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Higher Dimensional Algebra VII: Groupoidification

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John C. Baez, Alexander E. Hoffnung, and Christopher D. Walker

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 $A_2$ Dynkin diagram. In this
example we show that the solution of the Yang--Baxter equation built
into the $A_2$ Hecke algebra arises naturally from the axioms of
projective geometry applied to the projective plane over the finite
field $\mathbb{F}_q$. The third application is to Hall algebras. We
explain how the standard construction of the Hall algebra from the
category of $\mathbb{F}_q$ 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.

Keywords:
categorification, groupoid, Hecke algebra, Hall algebra,
quantum theory

2000 MSC:
17B37, 20C08, 20L05, 81R50, 81T18

*Theory and Applications of Categories,*
Vol. 24, 2010,
No. 18, pp 489-553.

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