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{{Short description|Map (arrow) between two objects of a category}}
In [[Mathematics|mathematics]], particularly in [[Category theory|category theory]], a '''morphism''' is a structure-preserving [[Map (mathematics)|map]] from one [[Mathematical structure|mathematical structure]] to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In [[Set theory|set theory]], morphisms are functions; in [[Linear algebra|linear algebra]], linear transformations; in [[Group theory|group theory]], [[Group homomorphism|group homomorphism]]s; in [[Mathematical analysis|analysis]] and [[Topology|topology]], continuous functions, and so on.

In [[Category theory|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|function composition]]. A morphism in category theory is an abstraction of a [[Homomorphism|homomorphism]].<ref>{{cite web|url=https://ncatlab.org/nlab/show/morphism|access-date=2019-06-12|title=morphism|publisher=nLab}}</ref>

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 category|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 (mathematics)|category]] ''C'' consists of two [[Class (set theory)|classes]], one of {{em|objects}} and the other of {{em|morphisms}}. There are two objects that are associated to every morphism, the {{em|source}}  and the {{em|target}}. A ''morphism'' ''f'' ''from'' ''X'' ''to'' ''Y'' is a morphism with source ''X'' and target ''Y''; it is commonly written as {{nowrap|''f'' : ''X'' → ''Y''}} or {{nowrap|''X'' {{overset|lh=0.6em|''f''|→}} ''Y''}} the latter form being better suited for [[Commutative diagram|commutative diagram]]s.

For many common categories, objects are [[Set (mathematics)|sets]] (often with some additional structure) and morphisms are [[Function (mathematics)|functions]] from an object to another object. Therefore, the source and the target of a morphism are often called {{em|{{visible anchor|domain|Domain|Domain of a morphism}}}} and {{em|{{visible anchor|codomain|Codomain|Codomain of a morphism}}}} respectively.

Morphisms are equipped with a partial binary operation, called {{em|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 {{nowrap|''g'' ∘ ''f''}} (or sometimes simply ''gf''). The source of {{nowrap|''g'' ∘ ''f''}} is the source of ''f'', and the target of {{nowrap|''g'' ∘ ''f''}} is the target of ''g''. The composition satisfies  two [[Axiom|axiom]]s:
; {{vanchor|Identity}} : For every object ''X'', there exists a morphism {{nowrap|id<sub>''X''</sub> : ''X'' → ''X''}} called the '''identity morphism''' on ''X'', such that for every morphism {{nowrap|''f'' : ''A'' → ''B''}} we have {{nowrap|1=id<sub>''B''</sub> ∘ ''f'' = ''f'' = ''f'' ∘ id<sub>''A''</sub>}}.
; Associativity : {{nowrap|1=''h'' ∘ (''g'' ∘ ''f'') = (''h'' ∘ ''g'') ∘ ''f''}}  whenever all the compositions are defined, i.e. when the target of ''f'' is the source of ''g'', and the target of ''g'' is the source of ''h''.

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|identity function]], and composition is just ordinary composition of functions.

The composition of morphisms is often represented by a [[Commutative diagram|commutative diagram]]. For example,
: [[File:Commutative diagram for morphism.svg]]

{{Anchor|Hom-set}}
The collection of all morphisms from ''X'' to ''Y'' is denoted {{nowrap|Hom<sub>''C''</sub>(''X'', ''Y'')}} or simply {{nowrap|Hom(''X'', ''Y'')}} and called the '''hom-set''' between ''X'' and ''Y''. Some authors write {{nowrap|Mor<sub>''C''</sub>(''X'', ''Y'')}}, {{nowrap|Mor(''X'', ''Y'')}} or {{nowrap|C(''X'', ''Y'')}}. The term hom-set is something of a misnomer, as the collection of morphisms is not required to be a set; a category where {{nowrap|Hom(''X'', ''Y'')}} is a set for all objects ''X'' and ''Y'' is called locally small. Because hom-sets may not be sets, some people prefer to use the term "hom-class".

The domain and codomain are in fact part of the information determining a morphism. For example, in the [[Category of sets|category of sets]], where morphisms are functions, two functions may be identical as sets of ordered pairs (may have the same [[Range of a function|range]]), while having different codomains. The two functions are distinct from the viewpoint of category theory. Thus many authors require that the hom-classes {{nowrap|Hom(''X'', ''Y'')}} be [[Disjoint sets|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).

== <span class="anchor" id="Some specific morphisms"></span> Some special morphisms ==

=== Monomorphisms and epimorphisms ===
A morphism {{nowrap|''f'' : ''X'' → ''Y''}} is called a [[Monomorphism|monomorphism]] if {{nowrap|1=''f'' ∘ ''g''<sub>1</sub> = ''f'' ∘ ''g''<sub>2</sub>}} implies {{nowrap|1=''g''<sub>1</sub> = ''g''<sub>2</sub>}} for all morphisms ''g''<sub>1</sub>, {{nowrap|''g''<sub>2</sub> : ''Z'' → ''X''}}. A monomorphism can be called a ''mono'' for short, and we can use ''monic'' as an adjective.<ref name="jacobson:morphisms">Jacobson (2009), p. 15.</ref> A morphism ''f'' has a '''left inverse''' or is a '''split monomorphism''' if there is a morphism {{nowrap|''g'' : ''Y'' → ''X''}} such that {{nowrap|1=''g'' ∘ ''f'' = id<sub>''X''</sub>}}. Thus {{nowrap|''f'' ∘ ''g'' : ''Y'' → ''Y''}} is idempotent; that is, {{nowrap|1=(''f'' ∘ ''g'')<sup>2</sup> = ''f'' ∘ (''g'' ∘ ''f'') ∘ ''g'' = ''f'' ∘ ''g''}}. The left inverse ''g'' is also called a '''[[Section (category theory)|retraction]]''' of ''f''.<ref name="jacobson:morphisms" />

