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Frobenius Structures in Star-Autonomous Categories

Authors Cédric de Lacroix, Luigi Santocanale



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Author Details

Cédric de Lacroix
  • LIS, CNRS UMR 7020, Aix-Marseille Université, Marseille, France
Luigi Santocanale
  • LIS, CNRS UMR 7020, Aix-Marseille Université, Marseille, France

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Cédric de Lacroix and Luigi Santocanale. Frobenius Structures in Star-Autonomous Categories. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 18:1-18:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CSL.2023.18

Abstract

It is known that the quantale of sup-preserving maps from a complete lattice to itself is a Frobenius quantale if and only if the lattice is completely distributive. Since completely distributive lattices are the nuclear objects in the autonomous category of complete lattices and sup-preserving maps, we study the above statement in a categorical setting. We introduce the notion of Frobenius structure in an arbitrary autonomous category, generalizing that of Frobenius quantale. We prove that the monoid of endomorphisms of a nuclear object has a Frobenius structure. If the environment category is star-autonomous and has epi-mono factorizations, a variant of this theorem allows to develop an abstract phase semantics and to generalise the previous statement. Conversely, we argue that, in a star-autonomous category where the monoidal unit is a dualizing object, if the monoid of endomorphisms of an object has a Frobenius structure and the monoidal unit embeds into this object as a retract, then the object is nuclear.

Subject Classification

ACM Subject Classification
  • Theory of computation → Linear logic
  • Theory of computation → Constructive mathematics
  • Theory of computation → Proof theory
Keywords
  • Quantale
  • Frobenius quantale
  • Girard quantale
  • associative algebra
  • star-autonomous category
  • nuclear object
  • adjoint

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