Euler characteristics of categories and homotopy colimits

  • Thomas M. Fiore

    Wolfgang Lück Department of Mathematics Mathematisches Institut and Statistics der Universität Bonn University of Michigan-Dearborn Endenicher Allee 60 4901 Evergreen Road 53115 Bonn Dearborn, MI 48128 Germany U.S.A
  • Wolfgang Lück

  • Roman Sauer

    Fakultät für Mathematik Universität Regensburg Universitätsstr. 31 93053 Regensburg Germany
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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 \( \cali \)-indexed categories where \( \cali \) is any small category admitting a finite \( \cali-CW \)-model for its \( \cali \)-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.

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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