24,423 research outputs found
Coherent Presentations of Monoidal Categories
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
Presentations for the punctured mapping class groups in terms of Artin groups
Consider an oriented compact surface F of positive genus, possibly with
boundary, and a finite set P of punctures in the interior of F, and define the
punctured mapping class group of F relatively to P to be the group of isotopy
classes of orientation-preserving homeomorphisms h: F-->F which pointwise fix
the boundary of F and such that h(P) = P. In this paper, we calculate
presentations for all punctured mapping class groups. More precisely, we show
that these groups are isomorphic with quotients of Artin groups by some
relations involving fundamental elements of parabolic subgroups.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-5.abs.htm
Partial mirror symmetry, lattice presentations and algebraic monoids
This is the second in a series of papers that develops the theory of
reflection monoids, motivated by the theory of reflection groups. Reflection
monoids were first introduced in arXiv:0812.2789. In this paper we study their
presentations as abstract monoids. Along the way we also find general
presentations for certain join-semilattices (as monoids under join) which we
interpret for two special classes of examples: the face lattices of convex
polytopes and the geometric lattices, particularly the intersection lattices of
hyperplane arrangements. Another spin-off is a general presentation for the
Renner monoid of an algebraic monoid, which we illustrate in the special case
of the "classical" algebraic monoids.Comment: 41 page
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