Categorical and convergent properties of convex spaces

A convex structure (dually, a concave structure) and a topological structure have many common characters. This paper aims to apply the topological methods to the theory of convex structures. From a categorical aspect, this paper first deals with the extensionality and productivity of quotient maps in the category of convex spaces. It is shown that the category of convex spaces is not extensional, but productive for finite quotient maps. Then the paper introduced the convergence approach via co-Scott closed sets on powerset and proposed the concept of (preconcave, concave) convergence structures in concave spaces. It is proved that the category of concave convergence spaces is isomorphic to that of concave spaces and the latter can be embedded in the category of convergence spaces as a full and reflective subcategory. Finally, it is shown that the category of convergence spaces is extensional and productive for finite quotient maps.