Higher-Dimensional Algebra VII: Groupoidification

摘要: Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators spans groupoids. We introduce this idea with detailed exposition "degroupoidification": systematic process that turns groupoids into operators. Then we present three applications groupoidification. The first to Feynman diagrams. Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying groupoid finite sets bijections. This allows purely combinatorial interpretation creation annihilation operators, their commutation relations, field normal-ordered powers, finally second application Hecke algebras. explain how groupoidify algebra associated Dynkin diagram whenever deformation parameter q prime power. illustrate simplest nontrivial example, coming A2 diagram. In example show solution Yang-Baxter equation built axioms projective geometry applied plane over elements. third Hall standard construction category representations simply-laced quiver can be seen as an degroupoidification. turn provides new way categorify - or more precisely, positive part group quiver.