Twisted tensor product of dg categories and Kontsevich's Swiss Cheese conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916524704399360 |
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| author | Batanin, Michael Shoikhet, Boris |
| author_facet | Batanin, Michael Shoikhet, Boris |
| contents | Let $A$ be a $1$-algebra. The Kontsevich Swiss Cheese conjecture [K2] states that the homotopy category $\mathrm{Ho}(\mathrm{Act}(A))$ of actions of $2$-algebras on $A$ has a final object and that this object is weakly equivalent to the pair $(\mathrm{Hoch}(A), A),$ where $\mathrm{Hoch}(A)$ is the Hochschild complex of $A$. Here the category of actions is the category whose objects are pairs $(B,A)$ which are algebras of the chain Swiss Cheese operad such that the induced action of the little interval operad on the component $A$ coincides with the $1$-structure on $A$.
We prove that there is a colored dg operad ${O}$ with 2 colors, weakly equivalent to the chain Swiss Cheese operad for which the following ``stricter" version of the Kontsevich Swiss Cheese conjecture holds. Denote the two colors of ${O}$ by $a$ (for the 1-algebra argument) and $b$ (for the 2-algebra argument), denote by $E_1^{O}$ the restriction of ${O}$ to the color $a$, and by $E_2^{O}$ the restriction of ${O}$ to the color $b$. Let $\mathrm{Alg}({O})$ be the category of dg algebras over ${O}$. For a fixed $1$-algebra $A$ we also have the the category of action $\mathrm{Alg}({O})_A$ (equal to $\mathrm{Act}(A)$ in case of the Swiss Cheese operad). We prove that there is an equivalence of categories $$\mathrm{Alg}({O})_A \cong \mathrm{Alg}(E_2^{O})/\mathrm{Hoch}(A)$$ We stress that for this particular model of Swiss Cheese operad the statement holds on the chain level, without passage to the homotopy category. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03239 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Twisted tensor product of dg categories and Kontsevich's Swiss Cheese conjecture Batanin, Michael Shoikhet, Boris Category Theory K-Theory and Homology Quantum Algebra Let $A$ be a $1$-algebra. The Kontsevich Swiss Cheese conjecture [K2] states that the homotopy category $\mathrm{Ho}(\mathrm{Act}(A))$ of actions of $2$-algebras on $A$ has a final object and that this object is weakly equivalent to the pair $(\mathrm{Hoch}(A), A),$ where $\mathrm{Hoch}(A)$ is the Hochschild complex of $A$. Here the category of actions is the category whose objects are pairs $(B,A)$ which are algebras of the chain Swiss Cheese operad such that the induced action of the little interval operad on the component $A$ coincides with the $1$-structure on $A$. We prove that there is a colored dg operad ${O}$ with 2 colors, weakly equivalent to the chain Swiss Cheese operad for which the following ``stricter" version of the Kontsevich Swiss Cheese conjecture holds. Denote the two colors of ${O}$ by $a$ (for the 1-algebra argument) and $b$ (for the 2-algebra argument), denote by $E_1^{O}$ the restriction of ${O}$ to the color $a$, and by $E_2^{O}$ the restriction of ${O}$ to the color $b$. Let $\mathrm{Alg}({O})$ be the category of dg algebras over ${O}$. For a fixed $1$-algebra $A$ we also have the the category of action $\mathrm{Alg}({O})_A$ (equal to $\mathrm{Act}(A)$ in case of the Swiss Cheese operad). We prove that there is an equivalence of categories $$\mathrm{Alg}({O})_A \cong \mathrm{Alg}(E_2^{O})/\mathrm{Hoch}(A)$$ We stress that for this particular model of Swiss Cheese operad the statement holds on the chain level, without passage to the homotopy category. |
| title | Twisted tensor product of dg categories and Kontsevich's Swiss Cheese conjecture |
| topic | Category Theory K-Theory and Homology Quantum Algebra |
| url | https://arxiv.org/abs/2412.03239 |