Morphism

Inmathematics,amorphismis a concept ofcategory theorythat generalizes structure-preservingmapssuch ashomomorphismbetweenalgebraic structures,functionsfrom a set to another set, andcontinuous functionsbetweentopological 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 andobjectsare constituents of acategory.Morphisms, also calledmapsorarrows,relate two objects called thesourceand thetargetof 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 (associativityof composition when it is defined, and existence of anidentity morphismfor every object).

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

Definition

AcategoryCconsists of twoclasses,one ofobjectsand the other ofmorphisms.There are two objects that are associated to every morphism, thesourceand thetarget.AmorphismffromXtoYis a morphism with sourceXand targetY;it is commonly written asf:X→YorXf→Ythe latter form being better suited forcommutative diagrams.

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

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

Identity: For every objectX,there exists a morphismid_X:X→Xcalled theidentity morphismonX,such that for every morphismf:A→Bwe haveid_B∘f=f=f∘ id_A.
Associativity: h∘ (g∘f) = (h∘g) ∘fwhenever all the compositions are defined, i.e. when the target offis the source ofg,and the target ofgis 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 fromXtoYis denotedHom_C(X,Y)or simplyHom(X,Y)and called thehom-setbetweenXandY.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 objectsXandYis 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).

Some special morphisms

Monomorphisms and epimorphisms

A morphismf:X→Yis called amonomorphismiff∘g₁=f∘g₂impliesg₁=g₂for all morphismsg₁,g₂:Z→X.A monomorphism can be called amonofor short, and we can usemonicas an adjective.^[1]A morphismfhas aleft inverseor is asplit monomorphismif there is a morphismg:Y→Xsuch thatg∘f= id_X.Thusf∘g:Y→Yisidempotent;that is,(f∘g)²=f∘ (g∘f) ∘g=f∘g.The left inversegis also called aretractionoff.^[1]

Morphisms with left inverses are always monomorphisms, but theconverseis 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→Yis called anepimorphismifg₁∘f=g₂∘fimpliesg₁=g₂for all morphismsg₁,g₂:Y→Z.An epimorphism can be called anepifor short, and we can useepicas an adjective.^[1]A morphismfhas aright inverseor is asplit epimorphismif there is a morphismg:Y→Xsuch thatf∘g= id_Y.The right inversegis also called asectionoff.^[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 monomorphismfsplits with left inverseg,thengis 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.

Isomorphisms

A morphismf:X→Yis called anisomorphismif there exists a morphismg:Y→Xsuch thatf∘g= id_Yandg∘f= id_X.If a morphism has both left-inverse and right-inverse, then the two inverses are equal, sofis an isomorphism, andgis called simply theinverseoff.Inverse morphisms, if they exist, are unique. The inversegis also an isomorphism, with inversef.Two objects with an isomorphism between them are said to beisomorphicor equivalent.

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

Endomorphisms and automorphisms

A morphismf:X→X(that is, a morphism with identical source and target) is anendomorphismofX.Asplit endomorphismis an idempotent endomorphismfiffadmits a decompositionf=h∘gwithg∘h= id.In particular, theKaroubi envelopeof a category splits every idempotent morphism.

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

Examples

Foralgebraic structurescommonly 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 epimorphismsthat are not surjective (e.g., when embedding theintegersin therational numbers).
In thecategory of topological spaces,the morphisms are thecontinuous functionsand isomorphisms are calledhomeomorphisms.There arebijections(that is, isomorphisms of sets) that are not homeomorphisms.
In the category ofsmooth manifolds,the morphisms are thesmooth functionsand isomorphisms are calleddiffeomorphisms.
In the category ofsmall categories,the morphisms arefunctors.
In afunctor category,the morphisms arenatural transformations.

For more examples, seeCategory theory.

Notes

^^a ^b ^c ^dJacobson (2009), p. 15.

References

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).

External links

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

[jacobson:morphisms-1] Jacobson (2009), p. 15.

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