Direct Limits in the Category of Hv-Groups: Construction, Properties and Quotient Direct Limit

This paper investigates the notion of direct limits in the category of [Formula: see text]-groups. We review the fundamental definitions and properties of [Formula: see text]-groups, including weak homomorphisms, fundamental relations and regular equivalence relations. A direct system of [Formula: see text]-groups is defined, and its direct limit is constructed as a quotient of the disjoint union of the components. It is shown that the direct limit inherits an [Formula: see text]-group structure. We also study compatible families of subhypergroups and prove that the direct limit of a compatible family is an [Formula: see text]-subgroup of the direct limit. Regular and strongly regular relations are lifted to the direct limit, and it is proved that the direct limit of the fundamental groups is isomorphic to the fundamental group of the direct limit. Finally, morphisms between direct systems are considered, and it is shown that they induce corresponding homomorphisms between the direct limits, with a description of the kernel.