Hom functor

In category theory denotes (or simply , if the relation to the category is clear, or also or ) the set of homomorphisms (or morphisms ) from an object to an object of a category and is therefore one of the basic data of a category. The respective mapping is the Hom functor for the category . ${\displaystyle \operatorname {Hom} _{C}(A,B)}$ ${\displaystyle \operatorname {Hom} (A,B)}$ ${\displaystyle \operatorname {Mor} _{C}(A,B)}$ ${\displaystyle C(A,B)}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle \operatorname {Hom} _{C}}$ ${\displaystyle C}$

If, for example, the objects of the category consist of “sets with additional properties” (e.g. groups , topological spaces ), the associated morphisms are generally exactly the mappings compatible with these properties (e.g. group homomorphisms, continuous mappings).

Hom as functor

However, it can also be viewed as a mapping that assigns a set to each pair of -objects . But you have even more: If a morphism, i.e. an element of , you can assign the homomorphism to everyone and thus get a mapping ${\displaystyle \operatorname {Hom} }$ ${\displaystyle (A,B)}$ ${\displaystyle C}$ ${\displaystyle \operatorname {Hom} (A,B)}$ ${\displaystyle f\colon A'\to A}$ ${\displaystyle C}$ ${\displaystyle \operatorname {Hom} (A',A)}$ ${\displaystyle h\in \operatorname {Hom} (A,B)}$ ${\displaystyle h\circ f\in \operatorname {Hom} (A',B)}$

${\displaystyle \operatorname {Hom} (f,B)\colon \operatorname {Hom} (A,B)\to \operatorname {Hom} (A',B).}$

A mapping is also obtained for a homomorphism ${\displaystyle g\in \operatorname {Hom} (B,B')}$

${\displaystyle \operatorname {Hom} (A,g)\colon \operatorname {Hom} (A,B)\to \operatorname {Hom} (A,B'),}$

by on maps. Combined, you get an illustration ${\displaystyle h}$ ${\displaystyle g\circ h}$

${\displaystyle \operatorname {Hom} (f,g)\colon \operatorname {Hom} (A,B)\to \operatorname {Hom} (A',B').}$

It is easy to verify the following properties:

• ${\displaystyle \operatorname {Hom} (\operatorname {id} _{A},\operatorname {id} _{B})=\operatorname {id} _{\operatorname {Hom} (A,B)}}$, where etc. denotes the identity of the respective object. ${\displaystyle \operatorname {id} _{A}}$
• ${\displaystyle \operatorname {Hom} (f,g)\circ \operatorname {Hom} (f',g')=\operatorname {Hom} (f'\circ f,g\circ g')}$as long as the links are defined (i.e. corresponding definition and target areas match).

In the language of category theory , this can be expressed using the terms of the dual category and the product category :

${\displaystyle \operatorname {Hom} }$is a functor from the category Set of sets. Note: Objects from are pairs of -objects, morphisms from to are pairs of morphisms, where and is, and it is as far as defined. ${\displaystyle C^{op}\times C}$ ${\displaystyle C^{op}\times C}$ ${\displaystyle (A,B)}$ ${\displaystyle C}$ ${\displaystyle (A,B)}$ ${\displaystyle (A',B')}$ ${\displaystyle (f,g)}$ ${\displaystyle g\in \operatorname {Hom} _{C}(B,B')}$ ${\displaystyle f\in \operatorname {Hom} _{C^{op}}(A,A')=\operatorname {Hom} _{C}(A',A)}$ ${\displaystyle (f,g)\circ (f',g')=(f'\circ f,g\circ g')}$

In particular, a covariant functor and a contravariant functor from to Set , the so-called partial Hom functors , are obtained for a fixed object . ${\displaystyle A\in \operatorname {Ob} (C)}$ ${\displaystyle \operatorname {Hom} (A,-)}$ ${\displaystyle \operatorname {Hom} (-,A)}$ ${\displaystyle C}$

Compatibility with additional structures

In general, it is only a set (if the category is locally small) and does not automatically carry an additional structure itself, apart from the fact that the endomorphisms under composition form a monoid with a neutral element. However, for example, the properties of abelian groups or R - moduli for a ring R , as homomorphisms can pointwise added and / or with elements of R are multiplied, and thus forms then even an abelian group and an R module. One then immediately checks that the above-defined assignments are compatible with this and that in these cases it can even be interpreted as a functor in the Abelian group's category Ab or the R - Mod category of R -modules. ${\displaystyle \operatorname {Hom} (A,B)}$ ${\displaystyle \operatorname {End} (A):=\operatorname {Hom} (A,A)}$ ${\displaystyle \operatorname {id} _{A}}$ ${\displaystyle C}$ ${\displaystyle \operatorname {Hom} (A,B)}$ ${\displaystyle \operatorname {Hom} }$

Depending on the category under consideration , further such additional structures are possible. That is, is conceived as an object of a category that is not necessarily the category of sets. Generally, one speaks of a more than one category enriched category (also: category) when the hom functor on a functor in the category , and has a certain tolerance, which can be chosen differently, for instance with a selected monoidal structure on . Each locally small category is enriched with the Cartesian product as a monoidal structure above the category of the sets . A pre-additive category is a category enriched with the usual tensor product above the category of the Abelian groups . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \operatorname {Hom} (A,B)}$ ${\displaystyle \operatorname {Hom} (A,B)}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {D}}}$

You can also enrich it with very simple categories whose objects are not sets. The category has two objects and next to the identities an interesting arrow between the objects. It has finite products as a monoidal structure. Under this a category is a quasi-order . The quasi-order can be equipped with sums (" ") or maximums (" ") as a monoidal structure. Generalized metric spaces are obtained as -categories and sets with generalized ultrametrics as -categories . (The generalization is that symmetry is not required and points with a distance of zero do not have to be identical.) ${\displaystyle \mathbf {2} :=\{{\stackrel {0}{\bullet }}\to {\stackrel {1}{\bullet }}\}}$ ${\displaystyle \mathbf {2} }$ ${\displaystyle (\mathbb {R} ^{+},\geq )}$ ${\displaystyle \mathbb {R} _{+}^{+}}$ ${\displaystyle \mathbb {R} _{\mathrm {max} }^{+}}$ ${\displaystyle \mathbb {R} _{+}^{+}}$ ${\displaystyle \mathbb {R} _{\mathrm {max} }^{+}}$

Applications

The Ext function, the derived functor of Hom, also plays an important role in the investigation of Abelian categories .