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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.  [72]) is closely related to the categorical perspective on first-order logic introduced by Lawvere [66], 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 Σf𝒫fΠf (one such triple for every function f:AB). On the other hand, a bifibration only induces a family of adjoint pairs 𝗉𝗎𝗌𝗁f𝗉𝗎𝗅𝗅f (again, one such pair for every f:AB). In [33], 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 𝐒𝐞𝐭 to the symmetric monoidal closed category 𝐑𝐞𝐥. 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.