Hom functor

In category theory refers to (or simply, if the reference to the category is clear or or ) the set of homomorphisms (or morphisms ) from an object to an object of a category and is one of the fundamental data of a category. The relevant figure is the Hom - functor to the category.

For example, if the objects of the category of " quantities with additional properties " exist (eg, groups, topological spaces ), then the corresponding morphisms i a precisely with these properties acceptable pictures (eg, group homomorphisms, continuous maps ).

Hom functor as

However, it is also regarded as mapping that assigns to each pair of objects a lot. However, it has even more: If a morphism, ie an element of, so you can assign any of the homomorphism and obtains a mapping

Similarly, we obtain a homomorphism a picture

By on maps. Combined yields a figure

One easily verifies the following properties:

• Where so denotes the identity of the respective object.
• As far as the links are defined (ie corresponding definition and target areas coincide ).

In the category theoretical language can express this using the concepts of the dual category and product category as follows:

Is a functor from the category Set of sets. Note: Objects of pairs of objects, morphisms are pairs of morphisms from to, where and is, and it is, if defined.

In particular, we have then to a fixed object a covariant functor and a contravariant functor from on the set, the so-called partial Hom - functors.

Compatibility with additional structures

In general, only an amount ( if the category is locally small) and itself carries an additional structure not automatically apart about the fact that the linear operators form a monoid with the neutral element at composition. However, if for example, the properties of abelian groups or R-modules for a ring R such homomorphisms can pointwise added and / or multiplied by elements of R, and thus then itself forms an abelian group or an R- module. It then checks immediately that the above- defined mappings are hereby tolerated and that therefore in these cases the R-modules can be understood even as a functor into the category Ab of abelian groups and the category R -Mod.

Depending on the observed category are more such additional structures possible. That is, as an object of a category which is not necessarily the category of sets, conceived. Generally one speaks of an enriched over a category category (also: category) if the Hom - functor is a functor in the category and has a certain compatibility, which can be chosen differently, for example with a selected monoidal structure. Each locally small category enriched over the category of sets with cartesian product as monoidaler structure. A präadditive category is enriched over the category of abelian groups with the usual tensor category.

Even on very simple categories whose objects are not sets, one can accumulate. The category has two objects and an interesting arrow between objects in addition to the identities. She has finite products as monoidale structure. Under this, a category is a quasi-ordering. The quasi-ordering can be sum ( "") or maximum generation ("") are equipped as monoidale structure. Is obtained as Categories generalized metric spaces, and as categories quantities with generalized ultrametric. ( The generalization is that symmetry is not required and dots must not be identical with a pitch of zero. )

Applications

In the investigation of abelian categories and the Ext- functor the derived functor Hom to plays an important role.

• Algebra
• Category theory