The universal property of the decomposition space of quasisymmetric functions

Abstract

Aguiar, Bergeron, and Sottile showed that the coalgebra of quasisymmetric functions is terminal among connected graded coalgebras with a zeta function. After developing the notions of gradings and zeta functions for decomposition spaces, we establish an objective homotopic version of this result. Namely, the decomposition space $Q$ of quasisymmetric functions is terminal: for any conservatively graded decomposition space $X$ with zeta function $F$, there is a unique graded ikeo-culf span $X\leftarrow W\to Q$ inducing $F$ from the special zeta function on $Q$. More generally, for gradings in any complete decomposition space $M$ with indecomposable identities, the terminal object is $J(M)$, the free decomposition space on the nondegenerate simplices of $M$; the quasisymmetric case is $Q=J(B\mathbb{N})$.

arXiv comment44 pages. comments welcome!