Modules over Monads and their Algebras

Maciej Pirog, Nicolas Wu, Jeremy Gibbons

Research output: Chapter in Book/Report/Conference proceedingConference Contribution (Conference Proceeding)

2 Citations (Scopus)
138 Downloads (Pure)


Modules over monads (or: actions of monads on endofunctors) are structures in which a monad interacts with an endofunctor, composed either on the left or on the right. Although usually not explicitly identified as such, modules appear in many contexts in programming and semantics. In this paper, we investigate the elementary theory of modules. In particular, we identify the monad freely generated by a right module as a generalisation of Moggi’s resumption monad and characterise its algebras, extending previous results by Hyland, Plotkin and Power, and by Filinski and Støvring. Moreover, we discuss a connection between modules and algebraic effects: left modules have a similar feeling to Eilenberg Moore algebras, and can be seen as handlers that are natural in the variables, while right modules can be seen as functions that run effectful computations in an appropriate context (such as an initial state for a stateful computation).
Original languageEnglish
Title of host publication6th International Conference on Algebra and Coalgebra in Computer Science (CALCO’15)
EditorsLarry Moss, Paweł Sobociński
Place of PublicationDagstuhl, Germany
PublisherSchloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany
Number of pages24
ISBN (Print)9783939897842
Publication statusPublished - 21 Oct 2015

Publication series

NameLiebniz International Proceedings in Informatics (LIPIcs)
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
ISSN (Print)1868-8969


  • monad
  • module over monad
  • resumptions
  • free object
  • handler

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