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DOI: 10.4230/LIPIcs.FSCD.2021.26

Polymorphic Automorphisms and the Picard Group

Authors: Pieter Hofstra, Jason Parker, and Philip J. Scott

Published in: LIPIcs, Volume 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)

Abstract

We investigate the concept of definable, or inner, automorphism in the logical setting of partial Horn theories. The central technical result extends a syntactical characterization of the group of such automorphisms (called the covariant isotropy group) associated with an algebraic theory to the wider class of quasi-equational theories. We apply this characterization to prove that the isotropy group of a strict monoidal category is precisely its Picard group of invertible objects. Furthermore, we obtain an explicit description of the covariant isotropy group of a presheaf category.

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Pieter Hofstra, Jason Parker, and Philip J. Scott. Polymorphic Automorphisms and the Picard Group. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hofstra_et_al:LIPIcs.FSCD.2021.26,
  author =	{Hofstra, Pieter and Parker, Jason and Scott, Philip J.},
  title =	{{Polymorphic Automorphisms and the Picard Group}},
  booktitle =	{6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)},
  pages =	{26:1--26:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-191-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{195},
  editor =	{Kobayashi, Naoki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.26},
  URN =		{urn:nbn:de:0030-drops-142646},
  doi =		{10.4230/LIPIcs.FSCD.2021.26},
  annote =	{Keywords: Partial Horn Theories, Monoidal Categories, Definable Automorphisms, Polymorphism, Indeterminates, Normal Forms}
}

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Polymorphic Automorphisms and the Picard Group

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