Morphisms of pre-Calabi-Yau categories and morphisms of cyclic $A_{\infty}$-categories
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914741424750592 |
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| author | Boucrot, Marion |
| author_facet | Boucrot, Marion |
| contents | In this article we prove that there exists a relation between $d$-pre-Calabi-Yau morphisms introduced by M. Kontsevich, A. Takeda and Y. Vlassopoulos and cyclic $A_{\infty}$-morphisms, extending a result proved by D. Fernández and E. Herscovich. This leads to a functor between the category of $d$-pre-Calabi-Yau structures and the partial category of $A_{\infty}$-categories of the form $\mathcal{A}\oplus\mathcal{A}^*[d-1]$ with $\mathcal{A}$ a graded quiver and whose morphisms are the data of an $A_{\infty}$-structure on $\mathcal{A}\oplus\mathcal{B}^*[d-1]$ together with $A_{\infty}$-morphisms $\mathcal{A}[1]\oplus\mathcal{B}^*[d]\rightarrow \mathcal{A}[1]\oplus\mathcal{A}^*[d]$ and $\mathcal{A}[1]\oplus\mathcal{B}^*[d]\rightarrow \mathcal{B}[1]\oplus\mathcal{B}^*[d]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_13661 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Morphisms of pre-Calabi-Yau categories and morphisms of cyclic $A_{\infty}$-categories Boucrot, Marion K-Theory and Homology Category Theory In this article we prove that there exists a relation between $d$-pre-Calabi-Yau morphisms introduced by M. Kontsevich, A. Takeda and Y. Vlassopoulos and cyclic $A_{\infty}$-morphisms, extending a result proved by D. Fernández and E. Herscovich. This leads to a functor between the category of $d$-pre-Calabi-Yau structures and the partial category of $A_{\infty}$-categories of the form $\mathcal{A}\oplus\mathcal{A}^*[d-1]$ with $\mathcal{A}$ a graded quiver and whose morphisms are the data of an $A_{\infty}$-structure on $\mathcal{A}\oplus\mathcal{B}^*[d-1]$ together with $A_{\infty}$-morphisms $\mathcal{A}[1]\oplus\mathcal{B}^*[d]\rightarrow \mathcal{A}[1]\oplus\mathcal{A}^*[d]$ and $\mathcal{A}[1]\oplus\mathcal{B}^*[d]\rightarrow \mathcal{B}[1]\oplus\mathcal{B}^*[d]$. |
| title | Morphisms of pre-Calabi-Yau categories and morphisms of cyclic $A_{\infty}$-categories |
| topic | K-Theory and Homology Category Theory |
| url | https://arxiv.org/abs/2304.13661 |