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An unpublished result by the first author states that there exists a Hopf algebra $H$ such that for any Möbius category $\cal C$ (in the sense of Leroux) there exists a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.
@article{TAC_2010_24_a9,
author = {F. W. Lawvere and M. Menni},
title = {The {Hopf} algebra of {M\"obius} intervals},
journal = {Theory and applications of categories},
pages = {221--265},
publisher = {mathdoc},
volume = {24},
year = {2010},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2010_24_a9/}
}
F. W. Lawvere; M. Menni. The Hopf algebra of Möbius intervals. Theory and applications of categories, Tome 24 (2010), pp. 221-265. http://geodesic.mathdoc.fr/item/TAC_2010_24_a9/