Morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological 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 to function composition.
Morphisms and
Morphisms and categories recur in much of contemporary mathematics. Originally, they were introduced for
Definition
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 from X to Y is a morphism with source X and target Y; it is commonly written as f : X → Y or X Y the latter form being better suited for commutative diagrams.
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
- 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 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
The composition of morphisms is often represented by a commutative diagram. For example,
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). 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
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 special morphisms
Monomorphisms and epimorphisms
A morphism f : X → Y is called a
Morphisms with left inverses are always monomorphisms, but the
Dually to monomorphisms, a morphism f : X → Y is called an epimorphism if g1 ∘ f = g2 ∘ f implies g1 = g2 for all morphisms g1, g2 : Y → Z. An epimorphism can be called an epi for short, and we can use epic as an adjective.[1] A morphism f has a right inverse or is a split epimorphism if there is a morphism g : Y → X such that f ∘ g = idY. 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 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
A morphism that is both an epimorphism and a monomorphism is called a bimorphism.
Isomorphisms
A morphism f : X → Y is called an
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 a Set, in which every bimorphism is an isomorphism is known as a balanced category.
Endomorphisms and automorphisms
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.
Examples
- For ring epimorphisms that are not surjective (e.g., when embedding the integers in the rational numbers).
- In the category of topological spaces, the morphisms are the continuous functions and isomorphisms are called homeomorphisms. There are bijections (that is, isomorphisms of sets) that are not homeomorphisms.
- In the category of 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 Category theory.
See also
Notes
References
- 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]