An enriched view on the extended finitary monad–Lawvere theory correspondence

Richard Garner, John Power

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    Abstract

    We give a new account of the correspondence, first established by Nishizawa-Power, between finitary monads and Lawvere theories over an arbitrary locally finitely presentable base. Our account explains this correspondence in terms of enriched category theory: the passage from a finitary monad to the corresponding Lawvere theory is exhibited as an instance of free completion of an enriched category under a class of absolute colimits. This extends work of the first author, who established the result in the special case of finitary monads and Lawvere theories over the category of sets; a novel aspect of the generalisation is its use of enrichment over a bicategory, rather than a monoidal category, in order to capture the monad-theory correspondence over all locally finitely presentable bases simultaneously.

    Original languageEnglish
    Article number16
    Pages (from-to)1-23
    Number of pages23
    JournalLogical Methods in Computer Science
    Volume14
    Issue number1
    DOIs
    Publication statusPublished - 27 Feb 2018

    Bibliographical note

    Copyright the Author(s) 2018. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

    Keywords

    • Absolute colimits
    • Enrichment in a bicategory
    • Finitary monad
    • Lawvere theory
    • Locally finitely presentable category

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