Pseudo-distributive laws and axiomatics for variable binding

  • Authors:

  • Miki Tanaka;John Power

  • Affiliations:

  • National Institute of Information and Communications Technology, Tokyo;School of Informatics, University of Edinburgh, Edinburgh, Scotland EH9 3JZ

  • Venue:
  • Higher-Order and Symbolic Computation
  • Year:
  • 2006

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Abstract

We give a general category theoretic formulation of the substitution structure underlying the category theoretic study of variable binding proposed by Fiore, Plotkin, and Turi. This general formulation provides the foundation for their work on variable binding, as well as Tanaka's linear variable binding and variable binding for other binders and for mixtures of binders as for instance in the Logic of Bunched Implications. The key structure developed by Fiore et al. was a substitution monoidal structure, from which their formulation of binding was derived; so we give an abstract formulation of a substitution monoidal structure, then, at that level of generality, derive the various category theoretic structures they considered. The central construction we use is that of a pseudo-distributive law between 2-monads on Cat, which suffices to induce a pseudo-monad on Cat, and hence a substitution monoidal structure on the free object on 1. We routinely generalise that construction to account for types.