Decompositions spaces, incidence algebras, and Moebius inversion
I'll explain how the classical Leroux theory of incidence
algebras of Moebius categories (which covers both locally
finite posets (Rota) and finite-decomposition monoids
(Cartier-Foata)) admits a far-reaching generalisation in terms
of what we call decomposition spaces (introduced independently
by Dyckerhoff-Kapranov). They are simplicial (infinity)
groupoids satisfying an exactness condition weaker than the
Segal condition. Just as the Segal condition expresses
up-to-homotopy composition, the new condition expresses
decomposition. Specific new examples covered by the theory
include the Faa di Bruno and Connes-Kreimer bialgebras, and
the Lawvere-Menni Hopf algebra of Moebius intervals which
contains the universal Moebius function (but does not itself
come from a Moebius category). Generic classes of examples
include Schmitt incidence algebras of restriction species, and
Hall algebras: the Waldhausen S-construction of abelian (or
stable infinity) categories are decomposition spaces, and
their incidence algebras are Hall algebras.
While explaining the basic theory and the key examples
mentioned above, I will also spend some time explaining the
bigger programme of upgrading some aspects of enumerative and
algebraic combinatorics from finite sets to homotopy-finite
groupoids and (infinity-groupoids), and how recent progress in
higher category theory allows for this upgrade at a very
reasonable price.
This is joint work with Imma Galvez and Andy Tonks.