In

_{''X''} : ''X'' → ''X'' called the identity morphism on ''X'', such that for every morphism we have id_{''B''} ∘ ''f'' = ''f'' = ''f'' ∘ id_{''A''}.
;_{''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. Because hom-sets may not be sets, some people prefer to use the term "hom-class".
Note that the domain and codomain are in fact part of the information determining a morphism. For example, in the

_{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.Jacobson (2009), p. 15. 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

_{''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

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, particularly in category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...

, 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
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...

, morphisms are functions; in linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mat ...

, linear transformations; in group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, group homomorphism
In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...

s; in topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, continuous functions, and so on.
In category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...

, ''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
In mathematics, function composition is an operation that takes two function (mathematics), functions and , and produces a function such that . In this operation, the function is function application, applied to the result of applying the ...

. A morphism in category theory is an abstraction of a homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

.
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

Acategory
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
*Categories (Aristotle), ''Categories'' (Aristotle)
*Category (Kant)
...

''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 axiom
An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...

s:
;: For every object ''X'', there exists a morphism idAssociativity
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

: ''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
Graph of the identity function on the real numbers
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantitie ...

, and composition is just ordinary composition of functions.
The composition of morphisms is often represented by a commutative diagram
image:5 lemma.svg, 350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a Diagram (category theory), diagram such that all directed paths in the diagram wit ...

. For example,
:
The collection of all morphisms from ''X'' to ''Y'' is denoted Homcategory of sets
In the mathematical field of category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundatio ...

, 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''category of sets
In the mathematical field of category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundatio ...

, the statement that every surjection has a section is equivalent to the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

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

A morphism ''f'': ''X'' → ''Y'' is called anisomorphism
In mathematics, an isomorphism is a structure-preserving Map (mathematics), mapping between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are iso ...

if there exists a morphism ''g'': ''Y'' → ''X'' such that ''f'' ∘ ''g'' = idcommutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring proper ...

s 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 a 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 anendomorphism
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

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
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of map (mathematics), mapping the object to itself while preserving all of its structure. The Set (m ...

is a morphism that is both an endomorphism and an isomorphism. In every category, the automorphisms of an object always form a group
A group is a number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...

, called the automorphism group
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...

of the object.
Examples

* Foralgebraic structure
In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ...

s commonly considered in algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

, such as groups, rings
Ring may refer to:
* Ring (jewellery)
A ring is a round band, usually made of metal, worn as ornamental jewelry. The term "ring" by itself always denotes jewellery worn on the finger; when worn as an ornament elsewhere, the body part is specifi ...

, modules, etc., the morphisms are usually the homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

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
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

", although there are ring epimorphisms that are not surjective (e.g., when embedding the integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

s in the rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s).
* In the category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category (category theory), category whose object (category theory), objects are topological spaces and whose morphisms are continuous maps. This is a category because th ...

, the morphisms are the continuous function
In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value ...

s and isomorphisms are called homeomorphism
In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...

s. There are bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...

s (that is, isomorphisms of sets) that are not homeomorphisms.
* In the category of smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spa ...

s, the morphisms are the smooth function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...

s and isomorphisms are called diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...

s.
* In the category of small categories, the morphisms are functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

s.
* In a functor category In category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topolo ...

, the morphisms are natural transformation
In category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topo ...

s.
For more examples, see Category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...

.
See also

*Normal morphism
In category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topol ...

* Zero morphism In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object.
Definitions
Suppose C is a Category (mathematics), category, and ''f'' : ''X'' → ''Y'' ...

Notes

References

* . * Now available as free on-line edition (4.2MB PDF).External links

* {{Authority control