The Euler characteristic is among the
earliest and most elementary homotopy invariants. For a finite
simplicial complex, it is the alternating sum of the numbers of
simplices in each dimension. This combinatorially defined invariant
has remarkable connections to geometric notions, such as genus,
curvature, and area.
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(RΓ), 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.