Category of groups with homologisms

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Suppose \mathcal{V} is a subvariety of the variety of groups. The category of groups with \mathcal{V}-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 \mathcal{V} 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 \mathcal{V} 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 \mathcal{V}

Important functors