A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine

TL;DR

This paper reconstructs Lawvere's presheaf hyperdoctrine using bifibrations, revealing dualities, axiomatizing directed equality predicates, and connecting to string diagrams and linear logic.

Contribution

It introduces a bifibrational reconstruction of Lawvere's hyperdoctrine, unveiling dualities and providing a new axiomatic and diagrammatic framework.

Findings

Reconstruction of presheaf hyperdoctrine via bifibrations

Axiomatic treatment of directed equality predicates

Development of a string diagram calculus for presheaves

Abstract

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.