A BV-Category of Spacetime Interventions
James Hefford, Matt Wilson
TL;DR
The paper develops a categorical framework for spacetime interventions using BV-logic by constructing BV-categories from duoidal, closed symmetric monoidal bases via the Chu construction. It proves that Chu(C,⊥) yields BV-categories and that Chu is right 2-adjoint to the forgetful functor, enabling cofree BV-extensions over a fragment with $(igotimes,igolessthan)$ and a dualising $(-)^*$. It then defines the strong Hyland envelope $ ext{StEnv}( ext{C}) = ext{Chu}( ext{StProf}( ext{C}),1)$ to unify interventions and contexts as intervention-context pairs, yielding a canonical, higher-order model of spatio-temporal relations over any symmetric monoidal category. The framework supports local comb decompositions, provides a robust interpretation of higher-order maps (including unitary supermaps via $ ext{uQS}$ embeddings), and remains applicable to infinite-dimensional quantum theories and generalized probabilistic theories, offering a path toward comprehensive causal semantics in physics and computation.
Abstract
We use the Chu construction to functorially build BV-categories from duoidal categories, demonstrating that candidate models of BV-logic can be cofreely constructed from a fragment of a model of Retoré's sequencing operator. By using this construction to show that the strong Hyland envelope is a BV-category, we find a way to build a canonical model of spatio-temporal relationships between agents in spacetime from any symmetric monoidal category. The concrete physical interpretation of spacetime events in this model as intervention-context pairs resolves deficiencies in previous attempts to give a general categorical semantics to quantum supermaps.