A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine

Paul-André Melliès (1) , Noam Zeilberger (2)

1 PPS - Preuves, Programmes et Systèmes
2 PARSIFAL - Proof search and reasoning with logic specifications

Combining insights from the study of type refinement systems and of monoidal closed chiralities, we show how to reconstruct Lawvere's hyperdoctrine of presheaves using a full and faithful embedding into a monoidal closed bifibration living now over the compact closed category of small categories and distributors. Besides revealing dualities which are not immediately apparent in the traditional presentation of the presheaf hyperdoctrine, this reconstruction leads us to an axiomatic treatment of directed equality predicates (modelled by hom presheaves), realizing a vision initially set out by Lawvere (1970). It also leads to a simple calculus of string diagrams (representing presheaves) that is highly reminiscent of C. S. Peirce's existential graphs for predicate logic, refining an earlier interpretation of existential graphs in terms of Boolean hyperdoctrines by Brady and Trimble. Finally, we illustrate how this work extends to a bifibrational setting a number of fundamental ideas of linear logic.

Mots clés: Lawvere’s presheaf hyperdoctrine; monoidal closed bifibrations; type refinement systems; monoidal closed chiralities; linear logic

Domaines: Catégories et ensembles [math.CT]; Logique [math.LO]; Logique en informatique [cs.LO]

Paul-André Melliès, Noam Zeilberger. A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine. IEEE/ACM Logic in Computer Science (LICS) 2016, Jun 2016, New York, United States. ⟨hal-01261955⟩