Open Access
december 2012 Notions of Möbius inversion
Tom Leinster
Bull. Belg. Math. Soc. Simon Stevin 19(5): 909-933 (december 2012). DOI: 10.36045/bbms/1354031556


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.


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Tom Leinster. "Notions of Möbius inversion." Bull. Belg. Math. Soc. Simon Stevin 19 (5) 909 - 933, december 2012.


Published: december 2012
First available in Project Euclid: 27 November 2012

zbMATH: 1269.18002
MathSciNet: MR3009022
Digital Object Identifier: 10.36045/bbms/1354031556

Primary: 18A99
Secondary: 05A99 , 06A99 , 11A25 , ‎15A09 , 18C15

Keywords: enriched category , Euler characteristic of category , incidence algebra , matrix of category , Möbius inversion , Möbius--Rota inversion , poset , pullback-homomorphism , unique lifting of factorizations

Rights: Copyright © 2012 The Belgian Mathematical Society

Vol.19 • No. 5 • december 2012
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