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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.
This work has been done with the support of EPSRC grant GR/586372/01, A Theory of Effects for Programming Languages. 相似文献
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