Abstract: Sub-topical functions, as an important class of abstract convex functions, have significant research value in fields such as optimization theory. In view of the limitation that scalar sub-topical functions in abstract convex analysis cannot directly handle set-valued optimization problems with order structures, this paper extends the relevant concepts of sub-topical functions from the scalar case to the set-valued case, aiming to address a wider range of optimization problems. To this end, by employing the notion of weak supremum in vector spaces, we first construct a class of set-valued elementary mappings with specific properties to serve as supporting elements. Then, we introduce the support set of set-valued sub-topical mappings and, on this basis, establish an upper envelope characterization with respect to this family of elementary mappings. Finally, we build a corresponding abstract convex theory for set-valued sub-topical mappings, laying a foundation for subsequent applications in areas such as set-valued optimization.