Monads need not be endofunctors

Altenkirch, Thorsten; Chapman, James; Uustalu, Tarmo

Abstract

We introduce a generalization of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed λ-calculus syntax and indexed containers. We show that the Kleisli and Eilenberg-Moore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions, relative monads are monoids in the functor category concerned and extend to monads, giving rise to a coreflection between relative monads and monads. Arrows are also an instance of relative monads.

Citation

Altenkirch, T., Chapman, J., & Uustalu, T. (2015). Monads need not be endofunctors. Logical Methods in Computer Science, 11(1), 1-40. https://doi.org/10.2168/LMCS-11%281%3A3%292015

• Journal Article Type: Article
• Online Publication Date: Mar 6, 2015
• Publication Date: Mar 6, 2015
• Deposit Date: Oct 12, 2015
• Publicly Available Date: Oct 12, 2015
• Journal: Logical Methods in Computer Science
• Print ISSN: 1860-5974
• Electronic ISSN: 1860-5974
• Publisher: Logical Methods in Computer Science
• Peer Reviewed: Peer Reviewed
• Volume: 11
• Issue: 1
• Article Number: 3
• Pages: 1-40
• DOI: https://doi.org/10.2168/LMCS-11%281%3A3%292015
• Keywords: monads, adjunctions, monoids, skew-monoidal categories, functional programming
• Public URL: https://nottingham-repository.worktribe.com/output/747987
• Publisher URL: https://lmcs.episciences.org/928