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By Imma Gálvez, Joachim Kock and Andy Tonks
ArXiv:1404.3202

#### Abstract

We introduce the notion of decomposition space as a general
framework for incidence algebras and Möbius inversion. A
decomposition space is a simplicial infinity-groupoid satisfying
an exactness condition weaker than the Segal condition,
expressed in terms of generic and free maps in Delta. Just as
the Segal condition expresses up-to-homotopy composition, the
new condition expresses decomposition. We work as much as
possible on the objective level of linear algebra with
coefficients in infinity-groupoids, and develop the necessary
homotopy linear algebra along the way. Independently of
finiteness conditions, to any decomposition space there is
associated an incidence (co)algebra (with coefficients in
infinity-groupoids), and under a completeness condition (weaker
than the Rezk condition) this incidence algebra is shown to
satisfy a sign-free version of the Möbius inversion principle.
Examples of decomposition spaces beyond Segal spaces are given
by the Waldhausen S-construction of an abelian (or stable
infinity) category. Their incidence algebras are various kinds
of Hall algebras. Another class of examples are Schmitt
restriction species. Imposing certain homotopy finiteness
conditions yields the notion of Möbius decomposition space,
covering the notion of Möbius category of Leroux (itself a
common generalisation of locally finite posets (Rota et al.)
and finite decomposition monoids (Cartier-Foata)), as well as
many constructions of Dür, including the Faà di Bruno and
Connes-Kreimer bialgebras. We take a functorial viewpoint
throughout, emphasising conservative ULF functors, and show that
most reduction procedures in the classical theory of incidence
coalgebras are examples of this notion, and in particular that
many are an example of decalage of decomposition spaces. Our
main theorem concerns the Lawvere-Menni Hopf algebra of Möbius
intervals, which contains the universal Möbius function (but
does not come from a Möbius category): we establish that Möbius
intervals (in the infinity-setting) form a decomposition space,
and that it has the universal property also with respect to
Möbius inversion in general decomposition spaces.

NOTE: The notion of decomposition space was arrived at
independently by Dyckerhoff and Kapranov (arXiv:1212.3563) who
call them unital 2-Segal spaces. Our theory is quite
orthogonal to theirs: the definitions are different in spirit
and appearance, and the theories differ in terms of motivation,
examples and directions. For the few overlapping results
('decalage of decomposition is Segal' and 'Waldhausen's S is
decomposition'), our approach seems generally simpler.

Last updated: 2014-04-19 by
Joachim Kock.