13 research outputs found
The Hecke Bicategory
We present an application of the program of groupoidification leading up to a
sketch of a categorification of the Hecke algebroid --- the category of
permutation representations of a finite group. As an immediate consequence, we
obtain a categorification of the Hecke algebra. We suggest an explicit
connection to new higher isomorphisms arising from incidence geometries, which
are solutions of the Zamolodchikov tetrahedron equation. This paper is
expository in style and is meant as a companion to Higher Dimensional Algebra
VII: Groupoidification and an exploration of structures arising in the work in
progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which
introduces the Hecke bicategory in detail.Comment: 27 pages, 11 .eps figures, Major revision of expositio
Higher-Dimensional Algebra VII: Groupoidification
Groupoidification is a form of categorification in which vector spaces are
replaced by groupoids, and linear operators are replaced by spans of groupoids.
We introduce this idea with a detailed exposition of "degroupoidification": a
systematic process that turns groupoids and spans into vector spaces and linear
operators. Then we present three applications of groupoidification. The first
is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator
arises naturally from degroupoidifying the groupoid of finite sets and
bijections. This allows for a purely combinatorial interpretation of creation
and annihilation operators, their commutation relations, field operators, their
normal-ordered powers, and finally Feynman diagrams. The second application is
to Hecke algebras. We explain how to groupoidify the Hecke algebra associated
to a Dynkin diagram whenever the deformation parameter q is a prime power. We
illustrate this with the simplest nontrivial example, coming from the A2 Dynkin
diagram. In this example we show that the solution of the Yang-Baxter equation
built into the A2 Hecke algebra arises naturally from the axioms of projective
geometry applied to the projective plane over the finite field with q elements.
The third application is to Hall algebras. We explain how the standard
construction of the Hall algebra from the category of representations of a
simply-laced quiver can be seen as an example of degroupoidification. This in
turn provides a new way to categorify - or more precisely, groupoidify - the
positive part of the quantum group associated to the quiver.Comment: 67 pages, 14 eps figures; uses undertilde.sty. This is an expanded
version of arXiv:0812.486
Convenient Categories of Smooth Spaces
A "Chen space" is a set X equipped with a collection of "plots" - maps from
convex sets to X - satisfying three simple axioms. While an individual Chen
space can be much worse than a smooth manifold, the category of all Chen spaces
is much better behaved than the category of smooth manifolds. For example, any
subspace or quotient space of a Chen space is a Chen space, and the space of
smooth maps between Chen spaces is again a Chen space. Souriau's "diffeological
spaces" share these convenient properties. Here we give a unified treatment of
both formalisms. Following ideas of Dubuc, we show that Chen spaces,
diffeological spaces, and even simplicial complexes are examples of "concrete
sheaves on a concrete site". As a result, the categories of such spaces are
locally cartesian closed, with all limits, all colimits, and a weak subobject
classifier. For the benefit of differential geometers, our treatment explains
most of the category theory we use.Comment: 43 pages, version to be published; includes corrected definition of
"concrete site
Nilpotency in type A cyclotomic quotients
We prove a conjecture made by Brundan and Kleshchev on the nilpotency degree
of cyclotomic quotients of rings that categorify one-half of quantum sl(k).Comment: 19 pages, 39 eps files. v3 simplifies antigravity moves and corrects
typo
Convenient categories of smooth spaces
A 'Chen space' is a set X equipped with a collection of 'plots', i.e., maps from convex sets to X, satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau's 'diffeological spaces' share these convenient properties. Here we give a unified treatment of both formalisms. Following ideas of Penon and Dubuc, we show that Chen spaces, diffeological spaces, and even simplicial complexes are examples of 'concrete sheaves on a concrete site'. As a result, the categories of such spaces are locally Cartesian closed, with all limits, all colimits, and a weak subobject classifier. For the benefit of differential geometers, our treatment explains most of the category theory we use. © 2011 John C. Baez and Alexander E. Hoffnung
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Categorified Symplectic Geometry and the Classical String
A Lie 2-algebra is a ‘categorified’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an ‘n-plectic manifold’: a finite-dimensional manifold equipped with a closed nondegenerate (n + 1)-form. Here we consider the case n = 2. For any 2-plectic manifold, we construct a Lie 2-algebra of observables. We then explain how this Lie 2-algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2-plectic structure for the string