Euler Characteristics of Categories and Homotopy Colimits

In this talk I will present a topological approach to Euler characteristics of categories in terms of finiteness obstructions. This approach is compatible with almost anything one would want, for example products, coproducts, covering maps, isofibrations, and homotopy colimits. Classical constructions are special cases, for example, under appropriate hypotheses the functorial $L^2$-Euler characteristic of the proper orbit category for a group $G$ is the equivariant Euler characteristic of the classifying space for proper $G$-actions. Our Homotopy Colimit Formula for Euler characteristics provides formulas for products, homotopy pushouts, homotopy orbits, and transport groupoids. This is joint work with Wolfgang Lueck and Roman Sauer.