Free decomposition spaces

Philip Hackney; Joachim Kock

doi:10.1007/s13348-024-00446-8

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arXiv · 2210.11192

Free decomposition spaces

Philip Hackney·Joachim Kock

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Abstract

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.

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BibTeX RIS

Philip Hackney, Joachim Kock. 2022-10-20. Free decomposition spaces. https://doi.org/10.1007/s13348-024-00446-8

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