An Alternative to Homotopy Transfer for $A_\infty$-Algebras
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929390897594368 |
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| author | Cremonini, C. A. Marotta, V. E. |
| author_facet | Cremonini, C. A. Marotta, V. E. |
| contents | In this work, we propose a novel approach to the homotopy transfer procedure starting from a set of homotopy data such that the first differential complex is a differential graded module over the second one. We show that the module structure may be used to induce an $A_\infty$-algebra on the second differential complex, constructed in a similar fashion to the homotopy transfer $A_\infty$-algebra. We prove that, under certain conditions, the $A_\infty$-algebras obtained with this procedure are quasi-isomorphic to the homotopy transfer one. On the other hand, when the side conditions do not hold, we find that there are cases where the existence of an $A_\infty$-quasi-isomorphism with the homotopy transfer $A_\infty$-algebra is obstructed. In other words, we obtain a new $A_\infty$-algebra on the second complex, inequivalent to the homotopy transfer one. Lastly, we prove that these $A_\infty$-algebras are not infinitesimal Hochschild deformations of the homotopy transfer $A_\infty$-algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12508 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Alternative to Homotopy Transfer for $A_\infty$-Algebras Cremonini, C. A. Marotta, V. E. Algebraic Topology High Energy Physics - Theory Mathematical Physics In this work, we propose a novel approach to the homotopy transfer procedure starting from a set of homotopy data such that the first differential complex is a differential graded module over the second one. We show that the module structure may be used to induce an $A_\infty$-algebra on the second differential complex, constructed in a similar fashion to the homotopy transfer $A_\infty$-algebra. We prove that, under certain conditions, the $A_\infty$-algebras obtained with this procedure are quasi-isomorphic to the homotopy transfer one. On the other hand, when the side conditions do not hold, we find that there are cases where the existence of an $A_\infty$-quasi-isomorphism with the homotopy transfer $A_\infty$-algebra is obstructed. In other words, we obtain a new $A_\infty$-algebra on the second complex, inequivalent to the homotopy transfer one. Lastly, we prove that these $A_\infty$-algebras are not infinitesimal Hochschild deformations of the homotopy transfer $A_\infty$-algebra. |
| title | An Alternative to Homotopy Transfer for $A_\infty$-Algebras |
| topic | Algebraic Topology High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2406.12508 |