Free rigid commutative algebras
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911562914070528 |
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| author | Ramzi, Maxime |
| author_facet | Ramzi, Maxime |
| contents | We describe free rigid commutative algebras in $2$-presentably symmetric monoidal $(\infty,2)$-categories as oplax colimits over the $1$-dimensional framed cobordism category. The special case of the $(\infty,2)$-category $\mathrm{Pr}^\mathrm{L}$ itself provides a description of the free symmetric monoidal $(\infty,1)$-category with duals on a given $(\infty,1)$-category, while the case of $\mathrm{Mod}_{\mathcal{V}}(\mathrm{Pr}^\mathrm{L})$ provides a description of a similar object in the $\mathcal{V}$-enriched context, for $\mathcal{V}$ a presentably symmetric monoidal $(\infty,1)$-category.
As a byproduct, we obtain new proofs of some results about rigidification of locally rigid categories, as well as a proof that any rigid category over $\mathrm{Sp}$ embeds into a compactly-rigidly generated one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01854 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Free rigid commutative algebras Ramzi, Maxime Category Theory Algebraic Topology K-Theory and Homology We describe free rigid commutative algebras in $2$-presentably symmetric monoidal $(\infty,2)$-categories as oplax colimits over the $1$-dimensional framed cobordism category. The special case of the $(\infty,2)$-category $\mathrm{Pr}^\mathrm{L}$ itself provides a description of the free symmetric monoidal $(\infty,1)$-category with duals on a given $(\infty,1)$-category, while the case of $\mathrm{Mod}_{\mathcal{V}}(\mathrm{Pr}^\mathrm{L})$ provides a description of a similar object in the $\mathcal{V}$-enriched context, for $\mathcal{V}$ a presentably symmetric monoidal $(\infty,1)$-category. As a byproduct, we obtain new proofs of some results about rigidification of locally rigid categories, as well as a proof that any rigid category over $\mathrm{Sp}$ embeds into a compactly-rigidly generated one. |
| title | Free rigid commutative algebras |
| topic | Category Theory Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2604.01854 |