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Convexity theories 0. Foundations

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Abstract

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 Γ overR is introduced and the category ΓC of Γ-modules is defined. It is shown that the canonical functor R pnSmod 1 → ΓC 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 → ΓC has a left adjoint.

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Many more happy returns, Nico. Sixty is forever.

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Röhrl, H. Convexity theories 0. Foundations. Appl Categor Struct 2, 13–43 (1994). https://doi.org/10.1007/BF00878500

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Mathematics Subject Classifications (1991)

  • 52A01
  • 16Y60
  • 16D99
  • 46H25

Key words

  • Convexity theory
  • Γ-convex module
  • unit ball functor
  • semiring
  • semimodule
  • banach semiring
  • banach semimodule
  • pretopology