Localisable Monads

Authors Carmen Constantin , Nuiok Dicaire, Chris Heunen

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Author Details

Carmen Constantin
  • University of Edinburgh, UK
Nuiok Dicaire
  • University of Edinburgh, UK
Chris Heunen
  • University of Edinburgh, UK


We thank Rui Soares Barbosa, Robert Furber, and Nesta van der Schaaf for useful discussions. We also thank the reviewers for their helpful feedback.

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Carmen Constantin, Nuiok Dicaire, and Chris Heunen. Localisable Monads. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Monads govern computational side-effects in programming semantics. A collection of monads can be combined together in a local-to-global way to handle several instances of such effects. Indexed monads and graded monads do this in a modular way. Here, instead, we start with a single monad and equip it with a fine-grained structure by using techniques from tensor topology. This provides an intrinsic theory of local computational effects without needing to know how constituent effects interact beforehand. Specifically, any monoidal category decomposes as a sheaf of local categories over a base space. We identify a notion of localisable monads which characterises when a monad decomposes as a sheaf of monads. Equivalently, localisable monads are formal monads in an appropriate presheaf 2-category, whose algebras we characterise. Three extended examples demonstrate how localisable monads can interpret the base space as locations in a computer memory, as sites in a network of interacting agents acting concurrently, and as time in stochastic processes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Monad
  • Monoidal category
  • Presheaf
  • Central idempotent
  • Graded monad
  • Indexed monad
  • Formal monad
  • Strong monad
  • Commutative monad


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