Symmetric monoidal extensions and graph cobordisms between finite sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908561658871808 |
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| author | Bianchi, Andrea |
| author_facet | Bianchi, Andrea |
| contents | Given a symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ and a space $X$, we address the problem of explicitly describing the symmetric monoidal $(\infty,n)$-category freely obtained from $\mathcal{C}$ by adjoining $X$ new $n$-morphisms with prescribed sources and targets. We develop an apparatus of tools that allow one to detect in concrete situations such a free symmetric monoidal extension. As motivating application, we introduce a symmetric monoidal $(\infty,2)$-category ${\mathbb{G}\mathrm{r}}$ of graph cobordisms between finite sets, following classical constructions of Gersten, Culler--Vogtmann and Hatcher--Vogtmann, and we exhibit it as an extension of the symmetric monoidal $(\infty,1)$-category $\mathrm{Fin}$ of finite sets, obtained by freely adjoining a specific list of new 1-morphisms and 2-morphisms. We recover results of Barkan--Steinebrunner and of Galatius. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22575 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetric monoidal extensions and graph cobordisms between finite sets Bianchi, Andrea Category Theory Algebraic Topology 18A40 18B10 18N50 18N55 18N65 19D23 Given a symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ and a space $X$, we address the problem of explicitly describing the symmetric monoidal $(\infty,n)$-category freely obtained from $\mathcal{C}$ by adjoining $X$ new $n$-morphisms with prescribed sources and targets. We develop an apparatus of tools that allow one to detect in concrete situations such a free symmetric monoidal extension. As motivating application, we introduce a symmetric monoidal $(\infty,2)$-category ${\mathbb{G}\mathrm{r}}$ of graph cobordisms between finite sets, following classical constructions of Gersten, Culler--Vogtmann and Hatcher--Vogtmann, and we exhibit it as an extension of the symmetric monoidal $(\infty,1)$-category $\mathrm{Fin}$ of finite sets, obtained by freely adjoining a specific list of new 1-morphisms and 2-morphisms. We recover results of Barkan--Steinebrunner and of Galatius. |
| title | Symmetric monoidal extensions and graph cobordisms between finite sets |
| topic | Category Theory Algebraic Topology 18A40 18B10 18N50 18N55 18N65 19D23 |
| url | https://arxiv.org/abs/2509.22575 |