Category of groups with homoclinisms
This article describes a category (in the mathematical sense) where the notion of "object" is groupand the notion of morphism is homoclinism of groups. In other words, it gives a category structure to the collection of all groups.
View other category structures on groups
The category of groups with homoclinisms is defined as follows:
|objects||groups||A group is a set with associative binary operation admitting an identity element and inverse map.|
|morphisms||homoclinism of groups||A homoclinism of groups is a pair of homomorphisms, one between the respective inner automorphism groups and the other between the respective derived subgroups, that are compatible via the commutator map.|
|composition law for morphisms||?||Compose the morphisms separately for the inner automorphisms group side and for the derived subgroup side.|
Note that this is an unconventional category structure on groups. It is definitely not the default category structure. When people talk of the category of groups without mentioning what they mean by the morphisms, they typically do not mean this category and instead mean the usual category of groups where the morphisms are the usual homomorphisms of groups.
Constructs in this category
|Construct||Name in this category||Definition/description|
|isomorphism||isoclinism of groups||An isoclinism is a homoclinism for which both the component homomorphisms are isomorphisms.|
|categorical product||the usual external direct product||We take the usual external direct product and apply the coordinate projections separately on the inner automorphism group and derived subgroup.|
|zero object||any abelian group|
- There is a natural functor to this category from the category of groups with central homomorphisms.