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 ← W → 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(Bℕ).
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BibTeX
@misc{hackney2026universal,
title = {{The universal property of the decomposition space of quasisymmetric functions}},
author = {Philip Hackney and Joachim Kock and Jan Steinebrunner},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08063},
eprint = {2611.08063},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08063},
}
RIS
TY - UNPB
TI - The universal property of the decomposition space of quasisymmetric functions
AU - Philip Hackney
AU - Joachim Kock
AU - Jan Steinebrunner
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08063
UR - https://arxiv.org/abs/2611.08063
AB - 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 ← W → 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(Bℕ).
ER -