Free decomposition spaces
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Free decomposition spaces. / Hackney, Philip; Kock, Joachim.
2022.Research output: Working paper › Preprint › Research
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TY - UNPB
T1 - Free decomposition spaces
AU - Hackney, Philip
AU - Kock, Joachim
N1 - 31 pages
PY - 2022/10/20
Y1 - 2022/10/20
N2 - We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by their inert maps. We show that left Kan extension along the inclusion $j \colon \Delta_{\operatorname{inert}} \to \Delta$ takes general objects to M\"obius decomposition spaces and general maps to CULF maps. We establish an equivalence of $\infty$-categories $\mathbf{PrSh}(\Delta_{\operatorname{inert}}) \simeq \mathbf{Decomp}_{/B\mathbb{N}}$. Although free decomposition spaces are rather simple objects, they abound in combinatorics: it seems that all comultiplications of deconcatenation type arise from free decomposition spaces. We give an extensive list of examples, including quasi-symmetric functions. We show that the Aguiar--Bergeron--Sottile map to the decomposition space of quasi-symmetric functions, from any M\"obius decomposition space $X$, factors through the free decomposition space of nondegenerate simplices of $X$, and offer a conceptual explanation of the zeta function featured in the universal property of $\operatorname{QSym}$.
AB - We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by their inert maps. We show that left Kan extension along the inclusion $j \colon \Delta_{\operatorname{inert}} \to \Delta$ takes general objects to M\"obius decomposition spaces and general maps to CULF maps. We establish an equivalence of $\infty$-categories $\mathbf{PrSh}(\Delta_{\operatorname{inert}}) \simeq \mathbf{Decomp}_{/B\mathbb{N}}$. Although free decomposition spaces are rather simple objects, they abound in combinatorics: it seems that all comultiplications of deconcatenation type arise from free decomposition spaces. We give an extensive list of examples, including quasi-symmetric functions. We show that the Aguiar--Bergeron--Sottile map to the decomposition space of quasi-symmetric functions, from any M\"obius decomposition space $X$, factors through the free decomposition space of nondegenerate simplices of $X$, and offer a conceptual explanation of the zeta function featured in the universal property of $\operatorname{QSym}$.
KW - math.CT
KW - math.AT
KW - math.CO
KW - 18N50, 55U10, 18N60, 16T30, 05E05
M3 - Preprint
BT - Free decomposition spaces
ER -
ID: 373038370