Extending monoidal structures on fibered categories via embeddings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909424722903040 |
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| author | Terenzi, Luca |
| author_facet | Terenzi, Luca |
| contents | Let $\mathcal{S}$ be a small category, and suppose that we are given a full subcategory $\mathcal{U}$ such that every object of $\mathcal{S}$ can be embedded into some object of $\mathcal{U}$ in the same way as every quasi-projective algebraic variety admits a closed embedding into a smooth one. We show that every monoidal structure on a given $\mathcal{S}$-fibered category satisfying certain natural conditions is completely determined by its restriction to $\mathcal{U}$; in fact, any monoidal structure over $\mathcal{U}$ satisfying similar natural conditions admits an essentially unique extension to the whole of $\mathcal{S}$. For instance, this allows one to recover the unit constraint on the classical constructible derived categories from the abelian categories of perverse sheaves. The same principle applies to morphisms of $\mathcal{S}$-fibered categories and monoidality thereof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13517 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extending monoidal structures on fibered categories via embeddings Terenzi, Luca Category Theory Algebraic Geometry 18D30, 18M05 Let $\mathcal{S}$ be a small category, and suppose that we are given a full subcategory $\mathcal{U}$ such that every object of $\mathcal{S}$ can be embedded into some object of $\mathcal{U}$ in the same way as every quasi-projective algebraic variety admits a closed embedding into a smooth one. We show that every monoidal structure on a given $\mathcal{S}$-fibered category satisfying certain natural conditions is completely determined by its restriction to $\mathcal{U}$; in fact, any monoidal structure over $\mathcal{U}$ satisfying similar natural conditions admits an essentially unique extension to the whole of $\mathcal{S}$. For instance, this allows one to recover the unit constraint on the classical constructible derived categories from the abelian categories of perverse sheaves. The same principle applies to morphisms of $\mathcal{S}$-fibered categories and monoidality thereof. |
| title | Extending monoidal structures on fibered categories via embeddings |
| topic | Category Theory Algebraic Geometry 18D30, 18M05 |
| url | https://arxiv.org/abs/2401.13517 |