Coherence of Gray Categories via Rewriting

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: Tietze transformations, 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.
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Simon Forest, Samuel Mimram. Coherence of Gray Categories via Rewriting. 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018), Jul 2018, Oxford, United Kingdom. ⟨10.4230/LIPIcs.FSCD.2018.15⟩. ⟨hal-02154822⟩

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