Category of polygroup objects. (Q1866634)

In this paper a generalization of the notion of a polygroup (i.e. a quasicanonical hypergroup) by categorical methods is given. The authors define the category of polygroup objects \(\text{PGrp}({\mathcal E})\) in an arbitrary category \(\mathcal E\) and they show that a faithful functor \(U_1\), from \(\text{PGrp}({\mathcal E})\) into the category of monoid objects in \(\mathcal E\) (\(\text{Mon}({\mathcal E})\)), exists. It is proved that the concrete category \(\text{PGrp}({\mathcal E})\) over \(\text{Mon}({\mathcal E})\) has free objects and so the previous functor preserves and reflects monomorphisms and reflects epimorphisms. On the other hand, by defining a functor \(F_1\) from \(\text{Mon}({\mathcal E})\) into \(\text{PGrp}({\mathcal E})\), they show that there is an adjoint situation between \(F_1\) and \(U_1\). Finally, the pair \((F_1,U_1)\), where \(F_1\lvdash U_1\), is a geometric morphism, in the case that \(\mathcal E\) has finite limits.