Paper: Monads and Distributive Laws in Substructural Contexts (at LICS 2026)

Authors: Soichiro Fujii Yun Chen Tsai Yoàv Montacute Ichiro Hasuo

Open access: https://doi.org/10.4230/LIPIcs.LICS.2026.45

Abstract

We present a categorical theory of monads and distributive laws in substructural contexts. In the study of distributive laws, the roles of (the absence of) structural rules for variable contexts have been recognized; our theory formalizes these substructural situations using Tronin’s verbal categories W, in a uniform and presentation-independent manner. We introduce the classes of W-operadic monads (those defined via the structural rules in W) and of W-commutative monads (those invariant under the structural rules in W). We give a canonical construction of a distributive law ST → TS of monads on Set; it is applicable when S is W-operadic and T is W-commutative (under mild conditions). This accounts for many known and new distributive laws. Even when S fails to be W-operadic, we can refine S and force W-operadicity; this captures Varacca and Winskel’s construction of indexed valuations.

BibTeX

  @InProceedings{FujiiTsaiMontacuteH-MonadsandDistributi,
    author = 	 {Soichiro Fujii and Yun Chen Tsai and Yoàv Montacute and Ichiro Hasuo},
    title = 	 {Monads and Distributive Laws in Substructural Contexts},
    booktitle =  {Proceedings of the Forty-First Annual Symposium on Logic in Computer Science (LICS 2026)},
    year =	 {2026},
    month =	 {July}, 
    pages =      {45:1--45:28},
    location =   {Lisbon, Portugal}, 
    publisher =	 {Schloss Dagstuhl -- Leibniz-Zentrum für Informatik},
    doi =        {10.4230/LIPIcs.LICS.2026.45}
  }