Section: New Results
A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine
Participant : Noam Zeilberger.
In joint work with Paul-André Melliès, we have been investigating the categorical semantics of type refinement systems, which are type systems built “on top of” a typed programming language to specify and verify more precise properties of programs. The fibrational view of type refinement we have been developing (cf. ) is closely related to the categorical perspective on first-order logic introduced by Lawvere , but with some important conceptual and technical differences that provide an opportunity for reflection. For example, Lawvere's axiomatization of first-order logic (his theory of so-called “hyperdoctrines”) was based on the idea that existential and universal quantification can be described respectively as left and right adjoints to the operation of substitution, this giving rise to a family of adjoint triples (one such triple for every function ). On the other hand, a bifibration only induces a family of adjoint pairs (again, one such pair for every ). In , we resolved this and other apparent mismatches by applying ideas inspired by the semantics of linear logic and the shift from the cartesian closed category to the symmetric monoidal closed category . Two other applications of our analysis include an axiomatic treatment of directed equality predicates (which can be modelled as “hom” presheaves, realizing an early vision of Lawvere), as well as a simple calculus of string diagrams that is highly reminiscent of C. S. Peirce's “existential graphs” for predicate logic.