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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 ${\Sigma }_{f}⊣{𝒫}_{f}⊣{\Pi }_{f}$ (one such triple for every function $f:A\to B$). On the other hand, a bifibration only induces a family of adjoint pairs ${\mathrm{𝗉𝗎𝗌𝗁}}_{f}⊣{\mathrm{𝗉𝗎𝗅𝗅}}_{f}$ (again, one such pair for every $f:A\to B$). 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 $\mathrm{𝐒𝐞𝐭}$ to the symmetric monoidal closed category $\mathrm{𝐑𝐞𝐥}$. 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.