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DOI: 10.4230/LIPIcs.RTA.2015.89

Presenting a Category Modulo a Rewriting System

Authors: Florence Clerc and Samuel Mimram

Published in: LIPIcs, Volume 36, 26th International Conference on Rewriting Techniques and Applications (RTA 2015)

Abstract

Presentations of categories are a well-known algebraic tool to provide descriptions of categories by the means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations where the objects are considered modulo an equivalence relation (in the spirit of rewriting modulo), which is described by equational generators. When those form a convergent (abstract) rewriting system on objects, there are three very natural constructions that can be used to define the category which is described by the presentation: one is based on restricting to objects which are normal forms, one consists in turning equational generators into identities (i.e. considering a quotient category), and one consists in formally adding inverses to equational generators (i.e. localizing the category). We show that, under suitable coherence conditions on the presentation, the three constructions coincide, thus generalizing celebrated results on presentations of groups. We illustrate our techniques on a non-trivial example, and hint at a generalization for 2-categories.

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Florence Clerc and Samuel Mimram. Presenting a Category Modulo a Rewriting System. In 26th International Conference on Rewriting Techniques and Applications (RTA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 36, pp. 89-105, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{clerc_et_al:LIPIcs.RTA.2015.89,
  author =	{Clerc, Florence and Mimram, Samuel},
  title =	{{Presenting a Category Modulo a Rewriting System}},
  booktitle =	{26th International Conference on Rewriting Techniques and Applications (RTA 2015)},
  pages =	{89--105},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-85-9},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{36},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2015.89},
  URN =		{urn:nbn:de:0030-drops-51916},
  doi =		{10.4230/LIPIcs.RTA.2015.89},
  annote =	{Keywords: presentation of a category, quotient category, localization, residuation}
}

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Presenting a Category Modulo a Rewriting System

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