Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Forest, Simon; Mimram, Samuel http://www.dagstuhl.de/lipics License
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URN: urn:nbn:de:0030-drops-91855
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Coherence of Gray Categories via Rewriting

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Abstract

Over the recent years, the theory of rewriting has been extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to low-dimensional weak categories, and consider in details the first non-trivial case: presentations of tricategories. By a general result, those are equivalent to the stricter Gray categories, for which we introduce a notion of rewriting system, as well as associated tools: critical pairs, termination orders, etc. We show that a finite rewriting system admits a finite number of critical pairs and, as a variant of Newman's lemma in our context, that a convergent rewriting system is coherent, meaning that two parallel 3-cells are necessarily equal. This is illustrated on rewriting systems corresponding to various well-known structures in the context of Gray categories (monoids, adjunctions, Frobenius monoids). Finally, we discuss generalizations in arbitrary dimension.

BibTeX - Entry

@InProceedings{forest_et_al:LIPIcs:2018:9185,
  author =	{Simon Forest and Samuel Mimram},
  title =	{{Coherence of Gray Categories via Rewriting}},
  booktitle =	{3rd International Conference on Formal Structures for  Computation and Deduction (FSCD 2018)},
  pages =	{15:1--15:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-077-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{108},
  editor =	{H{\'e}l{\`e}ne Kirchner},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9185},
  URN =		{urn:nbn:de:0030-drops-91855},
  doi =		{10.4230/LIPIcs.FSCD.2018.15},
  annote =	{Keywords: rewriting, coherence, Gray category, polygraph, pseudomonoid, precategory}
}

Keywords: rewriting, coherence, Gray category, polygraph, pseudomonoid, precategory
Seminar: 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)
Issue date: 2018
Date of publication: 2018


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