作为发散级数和的范畴的欧拉示性数

范畴的欧拉示性数

The Euler characteristic of a category

定义了有限范畴的欧拉示性数，并证明了它与其他类型的物体（包括轨道）的欧拉示性数是相容的.证明了集图上极限的基数公式，推广了经典的容斥公式。两者都依赖于从偏序集到范畴的Cupus-Rota反演的推广。

The Euler characteristic of an enriched category

We define Euler characteristic of a category enriched by a monoidal model category. If a monoidal model category V is equipped with Euler characteristic that is compatible with weak equivalences and fibrations in V, then our Euler characteristic is also compatible with weak equivalences and fibrations in the model structure induced by that of V. In particular, we focus on the case of topological categories; that is, categories enriched by the category of topological spaces. As its application, we obtain the ordinary Euler characteristic of a cellular stratified space X by computing the Euler c…

The zeta function of a finite category and the series Euler characteristic

We prove that a certain conjecture holds true and the conjecture states a relationship between the zeta function of a finite category and the Euler characteristic of a finite category.

Finiteness obstructions and Euler characteristics of categories

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 i…

Euler Characteristics of Categories and Homotopy Colimits

In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L^2-Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of I-indexed categories where I is any small category admitting a finite I-CW-model for its I-classifying space. Special cases of our Homotopy Colimit Formula include formulas for products, homotopy pushouts, homotopy orbits, and transport groupoids. We also apply our formulas to Haefliger complexes of groups, which extend Bass--Serre graphs of groups to highe…

有限范畴的欧拉示性数

The Euler Characteristic of Finite Categories

本文为对象和态射集有限的范畴定义了有理数指派 $χ(\mathcal{A})$，证明其在不相交并下可加且保持积，并将其视为遵循容斥原理的范畴族上的欧拉测度。