Correspondences Between Codensity and Coupling-Based Liftings, a Practical Approach

Authors Samuel Humeau , Daniela Petrisan , Jurriaan Rot



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Author Details

Samuel Humeau
  • ENS de Lyon, CNRS, LIP, UMR 5668, 69342, Lyon cedex 07, France
Daniela Petrisan
  • CNRS, IRIF, Université Paris Diderot, Paris, France
Jurriaan Rot
  • Institute for Computing and Information Sciences, Radboud University, Nijmegen, The Netherlands

Acknowledgements

The authors would like to thank Pedro Nora for suggestions and discussions.

Cite As Get BibTex

Samuel Humeau, Daniela Petrisan, and Jurriaan Rot. Correspondences Between Codensity and Coupling-Based Liftings, a Practical Approach. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.29

Abstract

The Kantorovich distance is a widely used metric between probability distributions. The Kantorovich-Rubinstein duality states that it can be defined in two equivalent ways: as a supremum, based on non-expansive functions into [0,1], and as an infimum, based on probabilistic couplings.
Orthogonally, there are categorical generalisations of both presentations proposed in the literature, in the form of codensity liftings and what we refer to as coupling-based liftings. Both lift endofunctors on the category Set of sets and functions to that of pseudometric spaces, and both are parameterised by modalities from coalgebraic modal logic.
A generalisation of the Kantorovich-Rubinstein duality has been more nebulous - it is known not to work in some cases. In this paper we propose a compositional approach for obtaining such generalised dualities for a class of functors, which is closed under coproducts and products. Our approach is based on an explicit construction of modalities and also applies to and extends known cases such as that of the powerset functor.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
Keywords
  • Kantorovich distance
  • behavioural metrics
  • Kantorovich-Rubinstein duality
  • functor liftings

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