Free decomposition spaces
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| Acceso en línea: | |
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| _version_ | 1866911507079495680 |
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| author | Hackney, Philip Kock, Joachim |
| author_facet | Hackney, Philip Kock, Joachim |
| contents | 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 $j \colon Δ_{\operatorname{inert}} \to Δ$ takes general objects to Möbius decomposition spaces and general maps to CULF maps. We establish an equivalence of $\infty$-categories $\mathbf{PrSh}(Δ_{\operatorname{inert}}) \simeq \mathbf{Decomp}_{/B\mathbb{N}}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_11192 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Free decomposition spaces Hackney, Philip Kock, Joachim Category Theory Algebraic Topology Combinatorics 18N50, 55U10, 18N60, 16T30, 05E05 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 $j \colon Δ_{\operatorname{inert}} \to Δ$ takes general objects to Möbius decomposition spaces and general maps to CULF maps. We establish an equivalence of $\infty$-categories $\mathbf{PrSh}(Δ_{\operatorname{inert}}) \simeq \mathbf{Decomp}_{/B\mathbb{N}}$. 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. |
| title | Free decomposition spaces |
| topic | Category Theory Algebraic Topology Combinatorics 18N50, 55U10, 18N60, 16T30, 05E05 |
| url | https://arxiv.org/abs/2210.11192 |