Hochschild cohomology for functors on linear symmetric monoidal categories
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866908946284937216 |
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| author | Romero, Nadia |
| author_facet | Romero, Nadia |
| contents | Let $R$ be a commutative ring with unit. We develop a Hochschild cohomology theory in the category $\mathcal{F}$ of linear functors defined from an essentially small symmetric monoidal category enriched in $R$-Mod, to $R$-Mod. The category $\mathcal{F}$ is known to be symmetric monoidal too, so one can consider monoids in $\mathcal{F}$ and modules over these monoids, which allows for the possibility of a Hochschild cohomology theory. The emphasis of the article is in considering natural hom constructions appearing in this context. These homs, together with the abelian structure of $\mathcal{F}$ lead to nice definitions and provide effective tools to prove the main properties and results of the classical Hochschild cohomology theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_12269 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hochschild cohomology for functors on linear symmetric monoidal categories Romero, Nadia Representation Theory Category Theory K-Theory and Homology 18M05, 18D20, 18G90 Let $R$ be a commutative ring with unit. We develop a Hochschild cohomology theory in the category $\mathcal{F}$ of linear functors defined from an essentially small symmetric monoidal category enriched in $R$-Mod, to $R$-Mod. The category $\mathcal{F}$ is known to be symmetric monoidal too, so one can consider monoids in $\mathcal{F}$ and modules over these monoids, which allows for the possibility of a Hochschild cohomology theory. The emphasis of the article is in considering natural hom constructions appearing in this context. These homs, together with the abelian structure of $\mathcal{F}$ lead to nice definitions and provide effective tools to prove the main properties and results of the classical Hochschild cohomology theory. |
| title | Hochschild cohomology for functors on linear symmetric monoidal categories |
| topic | Representation Theory Category Theory K-Theory and Homology 18M05, 18D20, 18G90 |
| url | https://arxiv.org/abs/2311.12269 |