Coherence of Gray Categories via Rewriting

Authors Simon Forest, Samuel Mimram

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Author Details

Simon Forest
  • LIX, École Polytechnique, France
Samuel Mimram
  • LIX, École Polytechnique, France

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Simon Forest and Samuel Mimram. Coherence of Gray Categories via Rewriting. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Rewrite systems
  • rewriting
  • coherence
  • Gray category
  • polygraph
  • pseudomonoid
  • precategory


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