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Flexibly Graded Monads and Graded Algebras

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Mathematics of Program Construction (MPC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13544))

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Abstract

When modelling side-effects using a monad, we need to equip the monad with effectful operations. This can be done by noting that each algebra of the monad carries interpretations of the desired operations. We consider the analogous situation for graded monads, which are a generalization of monads that enable us to track quantitative information about side-effects. Grading makes a significant difference: while many graded monads of interest can be equipped with similar operations, the algebras often cannot. We explain where these operations come from for graded monads. To do this, we introduce the notion of flexibly graded monad, for which the situation is similar to the situation for ordinary monads. We then show that each flexibly graded monad induces a canonical graded monad in such a way that operations for the flexibly graded monad carry over to the graded monad. In doing this, we reformulate grading in terms of locally graded categories, showing in particular that graded monads are a particular kind of relative monad. We propose that locally graded categories are a useful setting for work on grading in general.

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Acknowledgements

We thank Nathanael Arkor, Shin-ya Katsumata and Nicolas Wu for helpful discussions. Both authors were supported by the Icelandic Research Fund grants no. 196323-053 and 228684-051.

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Authors and Affiliations

  1. Department of Computer Science, Reykjavik University, Reykjavik, Iceland

    Dylan McDermott & Tarmo Uustalu

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

    Tarmo Uustalu

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  1. Dylan McDermott
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  2. Tarmo Uustalu
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Correspondence to Dylan McDermott .

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  1. School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK

    Ekaterina Komendantskaya

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McDermott, D., Uustalu, T. (2022). Flexibly Graded Monads and Graded Algebras. In: Komendantskaya, E. (eds) Mathematics of Program Construction. MPC 2022. Lecture Notes in Computer Science, vol 13544. Springer, Cham. https://doi.org/10.1007/978-3-031-16912-0_4

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  • DOI: https://doi.org/10.1007/978-3-031-16912-0_4

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