Convexity theories 0. Foundations

For an arbitrary prenormed semiring, the closed unit ball functor from the category _R pnSmod ¹ ofR-prenormedR-semimodules with contractions to the category of sets has a left adjoint. For such a semiringR the notion of finitary convexity theory overR is introduced and the category C of -modules is defined. It is shown that the canonical functor _R pnSmod ¹ C has a left adjoint. In caseR is a banach semiring one has infinitary convexity theories, in addition to the finitary ones, and again the canonical functor _R bnSmod ¹ C has a left adjoint.Many more happy returns, Nico. Sixty is forever.