Duality for distributive space

The main source of inspiration for the present paper is the w ork of R. Rosebrugh and R.J. Wood on constructive complete distributive lattices where the aut hors employ elegantly the concepts of adjunction and module in their study of ordered sets. Both notions (suitably ad apted) are available in topology too, which permits us to investigate topological, metric and other kinds of space s in a similar spirit. Therefore, relative to a choice Φ of modules, we consider spaces which admit all colimits with we ight inΦ, as well as (suitably defined) Φ-distributive andΦ-algebraic spaces. We show that the category of Φ-distributive spaces and Φ-colimit preserving maps is dually equivalent to the idempotent splitting completion of a category of spaces and convergence relations between them. We explain the connection of these results to the tradi tional duality of spaces with frames, and conclude further duality theorems. Finally, we study properties and str uctures of the resulting categories, in particular monoida l (closed) structures.

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