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The Euler characteristic can refer to a number of different numerical invariants of different sorts of objects. But all of them bear a family resemblance and in some cases they can be shown to agree.
The Euler characteristic of a topological space can be defined as the alternating sum of the ranks of its homology groups (when this is finite).
The Euler characteristic of a dualizable object in a symmetric monoidal category is the trace of its identity morphism. Applied to suspension spectra in the stable homotopy category , this recovers the usual Euler characteristic of a manifold.manifold.
There is a definition, due to Tom Leinster, of the Euler characteristic of a (suitably finite) category, which generalizes the notion of groupoid cardinality.
Mike: Can the Euler characteristic of a category be recovered as the trace for a dualizable object in some symmetric monoidal category?
Fiore et al on Euler char of categories.
arXiv:1007.3868 Euler Characteristics of Categories and Homotopy Colimits from arXiv Front: math.CT by Thomas M. Fiore, Wolfgang Lück, Roman Sauer 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 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.
<http://mathoverflow.net/questions/35156/how-do-you-define-the-euler-characteristic-of-a-scheme>
<http://mathoverflow.net/questions/74843/how-to-interpret-the-euler-characteristic-of-complex-algebraic-varieties>
nLab page on Euler characteristic
Revision on May 23, 2009 at 15:06:50 by Toby Bartels See the history of this page for a list of all contributions to it.