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arXiv · 0908.3417

Finiteness obstructions and Euler characteristics of categories

Thomas M. Fiore·Wolfgang Lück·Roman Sauer
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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.

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Thomas M. Fiore, Wolfgang Lück, Roman Sauer. 2009-08-24. Finiteness obstructions and Euler characteristics of categories. https://arxiv.org/abs/0908.3417

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