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DOI: 10.4230/LIPIcs.FSCD.2018.15
URN: urn:nbn:de:0030-drops-91855
URL: https://drops.dagstuhl.de/opus/volltexte/2018/9185/
Forest, Simon ; Mimram, Samuel
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: | 04.07.2018 |
