# Category of groups with homoclinisms

From Groupprops

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.