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 Γ of type ( FP R ) is a class in the projective class group K 0 ( R Γ ) ; the functorial Euler characteristic and functorial L 2 -Euler characteristic are respectively its RΓ-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 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 Möbius–Rota inversion are special cases of the K-theoretic Möbius inversion.