Morphism
Table of Contents
1. Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type.
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: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism.[1]
1.1. Definition
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.
Note that 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 locally small.
1.2. Some special morphisms
1.2.1. Monomorphisms and epimorphisms
1.2.2. Isomorphisms
1.2.3. Endomorphisms and automorphisms
1.3. 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 , the morphisms are functors. In a functor category, the morphisms are natural transformations.
2. Summary of Common Morphisms
- Homomorphism: structure preserving map between two algebraic structure.
- Isomorphism: Homomorphism that can be reversed. (or a Homomprohism with a one to one mapping)
- Endomorphism: Homomorphism from a mathematical object to itself.
- Automorphism: Isomorphism from a mathematical object to itself.
