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Rewriting in Gray categories with applications to coherence

Published online by Cambridge University Press:  22 November 2022

Simon Forest  and
Samuel Mimram Open the ORCID record for Samuel Mimram [Opens in a new window]
Simon Forest
Affiliation:
LIX, École Polytechnique, Palaiseau, France
Samuel Mimram*
Affiliation:
LIX, École Polytechnique, Palaiseau, France
*
*Corresponding author. Email: samuel.mimram@lix.polytechnique.fr
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Abstract

Over the recent years, the theory of rewriting has been used and extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to Gray categories, which are known to be equivalent to tricategories. This requires us to develop the theory of rewriting in the setting of precategories, which are adapted to mechanized computations and include Gray categories as particular cases. We show that a finite rewriting system in precategories admits a finite number of critical pairs, which can be efficiently computed. We also extend Squier’s theorem to our context, showing that a convergent rewriting system is coherent, which means that any two parallel 3-cells are necessarily equal. This allows us to prove coherence results for several well-known structures in the context of Gray categories: monoids, adjunctions, and Frobenius monoids.

Keywords

RewritingcoherenceGray categorypolygraph
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 32 , Issue 5 , May 2022 , pp. 574 - 647
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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