arXiv 0908.34172009-08math.ATmath.CT

范畴的非线性障碍与欧拉示性数

Finiteness obstructions and Euler characteristics of categories

Thomas M. Fiore, Wolfgang Lück, Roman Sauer

备注：: 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

摘要：

我们引入了有限性障碍、欧拉示性数、L^2-欧拉示性数和广义范畴的莫比乌斯反演的概念。（FP）型范畴Gamma的有限性阻塞是投射类群K_0（RGGamma）中的一类，函子欧拉示性数和函子L^2-欧拉示性数分别是其RGGamma秩和L^2秩.我们还将第二作者的K-理论莫比乌斯反演从有限范畴推广到拟有限范畴。我们的主要例子是真轨道范畴，这些不变量是为真群作用分类空间的几何学和拓扑学中建立的概念。Baez-Dolan的群胚基数和Leinster的欧拉示性数是L^2-欧拉示性数的特例。Leinster关于Möbius-Rota反演的一些结果是K-理论Möbius反演的特例。

英文摘要：

We introduce notions of finiteness obstruction, Euler characteristic, L^2-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 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 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 L^2-Euler characteristic. Some of Leinster's results on Möbius-Rota inversion are special cases of the K-theoretic Möbius inversion.

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