10.23638/LMCS-13(3:31)2017
Curien, Pierre-Louis
Pierre-Louis
Curien
Mimram, Samuel
Samuel
Mimram
Coherent Presentations of Monoidal Categories
episciences.org
2017
Computer Science - Logic in Computer Science
Mathematics - Category Theory
68Q42
F.4.2
contact@episciences.org
episciences.org
2017-09-26T07:56:20+02:00
2017-09-26T08:06:17+02:00
2017-09-26
eng
Journal article
https://lmcs.episciences.org/3955
arXiv:1705.03553
1860-5974
PDF
1
Logical Methods in Computer Science ; Volume 13, Issue 3 ; 1860-5974
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.