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Sweedler Theory of Monads

Sweedler Theory of Monads

  • Dylan McDermott  ORCID: orcid.org/0000-0002-6705-144910,
  • Exequiel Rivas  ORCID: orcid.org/0000-0002-2114-624X11 &
  • Tarmo Uustalu  ORCID: orcid.org/0000-0002-1297-057910,11 
  • Conference paper
  • Open Access
  • First Online: 29 March 2022
  • 1258 Accesses

Part of the Lecture Notes in Computer Science book series (LNCS,volume 13242)

Abstract

Monad-comonad interaction laws are a mathematical concept for describing communication protocols between effectful computations and coeffectful environments in the paradigm where notions of effectful computation are modelled by monads and notions of coeffectful environment by comonads. We show that monad-comonad interaction laws are an instance of measuring maps from Sweedler theory for duoidal categories whereby the final interacting comonad for a monad and a residual monad arises as the Sweedler hom and the initial residual monad for a monad and an interacting comonad as the Sweedler copower. We then combine this with a (co)algebraic characterization of monad-comonad interaction laws to derive descriptions of the Sweedler hom and the Sweedler copower in terms of their coalgebras resp. algebras.

Keywords

  • (co)monads
  • (co)algebras
  • interaction laws
  • runners
  • duoidal categories
  • Sweedler operations

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Author information

Authors and Affiliations

  1. Dept. of Computer Science, Reykjavik University, Menntavegur 1, 102, Reykjavik, Iceland

    Dylan McDermott & Tarmo Uustalu

  2. Dept. of Software Science, Tallinn University of Technology, Tallinn, Estonia

    Exequiel Rivas & Tarmo Uustalu

Authors
  1. Dylan McDermott
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  2. Exequiel Rivas
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  3. Tarmo Uustalu
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Corresponding author

Correspondence to Dylan McDermott .

Editor information

Editors and Affiliations

  1. Université Paris-Saclay, CNRS, ENS Paris-Saclay, Gif-sur-Yvette, France

    Prof. Patricia Bouyer

  2. Friedrich-Alexander-Universität Erlangen, Erlangen, Germany

    Lutz Schröder

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McDermott, D., Rivas, E., Uustalu, T. (2022). Sweedler Theory of Monads. In: Bouyer, P., Schröder, L. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2022. Lecture Notes in Computer Science, vol 13242. Springer, Cham. https://doi.org/10.1007/978-3-030-99253-8_22

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  • DOI: https://doi.org/10.1007/978-3-030-99253-8_22

  • Published: 29 March 2022

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