Morphism

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In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures.

The most common example occurs when the process is a function or map which preserves the structure in some sense. In set theory, for example, morphisms are just functions; in group theory they are group homomorphisms; while in topology they are continuous functions. In the context of universal algebra morphisms are generically known as homomorphisms.

The abstract study of morphisms and the structures (or objects) between which they are defined forms part of category theory. In category theory, morphisms need not be functions at all and are usually thought as arrows between two different objects (which need not be sets). Rather than mapping elements of one set to another they simply represent some sort of relationship between the domain and codomain.

Despite the abstract nature of morphisms, most people's intuition about them (and indeed much of the terminology) comes from the case of concrete categories where the objects are simply sets with some additional structure and morphisms are functions preserving this structure.

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Definition

A category C is given by two pieces of data: a class of objects and a class of morphisms.

There are two operations defined on every morphism, the domain (or source) and the codomain (or target).

Morphisms are often depicted as arrows from their domain to their codomain, e.g. if a morphism f has domain X and codomain Y, it is denoted f : X → Y. The set 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) or Mor(X, Y)).

For every three objects X, Y, and Z, there exists a binary operation hom(X, Y) × hom(Y, Z) → hom(X, Z) called composition. The composite of f : X → Y and g : Y → Z is written <math>g\circ f</math> or gf (Some authors write it as fg.) Composition of morphisms is often denoted by means of a commutative diagram. For example,

Image:MorphismComposition-01.png

Morphisms must satisfy 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 <math>{\rm id}_B\circ f=f=f\circ{\rm id}_A</math>.
ASSOCIATIVITY: <math>h\circ(g\circ f)=(h\circ g)\circ f</math> whenever the operations are defined.

When C is a concrete category, composition is just ordinary composition of functions, the identity morphism is just the identity function, and associativity is automatic. (Functional composition is associative.)

Note that the domain and codomain are really part of the information determining the morphism. For example, in the category of sets, where morphisms are functions, two functions may be identical as set of ordered pairs, but have different codomains. These functions are considered distinct for the purposes of category theory. For this reason, many authors require that the hom-classes hom(X, Y) be disjoint. In practice, this is not a problem, because if they are not disjoint, the domain and codomain can be appended to the morphisms, (say, as the second and third components of an ordered triple), making them disjoint.

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Some remarkable morphisms

A morphism f : X → Y is called a monomorphism if <math>f\circ g_1=f\circ g_2</math> implies g₁ = g₂ for all morphisms g₁, g₂ : Z → X. It is also called a mono or a monic. The morphism f has left-inverse if there is a morphism g:Y → X such that <math>g\circ f={\rm id}_X</math>. The left-inverse g is also called a retraction of f. Morphisms with left-inverses are always monomorphisms, but the converse is not always true in every category; a monomorphism may fail to have a left-inverse. A monomorphism which does have a left-inverse is called a split monomorphism. In concrete categories, a function which has 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, a morphism f : X → Y is called an epimorphism if <math>g_1\circ f=g_2\circ f</math> implies g₁ = g₂ for all morphisms g₁, g₂ : Y → Z. It is also called an epi or an epic. The morphism f has right-inverse if there is a morphism g:Y → X such that <math>f\circ g={\rm id}_Y</math>. The right-inverse g is also called a section of f. Morphisms with right-inverse are always epimorphisms, but the converse is not always true in every category; an epimorphism may fail to have a right-inverse. An epimorphism which does have a right-inverse is called a split epimorphism. In concrete categories, a function which has 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, every surjection has a section. This result is equivalent to the axiom of choice.
Note that if a split monomorphism f has a left-inverse g, then g is a split epimorphism and has right-inverse f.
A morphism which 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 <math>f\circ g={\rm id}_Y</math> and <math>g\circ f={\rm id}_X</math>. 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. <p>Note that every isomorphism is a bimorphism but, in general, not every bimorphism is an isomorphism. For example, in the category of commutative rings the inclusion Z → Q is a bimorphism which 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 in which every bimorphism is an isomorphism is a balanced category. For example, Set is a balanced category.
Any morphism f : X → X is called an endomorphism of X.
An endomorphism that is also an isomorphism is called an automorphism.
If a split monomorphism h : X → Y has left-inverse g : Y → X, so that <math>g\circ h={\rm id}_X</math>, then <math>h\circ g:\, Y\to Y</math> is idempotent, which means that <math>(h\circ g)^2=h\circ g</math>. More generally, any idempotent endomorphism f is said to be split if it admits a decomposition <math>f=h\circ g</math> with <math>g\circ h=\mathrm{id}</math>. In particular, the Karoubi envelope of a category splits every idempotent.

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See also:

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Examples

In the concrete categories studied in universal algebra (such as those of groups, rings, modules, etc.), morphisms are called homomorphisms. The terms isomorphism, epimorphism, monomorphism, endomorphism, and automorphism are all used in that specialized context as well.