On the category of geometric spaces and the category of (geometric) hypergroups

In this paper first we define the morphism between geometric spaces in two different types. We construct two categories of $uu$ and $l$ from geometric spaces then investigate some properties of the two categories, for instance $uu$ is topological. The relation between hypergroups and geometric spaces is studied. By constructing the category $qh$ of $H_{v}$-groups we answer the question that which construction of hyperstructures on the category of sets has free object in the sense of universal property. At the end we define the category of geometric hypergroups and we study its relation with the category of hypergroup.