2009/08/24 by Thomas M. Fiore, Fiore, Thomas M., Wolfgang Lück +3
Mathematics · #18F30 #18G10 #19A22 #19A49 #19J05 (Primary) #46L10 (Secondary) #Advanced Operator Algebra Research #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #math.CT #msc:18F30 #msc:18G10 #msc:19A22 #msc:19A49 #msc:19J05 #msc:46L10
paper · pdf · doi:10.48550/arxiv.0908.3417
Final version, accepted for publication in the Advances in Mathematics. Notational change: what was called chi(Gamma) in version 1 is now called chi(BGamma), and chi(Gamma) now signifies the sum of the components of the functorial Euler characteristic chi_f(Gamma). Theorem 5.25 summarizes when all Euler characteristics are equal. Minor typos have been corrected. 88 pages
openalex publication_date 2009/08/24 · arxiv created 2010/09/21 · arxiv updated 2010/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We introduce notions of finiteness obstruction, Euler characteristic, L2-Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Gamma of type (FP) is a class in the projective class group K0(RGamma); the functorial Euler characteristic and functorial L2-Euler characteristic are respectively its RGamma-rank and L2-rank. We also extend the second author's K-theoretic Möbius 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 L2-Euler characteristic. Some of Leinster's results on Möbius-Rota inversion are special cases of the K-theoretic Möbius inversion.