Finiteness obstructions and Euler characteristics of categories

We introduce notions of finiteness obstruction, Euler characteristic, L-2-Euler characteristic, and Mobius inversion for wide classes of categories. The finiteness obstruction of a category Gamma of type (FPR) is a class in the projective class group K-0(R Gamma); the functorial Euler characteristic and functorial L-2-Euler characteristic are respectively its R Gamma-rank and L-2-rank. We also extend the second author's K-theoretic Mobius 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 and Dolan's groupoid cardinality and Leinster's Euler characteristic are special cases of the L-2-Euler characteristic. Some of Leinster's results on Mobius-Rota inversion are special cases of the K-theoretic Mobius inversion. (c) 2010 Elsevier Inc. All rights reserved.