Inmathematics, amorphism is a concept ofcategory theory that generalizes structure-preservingmaps such ashomomorphism betweenalgebraic structures,functions from a set to another set, andcontinuous functions betweentopological spaces. Although many examples of morphisms are structure-preserving maps, morphisms need not to be maps, but they can be composed in a way that is similar tofunction composition.

Morphisms andobjects are constituents of acategory. Morphisms, also calledmaps orarrows, relate two objects called thesource and thetarget of the morphism. There is apartial operation, calledcomposition, on the morphisms of a category that is defined if the target of the first object equals the source of the second object. The composition of morphisms behave like function composition (associativity of composition when it is defined, and existence of anidentity morphism for every object).

Morphisms and categories recur in much of contemporary mathematics. Originally, they were introduced forhomological algebra andalgebraic topology. They belong to the foundational tools ofGrothendieck'sscheme theory, a generalization ofalgebraic geometry that applies also toalgebraic number theory.

AcategoryC consists of twoclasses, one ofobjects and the other ofmorphisms. There are two objects that are associated to every morphism, thesource and thetarget. AmorphismffromXtoY is a morphism with sourceX and targetY; it is commonly written asf :X →Y orXf→Y the latter form being better suited forcommutative diagrams.

For many common categories, objects aresets (often with some additional structure) and morphisms arefunctions from an object to another object. Therefore, the source and the target of a morphism are often calleddomain andcodomain respectively.

Morphisms are equipped with apartial binary operation, calledcomposition. The composition of two morphismsf andg is defined precisely when the target off is the source ofg, and is denotedg ∘f (or sometimes simplygf). The source ofg ∘f is the source off, and the target ofg ∘f is the target ofg. The composition satisfies twoaxioms:

Identity

For every objectX, there exists a morphismid_X :X →X called theidentity morphism onX, such that for every morphismf :A →B we haveid_B ∘f =f =f ∘ id_A.

Associativity

h ∘ (g ∘f) = (h ∘g) ∘f whenever all the compositions are defined, i.e. when the target off is the source ofg, and the target ofg is the source ofh.

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 theidentity function, and composition is just ordinarycomposition of functions.

The composition of morphisms is often represented by acommutative diagram. For example,

The collection of all morphisms fromX toY is denotedHom_C(X,Y) or simplyHom(X,Y) and called thehom-set betweenX andY. Some authors writeMor_C(X,Y),Mor(X,Y) orC(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 whereHom(X,Y) is a set for all objectsX andY is calledlocally 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 thecategory of sets, where morphisms are functions, two functions may be identical as sets of ordered pairs (may have the samerange), while having different codomains. The two functions are distinct from the viewpoint of category theory. Thus many authors require that the hom-classesHom(X,Y) bedisjoint. 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 morphismf :X →Y is called amonomorphism iff ∘g₁ =f ∘g₂ impliesg₁ =g₂ for all morphismsg₁,g₂ :Z →X. A monomorphism can be called amono for short, and we can usemonic as an adjective.^[1] A morphismf has aleft inverse or is asplit monomorphism if there is a morphismg :Y →X such thatg ∘f = id_X. Thusf ∘g :Y →Y isidempotent; that is,(f ∘g)² =f ∘ (g ∘f) ∘g =f ∘g. The left inverseg is also called aretraction off.^[1]

Morphisms with left inverses are always monomorphisms, but theconverse is not true in general; a monomorphism may fail to have a left inverse. Inconcrete categories, a function that has a left inverse isinjective. 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 morphismf :X →Y is called anepimorphism ifg₁ ∘f =g₂ ∘f impliesg₁ =g₂ for all morphismsg₁,g₂ :Y →Z. An epimorphism can be called anepi for short, and we can useepic as an adjective.^[1] A morphismf has aright inverse or is asplit epimorphism if there is a morphismg :Y →X such thatf ∘g = id_Y. The right inverseg is also called asection off.^[1] 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 monomorphismf splits with left inverseg, theng is a split epimorphism with right inversef. Inconcrete categories, a function that has a right inverse issurjective. 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 thecategory of sets, the statement that every surjection has a section is equivalent to theaxiom of choice.

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

A morphismf :X →Y is called anisomorphism if there exists a morphismg :Y →X such thatf ∘g = id_Y andg ∘f = id_X. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, sof is an isomorphism, andg is called simply theinverse off. Inverse morphisms, if they exist, are unique. The inverseg is also an isomorphism, with inversef. Two objects with an isomorphism between them are said to beisomorphic or equivalent.

While every isomorphism is a bimorphism, a bimorphism is not necessarily an isomorphism. For example, in the category ofcommutative rings the inclusionZ →Q is a bimorphism that is not an isomorphism. However, any morphism that is both an epimorphism and asplit monomorphism, or both a monomorphism and asplit epimorphism, must be an isomorphism. A category, such as aSet, in which every bimorphism is an isomorphism is known as abalanced category.

A morphismf :X →X (that is, a morphism with identical source and target) is anendomorphism ofX. Asplit endomorphism is an idempotent endomorphismf iff admits a decompositionf =h ∘g withg ∘h = id. In particular, theKaroubi envelope of a category splits every idempotent morphism.

Anautomorphism is a morphism that is both an endomorphism and an isomorphism. In every category, the automorphisms of an object always form agroup, called theautomorphism group of the object.

• Foralgebraic structures commonly considered inalgebra, such asgroups,rings,modules, etc., the morphisms are usually thehomomorphisms, 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 arering epimorphisms that are not surjective (e.g., when embedding theintegers in therational numbers).
• In thecategory of topological spaces, the morphisms are thecontinuous functions and isomorphisms are calledhomeomorphisms. There arebijections (that is, isomorphisms of sets) that are not homeomorphisms.
• In the category ofsmooth manifolds, the morphisms are thesmooth functions and isomorphisms are calleddiffeomorphisms.
• In the category ofsmall categories, the morphisms arefunctors.
• In afunctor category, the morphisms arenatural transformations.

For more examples, seeCategory theory.

• Normal morphism
• Zero morphism

• Jacobson, Nathan (2009),Basic algebra, vol. 2 (2nd ed.), Dover,ISBN 978-0-486-47187-7.
• Adámek, Jiří; Herrlich, Horst; Strecker, George E. (1990).Abstract and Concrete Categories(PDF). John Wiley & Sons.ISBN 0-471-60922-6. Now available as free on-line edition (4.2MB PDF).

• "Morphism",Encyclopedia of Mathematics,EMS Press, 2001 [1994]