Categorified noncommutative manifolds
Martins, Rachel
International journal of modern physics A, 24(15) (2009), 2802-2819
http://dx.doi.org/10.1142/S0217751X09046175
We construct a noncommutative geometry with generalised 'tangent bundle' from Fell bundle C*-categories (E) beginning by replacing pair groupoid objects (points) with objects in E. This provides a categorification of a certain class of real spectral triples where the Dirac operator D is constructed from morphisms in a category. Applications for physics include quantization via the tangent groupoid and new constraints on Dfinite (the fermion mass matrix).
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