  The angle defect, which is the standard way to measure curvature at the
vertices of polyhedral surfaces, goes back at least as far as Descartes.
Although the angle defect has been widely studied, there does not appear to be
in the literature an axiomatic characterization of the angle defect. We give a
characterization of the angle defect for simplicial surfaces, and we show that
variants of the same characterization work for two known approaches to
generalizing the angle defect to arbitrary 2-dimensional simplicial complexes.
Simultaneously, we give a characterization of the Euler characteristic on
2-dimensional simplicial complexes in terms of being geometrically locally
determined.
