2 Search Results for "Pasquali, Fabio"

Document

DOI: 10.4230/LIPIcs.FSCD.2023.25

Quotients and Extensionality in Relational Doctrines

Authors: Francesco Dagnino and Fabio Pasquali

Published in: LIPIcs, Volume 260, 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)

Abstract

Taking a quotient roughly means changing the notion of equality on a given object, set or type. In a quantitative setting, equality naturally generalises to a distance, measuring how much elements are similar instead of just stating their equivalence. Hence, quotients can be understood quantitatively as a change of distance. Quotients are crucial in many constructions both in mathematics and computer science and have been widely studied using categorical tools. Among them, Lawvere’s doctrines stand out, providing a fairly simple functorial framework capable to unify many notions of quotient and related constructions. However, abstracting usual predicate logics, they cannot easily deal with quantitative settings. In this paper, we show how, combining doctrines and the calculus of relations, one can unify quantitative and usual quotients in a common picture. More in detail, we introduce relational doctrines as a functorial description of (the core of) the calculus of relations. Then, we define quotients and a universal construction adding them to any relational doctrine, generalising the quotient completion of existential elementary doctrine and also recovering many quantitative examples. This construction deals with an intensional notion of quotient and breaks extensional equality of morphisms. Then, we describe another construction forcing extensionality, showing how it abstracts several notions of separation in metric and topological structures.

Cite as

Francesco Dagnino and Fabio Pasquali. Quotients and Extensionality in Relational Doctrines. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 25:1-25:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dagnino_et_al:LIPIcs.FSCD.2023.25,
  author =	{Dagnino, Francesco and Pasquali, Fabio},
  title =	{{Quotients and Extensionality in Relational Doctrines}},
  booktitle =	{8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
  pages =	{25:1--25:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-277-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{260},
  editor =	{Gaboardi, Marco and van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.25},
  URN =		{urn:nbn:de:0030-drops-180090},
  doi =		{10.4230/LIPIcs.FSCD.2023.25},
  annote =	{Keywords: Quantitative Reasoning, Calculus of Relations, Hyperdoctrines, Metric Spaces}
}

Document

DOI: 10.4230/LIPIcs.CALCO.2021.10

On Doctrines and Cartesian Bicategories

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

Published in: LIPIcs, Volume 211, 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)

Abstract

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.

Cite as

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)

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@InProceedings{bonchi_et_al:LIPIcs.CALCO.2021.10,
  author =	{Bonchi, Filippo and Santamaria, Alessio and Seeber, Jens and Soboci\'{n}ski, Pawe{\l}},
  title =	{{On Doctrines and Cartesian Bicategories}},
  booktitle =	{9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)},
  pages =	{10:1--10:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-212-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{211},
  editor =	{Gadducci, Fabio and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2021.10},
  URN =		{urn:nbn:de:0030-drops-153656},
  doi =		{10.4230/LIPIcs.CALCO.2021.10},
  annote =	{Keywords: Cartesian bicategories, elementary existential doctrines, string diagram}
}

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1 Dagnino, Francesco
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2 Theory of computation → Categorical semantics
2 Theory of computation → Logic

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