On Doctrines and Cartesian Bicategories

Authors Filippo Bonchi , Alessio Santamaria , Jens Seeber, Paweł Sobociński

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Author Details

Filippo Bonchi
  • University of Pisa, Italy
Alessio Santamaria
  • University of Pisa, Italy
Jens Seeber
  • University of Pisa, Italy
Paweł Sobociński
  • Tallinn University of Technology, Estonia


We thank Pino Rosolini and Fabio Pasquali for discussions that improved our understanding of doctrines.

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Filippo Bonchi, Alessio Santamaria, Jens Seeber, and Paweł Sobociński. On Doctrines and Cartesian Bicategories. In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We study the relationship between cartesian bicategories and a specialisation of Lawvere’s hyperdoctrines, namely elementary existential doctrines. Both provide different ways of abstracting the structural properties of logical systems: the former in algebraic terms based on a string diagrammatic calculus, the latter in universal terms using the fundamental notion of adjoint functor. We prove that these two approaches are related by an adjunction, which can be strengthened to an equivalence by imposing further constraints on doctrines.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Categorical semantics
  • Cartesian bicategories
  • elementary existential doctrines
  • string diagram


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