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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.14545 |
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| _version_ | 1866908940271353856 |
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| author | Pimenov, Slava Toledo, Angel |
| author_facet | Pimenov, Slava Toledo, Angel |
| contents | Let $(\mathcal{C}, \otimes)$ be a monoidal dg-category. We construct a complex controlling the deformation of the monoidal structure on $\mathcal{C}$ together with the deformation of the underlying dg-category itself. We show that in the case of a semisimple category $\mathcal{C}$ it reduces to the Davydov-Yetter complex. Furthermore, we study this complex in several special cases, in particular, in the case of the category of $A$-modules over a commutative algebra $A$ we obtain a complex computing operadic $E_2$-cohomology of $A$. And in the case of the category of representations of an associative bialgebra we recover the Gerstenhaber-Schack complex. In the latter case our construction can be considered as a generalization of the Gerstenhaber-Schack complex to quasi-bialgebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14545 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tensor-Hochschild complex Pimenov, Slava Toledo, Angel Algebraic Geometry Category Theory Quantum Algebra 18M05 (Primary) 16E40, 16S80, 17B37 (Secondary) Let $(\mathcal{C}, \otimes)$ be a monoidal dg-category. We construct a complex controlling the deformation of the monoidal structure on $\mathcal{C}$ together with the deformation of the underlying dg-category itself. We show that in the case of a semisimple category $\mathcal{C}$ it reduces to the Davydov-Yetter complex. Furthermore, we study this complex in several special cases, in particular, in the case of the category of $A$-modules over a commutative algebra $A$ we obtain a complex computing operadic $E_2$-cohomology of $A$. And in the case of the category of representations of an associative bialgebra we recover the Gerstenhaber-Schack complex. In the latter case our construction can be considered as a generalization of the Gerstenhaber-Schack complex to quasi-bialgebras. |
| title | Tensor-Hochschild complex |
| topic | Algebraic Geometry Category Theory Quantum Algebra 18M05 (Primary) 16E40, 16S80, 17B37 (Secondary) |
| url | https://arxiv.org/abs/2505.14545 |