Map monoidales and duoidal $\infty$-categories
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917681751392256 |
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| author | Torii, Takeshi |
| author_facet | Torii, Takeshi |
| contents | In this paper we give an example of duoidal $\infty$-categories. We introduce map $\mathcal{O}$-monoidales in an $\mathcal{O}$-monoidal $(\infty,2)$-category for an $\infty$-operad $\mathcal{O}^{\otimes}$. We show that the endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is a coCartesian $(Δ^{\rm op},\mathcal{O})$-duoidal $\infty$-category. After that, we introduce a convolution product on the mapping $\infty$-category from an $\mathcal{O}$-comonoidale to an $\mathcal{O}$-monoidale. We show that the $\mathcal{O}$-monoidal structure on the duoidal endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is equivalent to the convolution product on the mapping $\infty$-category from the dual $\mathcal{O}$-comonoidale to the map $\mathcal{O}$-monoidale. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00223 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Map monoidales and duoidal $\infty$-categories Torii, Takeshi Category Theory Algebraic Topology 18N60 (primary), 18N70, 18M50, 55U40 (secondary) In this paper we give an example of duoidal $\infty$-categories. We introduce map $\mathcal{O}$-monoidales in an $\mathcal{O}$-monoidal $(\infty,2)$-category for an $\infty$-operad $\mathcal{O}^{\otimes}$. We show that the endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is a coCartesian $(Δ^{\rm op},\mathcal{O})$-duoidal $\infty$-category. After that, we introduce a convolution product on the mapping $\infty$-category from an $\mathcal{O}$-comonoidale to an $\mathcal{O}$-monoidale. We show that the $\mathcal{O}$-monoidal structure on the duoidal endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is equivalent to the convolution product on the mapping $\infty$-category from the dual $\mathcal{O}$-comonoidale to the map $\mathcal{O}$-monoidale. |
| title | Map monoidales and duoidal $\infty$-categories |
| topic | Category Theory Algebraic Topology 18N60 (primary), 18N70, 18M50, 55U40 (secondary) |
| url | https://arxiv.org/abs/2406.00223 |