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DOI: 10.4230/LIPIcs.CSL.2023.18
Frobenius Structures in Star-Autonomous Categories

Authors: Cédric de Lacroix and Luigi Santocanale

Published in: LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

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.
Cite as

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)

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@InProceedings{delacroix_et_al:LIPIcs.CSL.2023.18,
  author =	{de Lacroix, C\'{e}dric and Santocanale, Luigi},
  title =	{{Frobenius Structures in Star-Autonomous Categories}},
  booktitle =	{31st EACSL Annual Conference on Computer Science Logic (CSL 2023)},
  pages =	{18:1--18:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-264-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{252},
  editor =	{Klin, Bartek and Pimentel, Elaine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.18},
  URN =		{urn:nbn:de:0030-drops-174798},
  doi =		{10.4230/LIPIcs.CSL.2023.18},
  annote =	{Keywords: Quantale, Frobenius quantale, Girard quantale, associative algebra, star-autonomous category, nuclear object, adjoint}
}
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