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Notions of computation as monoids*

Part of: JFP Research Articles

Published online by Cambridge University Press:  05 October 2017

EXEQUIEL RIVAS  and
MAURO JASKELIOFF
EXEQUIEL RIVAS
Affiliation:
Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas, CONICET, Rosario, Santa Fe, Argentina FCEIA, Universidad Nacional de Rosario, Rosario, Santa Fe, Argentina (e-mails: rivas@cifasis-conicet.gov.ar, jaskelioff@cifasis-conicet.gov.ar)
MAURO JASKELIOFF
Affiliation:
Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas, CONICET, Rosario, Santa Fe, Argentina FCEIA, Universidad Nacional de Rosario, Rosario, Santa Fe, Argentina (e-mails: rivas@cifasis-conicet.gov.ar, jaskelioff@cifasis-conicet.gov.ar)
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Abstract

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There are different notions of computation, the most popular being monads, applicative functors, and arrows. In this article, we show that these three notions can be seen as instances of a unifying abstract concept: monoids in monoidal categories. We demonstrate that even when working at this high level of generality, one can obtain useful results. In particular, we give conditions under which one can obtain free monoids and Cayley representations at the level of monoidal categories, and we show that their concretisation results in useful constructions for monads, applicative functors, and arrows. Moreover, by taking advantage of the uniform presentation of the three notions of computation, we introduce a principled approach to the analysis of the relation between them.

Type
Articles
Information
Journal of Functional Programming , Volume 27 , 2017 , e21
Copyright
Copyright © Cambridge University Press 2017 

Footnotes

*

This work was partially funded by the Agencia Nacional de Promoción Científica y Tecnológica (PICT 2009-15) and Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET).

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