nLab Euler characteristic (Rev #3, changes)

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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.

• There is a definition, due to Tom Leinster, of the Euler characteristic of a (suitably finite)Tom Leinster, of the Euler characteristic of a (suitably finite) category, which generalizes the notion of groupoid cardinality.