Curien, Pierre-Louis and Mimram, Samuel - Coherent Presentations of Monoidal Categories

lmcs:3955 - Logical Methods in Computer Science, September 26, 2017, Volume 13, Issue 3
Coherent Presentations of Monoidal Categories

Authors: Curien, Pierre-Louis and Mimram, Samuel

Presentations of categories are a well-known algebraic tool to provide descriptions of categories by 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, 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 consists in turning equational generators into identities (i.e. considering a quotient category), one consists in formally adding inverses to equational generators (i.e. localizing the category), and one consists in restricting to objects which are normal forms. We show that, under suitable coherence conditions on the presentation, the three constructions coincide, thus generalizing celebrated results on presentations of groups, and we extend those conditions to presentations of monoidal categories.


Source : oai:arXiv.org:1705.03553
DOI : 10.23638/LMCS-13(3:31)2017
Volume: Volume 13, Issue 3
Published on: September 26, 2017
Submitted on: September 26, 2017
Keywords: Computer Science - Logic in Computer Science,Mathematics - Category Theory,68Q42,F.4.2


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