and that ''g''
O ''f'' : ''X'' → ''X'' is
idempotent. In this case ''f'' and ''g'' are said to be '''split'''. ''f'' is called a '''retraction''' of ''g'' and ''g'' is called a '''section''' of ''f''. Any morphism that is both an epimorphism and a split monomorphism, or both a monomorphism and a split epimorphism, must be an isomorphism.
''See also'':
* '''
zero morphisms'''
* '''
normal morphisms'''
== Examples ==
* In the concrete categories studied in
universal algebra (such as those of
groups,
rings,
modules, etc.), morphisms are called
homomorphisms. The terms isomorphism, epimorphism, monomorphism, endomorphism, and automorphism are all used in that specialized context as well.
* 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.
*
Functors can be thought of as morphisms in the category of
small categories.
* In a
functor category the morphisms are
natural transformations.
For more examples see the article on
category theory.
es:Morfismo
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