Notions of Möbius inversion

Abstract

Möbius inversion, originally a tool in number theory, was generalized to posets for use in group theory and combinatorics. It was later generalized to categories in two different ways, both of which are useful. We provide a unifying abstract framework. This allows us to compare and contrast the two theories of Möbius inversion for categories, and advance each of them. Among several side benefits is an improved understanding of the following fact: the Euler characteristic of the classifying space of a (suitably finite) category depends only on its underlying graph.

21 pages. Version 4: minor edits; final journal version

References in corpus (1)

• Sur l'existence d'une catégorie ayant une matrice strictement positive donnée

Cited by in corpus (5)

• Decomposition spaces, incidence algebras and Möbius inversion I: basic theory
• The magnitude of metric spaces
• Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration, and finiteness
• The zeta function of a finite category and the series Euler characteristic
• On the Categorification of the Möbius Function