Euler characteristics of categories and homotopy colimits
Thomas M. Fiore
Department of Mathematics and Statistics, University of Michigan-Dearborn, 4901 Evergreen Road, Dearborn, MI 48128 U.S.AWolfgang Lück
Mathematisches Institut der Universität Bonn Endenicher Allee 60, 53115 Bonn, GermanyRoman Sauer
Fakultät für Mathematik Universität Regensburg Universitätsstr. 31 93053 Regensburg Germany

Abstract
In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and -Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of -indexed categories where is any small category admitting a finite --model for its -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 higher dimensions. In particular, we obtain necessary conditions for developability of a finite complex of groups from an action of a finite group on a finite category without loops.
Cite this article
Thomas M. Fiore, Wolfgang Lück, Roman Sauer, Euler characteristics of categories and homotopy colimits. Doc. Math. 16 (2011), pp. 301–354
DOI 10.4171/DM/333