Morphism

In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. 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.

In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map.

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

A category C consists of two classes, one of objects and the other of morphisms.

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

If a morphism f has domain X and codomain Y, we write f : X → Y. Thus a morphism is represented by an arrow from its domain to its codomain. The collection of all morphisms from X to Y is denoted homC(X,Y) or simply hom(X, Y) and called the hom-set between X and Y. Some authors write MorC(X,Y), Mor(X, Y) or C(X, Y). Note that the term hom-set is a bit 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.

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 g ∘ f or gf. The composition of morphisms is often represented by a commutative diagram.

Morphisms satisfy two axioms:

Identity: for every object X, there exists a morphism idX : X → X called the identity morphism on X, such that for every morphism f : A → B we have idB ∘ f = f = f ∘ idA.
Associativity: h ∘ (g ∘ f) = (h ∘ g) ∘ f whenever the operations are defined.

When C is a concrete category, the identity morphism is just the identity function, and composition is just the ordinary composition of functions. Associativity then follows, because the composition of functions is associative.

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).

Some specific morphisms

• Monomorphism: f : X → Y is called a monomorphism if f ∘ g₁ = f ∘ g₂ implies g₁ = g₂ for all morphisms g₁, g₂ : Z → X. It is also called a mono or a monic.^[1]
  • The morphism f has a left inverse if there is a morphism g:Y → X such that g ∘ f = id_X. The left inverse g is also called a retraction of f.^[1] 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 split monomorphism h : X → Y is a monomorphism having a left inverse g : Y → X, so that g ∘ h = id_X. Thus h ∘ g : Y → Y is idempotent, so that (h ∘ g)² = h ∘ g.
  • 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.
• Epimorphism: Dually, f : X → Y is called an epimorphism if g₁ ∘ f = g₂ ∘ f implies g₁ = g₂ for all morphisms g₁, g₂ : Y → Z. It is also called an epi or an epic.^[1]
  • The morphism f has a right-inverse 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.^[1] Morphisms having a right inverse are always epimorphisms, but the converse is not always true in every category, as an epimorphism may fail to have a right inverse.
  • A split epimorphism is an epimorphism having a right inverse. Note that 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, every surjection has a section, a result equivalent to the axiom of choice.
• A bimorphism is a morphism that is both an epimorphism and a monomorphism.
• Isomorphism: 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. Note that 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.
• Endomorphism: f : X → X 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.

Examples

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, functors can be thought of as morphisms.

In a functor category, the morphisms are natural transformations.

For more examples, see the entry category theory.
See also

Normal morphism
Zero morphism

Notes

Jacobson (2009), p. 15.

References

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