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 $\cali$-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.

2010 Mathematics Subject Classification: Primary: 18F30, 19J05; Secondary: 18G10, 19A49, 55U35, 19A22, 46L10.

Keywords and Phrases: finiteness obstruction, Euler characteristic of a category, $L^2$-Euler characteristic, projective class group, homotopy colimits of categories, Grothendieck construction, spaces over a category, Grothendieck fibration, complex of groups, small category without loops.

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