Finiteness obstructions and Euler characteristics of categories

TL;DR

This paper develops new invariants like finiteness obstructions and Euler characteristics for categories, extending classical concepts and connecting them to K-theory, with applications to proper orbit categories and classifying spaces.

Contribution

It introduces generalized notions of finiteness and Euler characteristics for broad classes of categories, extending M"obius inversion and linking to K-theoretic and geometric invariants.

Findings

Finiteness obstruction of a category is a class in K_0(RGamma).

Established invariants for the proper orbit category in topology.

Unified various existing invariants as special cases of L^2-Euler characteristic.

Abstract

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.