A Unified Category-theoretic Semantics for Binding Signatures in Substructural Logics

Abstract

Generalizing Fiore et al.'s use of the category of finite sets to model untyped Cartesian contexts and Tanaka's use of the category of permutations to model untyped linear contexts, we let S be an arbitrary pseudo-monad on Cat and let S1 model untyped contexts in general: this generality includes contexts for sub-structural logics such as the Logic of Bunched Implications and variants. Given a pseudo-distributive law of S over the (partial) pseudo-monad for free cocompletions, we define a canonical substitution monoidal structure on the category [(S1)op, Set], generalizing substitution monoidal structures for Cartesian and linear contexts and providing a natural substitution structure for Bunched Implications and its variants. We give a concrete description of the substitution monoidal structure. We then give an axiomatic definition of a binding signature, again extending the definitions for Cartesian and linear contexts. We investigate examples in detail, then prove the central result of the paper, yielding initial algebra semantics for binding signatures at the level of generality we propose.