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

Definition

The category of groups with homoclinisms is defined as follows:

Aspect	Name	Definition/description
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.
monomorphism	?
epimorphism	?
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.
categorical coproduct	?
zero object	any abelian group

Important functors

• There is a natural functor to this category from the category of groups with central homomorphisms.