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On Exponentiable Morphisms in Classical Algebra

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Abstract

We study exponentiability of homomorphisms in varieties of universal algebras close to classical ones. After describing an “almost folklore” general result, we present a purely algebraic proof of “étale implies exponentiable”, alternative to the topologically motivated proof given in one of our previous papers, in a different context. We prove that only isomorphisms are exponentiable homomorphisms in ideal determined varieties and extend this to ideal determined categories. Finally, we give a complete characterization of exponentiable homomorphisms of semimodules over semirings.

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Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal

    Maria Manuel Clementino

  2. CIDMA, Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    Dirk Hofmann

  3. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa

    George Janelidze

Authors
  1. Maria Manuel Clementino
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  2. Dirk Hofmann
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  3. George Janelidze
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Corresponding author

Correspondence to Maria Manuel Clementino.

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Dedicated to the memory of Horst Herrlich.

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Clementino, M.M., Hofmann, D. & Janelidze, G. On Exponentiable Morphisms in Classical Algebra. Appl Categor Struct 24, 733–742 (2016). https://doi.org/10.1007/s10485-016-9458-7

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  • DOI: https://doi.org/10.1007/s10485-016-9458-7

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