Finiteness obstructions and Euler characteristics of categories

Abstract

We introduce notions of finiteness obstruction, Euler characteristic, L^2-Euler characteristic, and Möbius 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 Möbius 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 Möbius-Rota inversion are special cases of the K-theoretic Möbius inversion.

Final version, accepted for publication in the Advances in Mathematics. Notational change: what was called chi(Gamma) in version 1 is now called chi(BGamma), and chi(Gamma) now signifies the sum of the components of the functorial Euler characteristic chi_f(Gamma). Theorem 5.25 summarizes when all Euler characteristics are equal. Minor typos have been corrected. 88 pages