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Polymorphic Automorphisms and the Picard Group

Authors Pieter Hofstra, Jason Parker, Philip J. Scott

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ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Theory of computation → Algebraic semantics
  • Theory of computation → Equational logic and rewriting
Keywords
  • Partial Horn Theories
  • Monoidal Categories
  • Definable Automorphisms
  • Polymorphism
  • Indeterminates
  • Normal Forms

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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) https://doi.org/10.4230/LIPIcs.FSCD.2021.26

Author Details

Pieter Hofstra
  • Dept. of Mathematics & Statistics, University of Ottawa, Canada
Jason Parker
  • Department of Mathematics & Computer Science, Brandon University, Canada
Philip J. Scott
  • Dept. of Mathematics & Statistics, University of Ottawa, Canada

Funding

  • Hofstra, Pieter: Research funded by an NSERC Discovery Grant.
  • Parker, Jason: Postdoctoral research funded by NSERC grant of R. Lucyshyn-Wright (Brandon).
  • Scott, Philip J.: Research funded by an NSERC Discovery Grant.

Acknowledgements

Pieter Hofstra would like to acknowledge illuminating discussions with Martti Karvonen and Eugenia Cheng. We would also like to thank the three anonymous referees for their insightful comments, corrections, and suggestions.

References

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