A unified category-theoretic formulation of typed binding signatures
Miki Tanaka, John Power · 2005
We generalise Fiore et al's account of variable binding for untyped cartesian contexts and Tanaka's account of vari-able binding for untyped linear contexts to give an account of variable binding for simply typed axiomatically dened contexts. In line with earlier work by us, we axiomatise the notion of context by means of a pseudo-monad S on Cat: Fiore et al implicitly used the pseudo-monad Tfp for small categories with nite products, and Tanaka implicitly used the pseudo-monad Tsm for small symmetric monoidal categories. But here we also extend from untyped variable binding to typed variable binding. Given a set A of types, this involves generalising from Fiore et al's use of [; Set] to [(SA)op; SetA]. We dene a substitution monoidal struc-ture on [(SA)op; SetA], give a denition of binding signa-ture at this level of generality, and extend initial algebra semantics to this typed, axiomatic setting. This generalises and axiomatises previous work by Fiore et al and later au-thors in particular cases. In particular, it includes the Logic of Bunched Implications and variants, and it yields an im-proved axiomatic denition of binding signature even in the case of untyped binders.