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Distributive Laws for Relative Monads - Applied Categorical Structures

  • ️Lobbia, Gabriele
  • ️Wed Apr 05 2023

Abstract

We introduce the notion of a distributive law between a relative monad and a monad. We call this a relative distributive law and define it in any 2-category \(\mathcal {K}\). In order to do that, we introduce the 2-category of relative monads in a 2-category \(\mathcal {K}\) with relative monad morphisms and relative monad transformations as 1- and 2-cells, respectively. We relate our definition to the 2-category of monads in \(\mathcal {K}\) defined by Street. Using this perspective, we prove two Beck-type theorems regarding relative distributive laws. We also describe what does it mean to have Eilenberg–Moore and Kleisli objects in this context and give examples in the 2-category of locally small categories.

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Acknowledgements

The author is very grateful to Martin Hyland for the helpful conversation about the definition of extension to Kleisli in this particular case. This paper also owes a lot to Nicola Gambino’s suggestions and feedback. Discussions with Francesco Gallinaro and Giovanni Soldà helped the author dealing with some examples. This research is part of the author’s PhD project, supported by an EPSRC Scholarship. The second version of this paper was improved also thanks to suggestions and comments by Nathanael Arkor and John Bourke.

Funding

This research is part of the author’s PhD project, supported by an EPSRC Scholarship.

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  1. School of Mathematics, University of Leeds, Leeds, UK

    Gabriele Lobbia

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  1. Gabriele Lobbia

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Lobbia, G. Distributive Laws for Relative Monads. Appl Categor Struct 31, 19 (2023). https://doi.org/10.1007/s10485-023-09716-1

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  • Received: 01 July 2022

  • Accepted: 23 February 2023

  • Published: 05 April 2023

  • DOI: https://doi.org/10.1007/s10485-023-09716-1

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