# The Hopf algebra of Möbius intervals

## F. W. Lawvere and M. Menni

An unpublished result by the first author states that there exists a Hopf algebra $H$ such that for any Möbius category $\cal C$ (in the sense of Leroux) there exists a canonical algebra morphism from the dual $H^*$ of $H$ to the incidence algebra of $\cal C$. Moreover, the Möbius inversion principle in incidence algebras follows from a `master' inversion result in $H^*$. The underlying module of $H$ was originally defined as the free module on the set of iso classes of Möbius intervals, i.e. Möbius categories with initial and terminal objects. Here we consider a category of Möbius intervals and construct the Hopf algebra via the objective approach applied to a monoidal extensive category of combinatorial objects, with the values in appropriate rings being abstracted from combinatorial functors on the objects. The explicit consideration of a category of Möbius intervals leads also to two new characterizations of Möbius categories.

Keywords: Möbius category, incidence algebra

2000 MSC: 18A05, 13J05

Theory and Applications of Categories, Vol. 24, 2010, No. 10, pp 221-265.

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Revised 2010-09-20. Original version at
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