Morphisms with left inverses are always monomorphisms, but the [[Philosophy:Converse (logic)|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 {{nowrap|''f'' : ''X'' → ''Y''}} is called an [[Epimorphism|epimorphism]] if {{nowrap|1=''g''<sub>1</sub> ∘ ''f'' = ''g''<sub>2</sub> ∘ ''f''}} implies {{nowrap|1=''g''<sub>1</sub> = ''g''<sub>2</sub>}} for all morphisms ''g''<sub>1</sub>, {{nowrap|''g''<sub>2</sub> : ''Y'' → ''Z''}}. An epimorphism can be called an ''epi'' for short, and we can use ''epic'' as an adjective.<ref name="jacobson:morphisms" /> A morphism ''f'' has a '''right inverse''' or is a '''split epimorphism''' if there is a morphism {{nowrap|''g'' : ''Y'' → ''X''}} such that {{nowrap|1=''f'' ∘ ''g'' = id<sub>''Y''</sub>}}. The right inverse ''g'' is also called a '''section''' of ''f''.<ref name="jacobson:morphisms" />  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|category of sets]], the statement that every surjection has a section is equivalent to the [[Axiom of choice|axiom of choice]].

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

=== Isomorphisms ===
A morphism {{nowrap|''f'' : ''X'' → ''Y''}} is called an [[Isomorphism|isomorphism]] if there exists a morphism {{nowrap|''g'' : ''Y'' → ''X''}} such that {{nowrap|1=''f'' ∘ ''g'' = id<sub>''Y''</sub>}} and {{nowrap|1=''g'' ∘ ''f'' = id<sub>''X''</sub>}}. 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 ring|commutative ring]]s the inclusion {{nowrap|'''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 a '''Set''', in which every bimorphism is an isomorphism is known as a '''balanced category'''.

=== Endomorphisms and automorphisms ===
A morphism {{nowrap|''f'' : ''X'' → ''X''}} (that is, a morphism with identical source and target) is an [[Endomorphism|endomorphism]] of ''X''. A '''split endomorphism''' is an idempotent endomorphism ''f'' if ''f'' admits a decomposition {{nowrap|1=''f'' = ''h'' ∘ ''g''}} with {{nowrap|1=''g'' ∘ ''h'' = id}}.  In particular, the [[Karoubi envelope]] of a category splits every idempotent morphism.

An [[Automorphism|automorphism]] is a morphism that is both an endomorphism and an isomorphism.  In every category, the automorphisms of an object always form a [[Group (mathematics)|group]], called the automorphism group of the object.

== Examples ==
* For [[Algebraic structure|algebraic structure]]s commonly considered in [[Algebra|algebra]], such as [[Group (mathematics)|groups]], rings, [[Module (mathematics)|modules]], etc., the morphisms are usually the [[Homomorphism|homomorphism]]s, and the notions of isomorphism, automorphism, endomorphism, epimorphism, and monomorphism are the same as the above defined ones. However, in the case of rings, "epimorphism" is often considered as a synonym of "surjection", although there are ring epimorphisms that are not surjective (e.g., when embedding the [[Integer|integer]]s in the [[Rational number|rational number]]s).
* In the [[Category of topological spaces|category of topological spaces]], the morphisms are the [[Continuous function|continuous function]]s and isomorphisms are called [[Homeomorphism|homeomorphism]]s. There are [[Bijection|bijection]]s (that is, isomorphisms of sets) that are not homeomorphisms.
* In the category of smooth manifolds, the morphisms are the smooth functions and isomorphisms are called [[Diffeomorphism|diffeomorphism]]s.
* In the category of small categories, the morphisms are [[Functor|functor]]s.
* In a [[Functor category|functor category]], the morphisms are [[Natural transformation|natural transformation]]s.

For more examples, see [[Category theory]].

== See also ==
* [[Normal morphism]]
* [[Zero morphism]]

== Notes ==
{{Reflist}}

== References ==
* {{citation| last=Jacobson| first=Nathan|  date=2009| title=Basic algebra| edition=2nd| volume = 2 | publisher=Dover| isbn = 978-0-486-47187-7}}.
* {{cite book | last1 = Adámek | first1 = Jiří | first2 = Horst | last2 = Herrlich | first3 = George E. | last3 = Strecker  | year = 1990 | url = http://katmat.math.uni-bremen.de/acc/acc.pdf | title = Abstract and Concrete Categories | publisher = John Wiley & Sons | isbn = 0-471-60922-6}} Now available as free on-line edition (4.2MB PDF).

== External links ==
* {{springer|title=Morphism|id=p/m064940}}

{{Category theory}}

[[Category:Morphisms| ]]

{{Sourceattribution|Morphism}}