Euler characteristics are not only defined for simplicial complexes or manifolds, but for many other objects as well, such as posets and, more generally, categories. We propose in this talk a topological approach to Euler characteristics of categories. The idea, phrased in homological algebra, is the following. Given a category Γ and a ring R, we take a finite projective RΓ-module resolution P* of the constant module R (assuming such a resolution exists). The alternating sum of the modules Pi is the finiteness obstruction o(Γ,R). It is a class in the projective class group K_0(R&Gamma), which is the free abelian group on isomorphism classes of finitely generated projective RΓ-modules modulo short exact sequences. From the finiteness obstruction we obtain the Euler characteristic respectively L2-Euler characteristic, by adding the entries of the RΓ-rank respectively the L2-rank of the finiteness obstruction.
This topological approach has many advantages, several of which now follow. First of all, this approach is compatible with almost anything one would want, for example products, coproducts, covering maps, isofibrations, and homotopy colimits. It works equally well for infinite categores and finite categories. There are many examples. Classical constructions are special cases, for example, under appropriate hypotheses the functorial L2-Euler characteristic of the proper orbit category for a group G is the equivariant Euler characteristic of the classifying space for proper G-actions. The K-theoretic Möbius inversion has Möbius-Rota inversion and Leinster's Möbius inversion as special cases. The classical Burnside ring congruences arise from rational Möbius inversion.
This talk will focus on our Homotopy Colimit Formula for Euler characteristics.
In certain cases, the L2-Euler characteristic agrees with the groupoid cardinality of Baez-Dolan and the Euler characteristic of Leinster, and comparisons will be made.
This is joint work with Wolfgang Lück and Roman Sauer. Our
preprints are available online:
Finiteness obstructions and Euler characteristics of categories. Accepted at the Advances in Mathematics.
Euler characteristics of categories and homotopy colimits. Accepted at Documenta Mathematica.