Higher Dimensional Algebra VII: Groupoidification

By John C. Baez, Alexander E. Hoffnung, Christopher D. Walker. Theory and Applications of Categories, 2009

59 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/21ac24760c456d3ca568a04277ccc50b4126b9f3

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Citations in the corpus

Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory (2015, background): “In [24] we have developed the necessary ‘homotopy linear algebra’ and the notion of homotopy cardinality, extending many results of Baez–Hoffnung–Walker [3] who worked with 1-groupoids.”
Fusion 2-categories and a state-sum invariant for 4-manifolds (2018, background): “More generally, for a finite 1-groupoid (π0, π1), the dimension of the 2-category of (π0, π1)-graded 2-vector spaces 2Rep(π0, π1, ∗) is the groupoid cardinality [BHW10] of (π0, π1), that is the sum over components of the reciprocal of the size of the automorphism groups.”
Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness (2015, unknown): “In [10] we have developed the necessary ‘homotopy linear algebra’ and homotopy cardinality, extending and streamlining many results of Baez–Hoffnung– Walker [2] who worked with 1-groupoids.”