Incidence algebras and Moebius inversion in Rezk categories
and decomposition spaces
I'll explain how the classical theory of incidence algebras
of locally finite posets (Rota) and Moebius categories
(Leroux) can be generalised to higher categories, leading to
the new notion of decomposition space: it 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 (and as everybody knows from watch repairs, it
is much easier to decompose than to compose). New examples
covered by the theory include the Faa di Bruno and
Connes-Kreimer bialgebras, the Lawvere-Menni category of
Moebius intervals which contains the universal Moebius
function (but is not itself a Moebius category), and Hall
algebras: the Waldhausen S-construction of abelian (or stable
infinity) categories are decomposition spaces, and their
incidence algebras are Hall algebras. This is joint work
with Imma Galvez and Andy Tonks.