Finiteness obstructions and Euler characteristics of categories

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• 作者：Thomas M. Fiore ; Wolfgang Lü ; ck ; Roman Sauer
• 关键词：Finiteness obstruction ; Euler characteristic of a category ; Projective class group ; M&#xf6 ; bius inversion ; ; none ; color ; black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6W9F-515RHM8-2&_mathId=mml7&_user=
• 刊名：Advances in Mathematics
• 出版年：2011
• 出版时间：15 February 2011
• 年：2011
• 卷：226
• 期：3
• 页码：2371-2469
• 全文大小：639 K

文摘

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 Γ of type (FP_R) is a class in the projective class group K₀(RΓ); the functorial Euler characteristic and functorial L²-Euler characteristic are respectively its RΓ-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 and 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.