The graphical theory of monads

Hinze, Ralf; Marsden, Dan

Abstract

The formal theory of monads shows that much of the theory of monads can be developed in the abstract at the level of 2-categories. This means that results about monads can established once and for all, and simply instantiated in settings such as enriched category theory. Unfortunately, these results can be hard to reason about as they involve more abstract machinery. In this paper, we present the formal theory of monads in terms of string diagrams-a graphical language for 2-categorical calculations. Using this perspective, we show that many aspects of the theory of monads, such as the Eilenberg-Moore and Kleisli resolutions of monads, liftings, and distributive laws, can be understood in terms of systematic graphical calculational reasoning. This paper will serve as an introduction both to the formal theory of monads and to the use of string diagrams, in particular, their application to calculations in monad theory.

Citation

Hinze, R., & Marsden, D. (2025). The graphical theory of monads. Journal of Functional Programming, 35, Article e11. https://doi.org/10.1017/S095679682500005X

• Journal Article Type: Article
• Acceptance Date: Jan 20, 2025
• Online Publication Date: Apr 7, 2025
• Publication Date: 2025
• Deposit Date: Feb 28, 2025
• Publicly Available Date: Apr 9, 2025
• Journal: Journal of Functional Programming
• Print ISSN: 0956-7968
• Electronic ISSN: 1469-7653
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 35
• Article Number: e11
• DOI: https://doi.org/10.1017/S095679682500005X
• Public URL: https://nottingham-repository.worktribe.com/output/45861019
• Publisher URL: https://www.cambridge.org/core/journals/journal-of-functional-programming/article/graphical-theory-of-monads/15AD68F2BC02195A7A2F16075BF0A44D
• Additional Information: Copyright: © The Author(s), 2025. Published by Cambridge University Press; License: This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.; Free to read: This content has been made available to all.