Matrix Geometries Emergent from a Point

Abstract

We describe a categorical approach to finite noncommutative geometries. Objects in the category are spectral triples, rather than unitary equivalence classes as in other approaches. This enables to treat fluctuations of the metric and unitary equivalences on the same footing, as representatives of particular morphisms in this category. We then show how a matrix geometry (Moyal plane) emerges as a fluctuation from one point, and discuss some geometric aspects of this space.