Algebras for enriched $\infty$-operads
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arXiv
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866908626942164992 |
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| author | Haugseng, Rune |
| author_facet | Haugseng, Rune |
| contents | Using the description of enriched $\infty$-operads as associative algebras in symmetric sequences, we define algebras for enriched $\infty$-operads as certain modules in symmetric sequences. For $\mathbf{V}$ a symmetric monoidal model category and $\mathbf{O}$ a $Σ$-cofibrant operad in $\mathbf{V}$ for which the model structure on $\mathbf{V}$ can be lifted to one on $\mathbf{O}$-algebras, we then prove that strict algebras in $\mathbf{V}$ are equivalent to $\infty$-categorical algebras in the symmetric monoidal $\infty$-category associated to $\mathbf{V}$. We also show that for an $\infty$-operad $\mathcal{O}$ enriched in a suitable closed symmetric monoidal $\infty$-category $\mathcal{V}$, we can equivalently describe $\mathcal{O}$-algebras in $\mathcal{V}$ as morphisms of $\infty$-operads from $\mathcal{O}$ to a self-enrichment of $\mathcal{V}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_10042 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Algebras for enriched $\infty$-operads Haugseng, Rune Algebraic Topology Category Theory Using the description of enriched $\infty$-operads as associative algebras in symmetric sequences, we define algebras for enriched $\infty$-operads as certain modules in symmetric sequences. For $\mathbf{V}$ a symmetric monoidal model category and $\mathbf{O}$ a $Σ$-cofibrant operad in $\mathbf{V}$ for which the model structure on $\mathbf{V}$ can be lifted to one on $\mathbf{O}$-algebras, we then prove that strict algebras in $\mathbf{V}$ are equivalent to $\infty$-categorical algebras in the symmetric monoidal $\infty$-category associated to $\mathbf{V}$. We also show that for an $\infty$-operad $\mathcal{O}$ enriched in a suitable closed symmetric monoidal $\infty$-category $\mathcal{V}$, we can equivalently describe $\mathcal{O}$-algebras in $\mathcal{V}$ as morphisms of $\infty$-operads from $\mathcal{O}$ to a self-enrichment of $\mathcal{V}$. |
| title | Algebras for enriched $\infty$-operads |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/1909.10042 |