The coalgebra QSym of quasisymmetric functions was shown by Aguiar,
Bergeron, and Sottile to be the terminal object in the category of
graded coalgebras with a zeta function. I'll explain a categorified
version of that result, in the framework of simplicial homotopy theory:
QSym is the incidence coalgebra of a decomposition space Q of monotone
surjections, and its zeta function Z is given by the empty surjection
and the connected surjections. We show that for any graded decomposition
space X with a zeta function F, there is a unique graded span of
decomposition spaces X ← J → Q, where the backward map is ikeo (inner
Kan and equivalence on objects) and the forward map is culf
(conservative and unique lifting of factorisations) inducing F from Z.
(Such spans induce coalgebra homomorphisms, and conjecturally all.) In
fact, the result turns out to be much more general, leading to a
functoriality that also gives the universal property of QSym as a
bialgebra, and giving as a side effect Hoffmann's general construction
of quasi-shuffles. (Most of the talk will be about simplicial machinery,
and a secondary goal is to promote interactions between algebraic
combinatorics and homotopy theory/category theory.)

This is joint work with Philip Hackney and Jan Steinebrunner