Decomposition spaces, incidence algebras and Möbius inversion

I'll start rather leisurely with a review of incidence algebras
and Möbius inversion, starting with the classical Möbius function
in number theory, then incidence algebras and Möbius inversion
for locally finite posets (Rota) and monoids with the finite
decomposition property (Cartier-Foata), and finally their common
generalisation to Möbius categories (Leroux). From here I'll move
on to survey recent work with Gálvez and Tonks taking these
constructions into homotopy theory. On one hand we generalise
from categories to infinity-categories in the form of
Rezk-complete Segal spaces, taking an objective approach working
directly with coefficients in infinity-groupoids instead of
numbers. (Under certain finiteness conditions, numerical results
can be obtained by taking homotopy cardinality.) On the other
hand we show that the Segal condition is not needed for these
constructions: it can be replaced by a weaker exactness condition
formulated in terms of the active-inert factorisation system in
Delta. These new simplicial spaces we call decomposition spaces:
while the Segal condition expresses composition, the new
condition expresses decomposition. (An equivalent notion,
formulated in terms of triangulations of polygons, was discovered
independently by Dyckerhoff and Kapranov under the name unital
2-Segal space.) Many convolution algebras in combinatorics arise
as the incidence algebra of decomposition spaces which are not 
categories, for example Schmitt's chromatic Hopf algebra of
graphs or the Butcher-Connes-Kreimer Hopf algebra of trees. The
Waldhausen S-construction of an abelian (or stable infinity-)
category is another example of a decomposition space; the
associated incidence algebras are versions of (derived) Hall
algebras. I will finish with some recent applications to free
probability (joint work with Ebrahimi-Fard, Foissy, and Patras).