## 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* ${\Sigma}_{f}\u22a3{\mathcal{P}}_{f}\u22a3{\Pi}_{f}$ (one such triple for every function $f:A\to B$).
On the other hand, a bifibration only induces a family of *adjoint pairs* ${\mathrm{\U0001d5c9\U0001d5ce\U0001d5cc\U0001d5c1}}_{f}\u22a3{\mathrm{\U0001d5c9\U0001d5ce\U0001d5c5\U0001d5c5}}_{f}$ (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 $\mathrm{\mathbf{S}\mathbf{e}\mathbf{t}}$ to the symmetric monoidal closed category $\mathrm{\mathbf{R}\mathbf{e}\mathbf{l}}$.
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.