A categorical approach to abstract convex spaces and interval spaces

In this paper, we establish the axiomatic conditions of hull operators and introduce the category of interval spaces. We also investigate their relations with convex spaces from a categorical sense. It is shown that the category of convex spaces is isomorphic to the category of hull spaces, and they are all topological over . Also, it is proved that there is an adjunction between the category of interval spaces and the category of convex spaces. In particular, the category of arity 2 convex spaces can be embedded in as a reflective subcategory.