# Category of groups with homologisms

From Groupprops

## Definition

Suppose is a subvariety of the variety of groups. The **category of groups with -homologisms** 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 | homologism of groups with respect to the variety | A homologism of groups is a pair of homomorphisms, one between their quotient groups by their respective marginal subgroups, and the other between their respective verbal subgroups, that commute with the word map evaluations. |

composition law for morphisms | ? | Compose the morphisms separately for the quotient group by the marginal subgroup and for the verbal subgroup. |

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 | isologism of groups with respect to | An isologism is a homologism 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 quotient group by the marginal subgroup and on the verbal subgroup. |

categorical coproduct | ? | |

zero object | any group that is in the subvariety |

## Important functors

- There is a natural functor to this category from the category of groups with marginal homomorphisms.