An unpublished result by the first author states that there exists a Hopf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.

http://www.tac.mta.ca/tac/volumes/24/10/24-10.dvi

http://www.tac.mta.ca/tac/volumes/24/10/24-10.ps

http://www.tac.mta.ca/tac/volumes/24/10/24-10.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/24/10/24-10.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/24/10/24-10.ps

Revised 2010-09-20. Original version at

http://www.tac.mta.ca/tac/volumes/24/10/24-10a.dvi