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.