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Convexity theories 0. Foundations

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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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  1. 1.

    Kneser, H.: Konvexe Räume,Arch. d. Math. 3 (1952), 198–206.

  2. 2.

    MacLane, S.:Categories for the Working Mathematician, Springer-Verlag, 1971.

  3. 3.

    Manes, E.G.:Algebraic Theories, Springer-Verlag, 1976.

  4. 4.

    Pumplün, D. and Röhrl, H.: Banach spaces and totally convex spaces I,Comm. in Alg. 12 (1984), 953–1019.

  5. 5.

    Pumplün, D. and Röhrl, H.: Convexity Theories IV. Klein-Hilbert Parts in Convex Modules. In preparation.

  6. 6.

    Röhrl, H.:Convexity Theories I. Γ-Convex Spaces, Constantin Carathéodory: An International Tribute, World Sci. Publ. (1991), 1175–1209.

  7. 7.

    Semadeni, Z.:Banach Spaces of Continuous Functions, Polish Sci. Publ., 1971.

  8. 8.

    Stone, M.H.: Postulates for the barycentric calculus,Ann. Math. 29 (1949), 25–30.

  9. 9.

    Tholen, W.: Relative Bildzerlegungen und algebraische Kategorien, Diss. Univ. Münster, 1973.

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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