  We introduce notions of finiteness obstruction, Euler characteristic,
L^2-Euler characteristic, and M\"obius inversion for wide classes of
categories. The finiteness obstruction of a category Gamma of type (FP) is a
class in the projective class group K_0(RGamma); the functorial Euler
characteristic and functorial L^2-Euler characteristic are respectively its
RGamma-rank and L^2-rank. We also extend the second author's K-theoretic
M\"obius inversion from finite categories to quasi-finite categories. Our main
example is the proper orbit category, for which these invariants are
established notions in the geometry and topology of classifying spaces for
proper group actions. Baez-Dolan's groupoid cardinality and Leinster's Euler
characteristic are special cases of the L^2-Euler characteristic. Some of
Leinster's results on M\"obius-Rota inversion are special cases of the
K-theoretic M\"obius inversion.
