## The Euler characteristic of a category

### Summary

Summary: 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.

### Mathematics Subject Classification

18F99, 55U99, 05C50, 57N65

### Keywords/Phrases

Euler characteristic, finite category, inclusion-exclusion, Möbius inversion, cardinality of colimit