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.

- Definition
- Some special morphisms
- Monomorphisms and epimorphisms
- Isomorphisms
- Endomorphisms and automorphisms
- Examples
- See also
- Notes
- References
- External links

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.^{ [1] }

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**.

A category *C* consists of two classes, one of *objects* and the other of *morphisms*. There are two objects that are associated to every morphism, the *source* and the *target*. A morphism *f* with source *X* and target *Y* is written *f* : *X* → *Y*, and is represented diagrammatically by an *arrow* 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 *domain* and *codomain* respectively.

Morphisms are equipped with a partial binary operation, called *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 *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:

- Identity
- 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}∘*f*=*f*=*f*∘ id_{A}. - Associativity
*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, 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. 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 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).

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.^{ [2] } 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*.^{ [2] }

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.^{ [2] } 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*.^{ [2] } 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**.

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**.

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.

- In the concrete categories studied in universal algebra (groups, rings, modules, etc.), morphisms are usually 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, the morphisms are functors.
- In a functor category, the morphisms are natural transformations.

For more examples, see the entry category theory.

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 mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

**Category theory** formalizes mathematical structure and its concepts in terms of a labeled directed graph called a *category*, whose nodes are called *objects*, and whose labelled directed edges are called *arrows*. A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

In mathematics, an **endomorphism** is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space *V* is a linear map *f*: *V* → *V*, and an endomorphism of a group *G* is a group homomorphism *f*: *G* → *G*. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set *S* to itself.

In mathematics, given two groups, and, a **group homomorphism** from to is a function *h* : *G* → *H* such that for all *u* and *v* in *G* it holds that

In algebra, a **homomorphism** is a structure-preserving map between two algebraic structures of the same type. The word *homomorphism* comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German *ähnlich* meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

In ring theory, a branch of abstract algebra, a **ring homomorphism** is a structure-preserving function between two rings. More explicitly, if *R* and *S* are rings, then a ring homomorphism is a function *f* : *R* → *S* such that *f* is

In mathematics, a **category** is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.

In the context of abstract algebra or universal algebra, a **monomorphism** is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation .

In category theory, an **epimorphism** is a morphism *f* : *X* → *Y* that is right-cancellative in the sense that, for all objects *Z* and all morphisms *g*_{1}, *g*_{2}: *Y* → *Z*,

An **exact sequence** is a sequence of morphisms between objects such that the image of one morphism equals the kernel of the next.

The **cokernel** of a linear mapping of vector spaces *f* : *X* → *Y* is the quotient space *Y* / im(*f*) of the codomain of f by the image of f. The dimension of the cokernel is called the *corank* of f.

In category theory, a branch of mathematics, **duality** is a correspondence between the properties of a category *C* and the dual properties of the opposite category *C*^{op}. Given a statement regarding the category *C*, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category *C*^{op}. Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about *C*, then its dual statement is true about *C*^{op}. Also, if a statement is false about *C*, then its dual has to be false about *C*^{op}.

In category theory, a branch of abstract mathematics, an **equivalence of categories** is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

In mathematics, the **category of topological spaces**, often denoted **Top**, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of **Top** and of properties of topological spaces using the techniques of category theory is known as **categorical topology**.

In mathematics, especially in the field of category theory, the concept of **injective object** is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.

**Fibred categories** are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which *inverse images* of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space *X* to another topological space *Y* is associated the pullback functor taking bundles on *Y* to bundles on *X*. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories with "descent". Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories.

This is a glossary of properties and concepts in category theory in mathematics.

In category theory, a branch of mathematics, a **section** is a right inverse of some morphism. Dually, a **retraction** is a left inverse of some morphism. In other words, if *f* : *X* → *Y* and *g* : *Y* → *X* are morphisms whose composition *f*o*g* : *Y* → *Y* is the identity morphism on *Y*, then *g* is a section of *f*, and *f* is a retraction of *g*.

In category theory, an abstract mathematical discipline, a **nodal decomposition** of a morphism is a representation of as a product , where is a strong epimorphism, a bimorphism, and a strong monomorphism.

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*Abstract and Concrete Categories*(PDF). John Wiley & Sons. ISBN 0-471-60922-6. Now available as free on-line edition (4.2MB PDF).

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