The Euler characteristic of a category

Abstract

The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusion-exclusion formula. Both rest on a generalization of Rota's Möbius inversion from posets to categories. Euler characteristic, finite category, inclusion-exclusion, Möbius inversion, cardinality of colimit.