Decomposition spaces, incidence algebras and Mšbius inversion
I'll start rather leisurely with a review of incidence algebras
and Mšbius inversion, starting with the classical Mšbius
function in number theory, then incidence algebras and Mšbius
inversion for locally finite posets (Rota) and monoids with the
finite decomposition property (Cartier-Foata), and finally their
common generalisation to Mšbius categories (Leroux). From here
I'll move on to survey recent work with G‡lvez and Tonks taking
these constructions into homotopy theory. On one hand we
generalise from categories to 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 generic and free maps in Delta. This new notion we
call decomposition space: while the Segal condition expresses
up-to-homotopy composition, the new condition expresses
decomposition. (An equivalent notion, formulated in terms of
triangulations of polygons, was discovered independently by
Dyckerhoff-Kapranov under the name unital 2-Segal space.) Many
convolution algebras in combinatorics arise as the incidence
algebra of a decomposition space which is not a category, for
example those constructed by Schmitt from restriction species.
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. To finish I'll explain how the Lawvere-Menni Hopf
algebra, which contains the universal Mšbius function, arises
from a decomposition space of Mšbius intervals. [Reference:
arXiv:1404.3202.]