Decomposition spaces, incidence algebras, and Moebius inversion
I'll survey recent work with Imma Galvez and Andy Tonks
[arXiv:1404.3202] developing an infinity-version of the theory
of incidence algebras and Moebius inversion. The 'combinatorial
objects' playing the role of posets and Moebius categories are
DECOMPOSITION SPACES, simplicial $\infty$-groupoids 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. The role of vector
spaces is played by slices over infinity-groupoids, eventually
with homotopy finiteness conditions imposed. To any
decomposition space (subject to a completeness condition weaker
than the Rezk condition) there is associated an incidence
(co)algebra (with coefficients in infinity-groupoids), which
satisfies an objective Moebius inversion principle in the style
of Lawvere-Menni. Generic examples of decomposition spaces
beyond Segal spaces are given by the Waldhausen S-construction
(yielding Hall algebras) and by Schmitt restriction species, and
many examples from classical combinatorics admit uniform
descriptions in this framework. My focus will be on a specific
example, namely the Lawvere-Menni Hopf algebra of Moebius
intervals, which contains the universal Moebius function, but
does not itself come from a Moebius category in the classical
sense. It turns out that it DOES come from a decomposition
space, which is in some sense universal.
(The notion of decomposition space is equivalent to the notion
of unital 2-Segal space of Dyckerhoff-Kapranov [arXiv:1212.3563],
who develop other aspects of the theory, motivated by geometry,
representation theory and homological algebra.)