Tensor structures on fibered categories
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914947577937920 |
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| author | Terenzi, Luca |
| author_facet | Terenzi, Luca |
| contents | Let $\mathcal{S}$ be a small category admitting binary products. We show that the whole theory of monoidal $\mathcal{S}$-fibered categories, which is customarily formulated in terms of the usual internal tensor product, can be rephrased purely in terms of the associated external tensor product. More precisely, we construct a canonical dictionary relating the classical structures and properties of the internal tensor product to analogous structures and properties of the external tensor product: this applies to associativity, commutativity, and unit constraints, to projection formulae, as well as to monoidality of morphisms between monoidal $\mathcal{S}$-fibered categories. For instance, we show how Mac Lane's classical pentagon and hexagon axioms can be stated using the external tensor product. Our results provide a satisfactory abstract framework to study monoidal structures in the setting of perverse sheaves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13491 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tensor structures on fibered categories Terenzi, Luca Category Theory Algebraic Geometry 18D30, 18M05 Let $\mathcal{S}$ be a small category admitting binary products. We show that the whole theory of monoidal $\mathcal{S}$-fibered categories, which is customarily formulated in terms of the usual internal tensor product, can be rephrased purely in terms of the associated external tensor product. More precisely, we construct a canonical dictionary relating the classical structures and properties of the internal tensor product to analogous structures and properties of the external tensor product: this applies to associativity, commutativity, and unit constraints, to projection formulae, as well as to monoidality of morphisms between monoidal $\mathcal{S}$-fibered categories. For instance, we show how Mac Lane's classical pentagon and hexagon axioms can be stated using the external tensor product. Our results provide a satisfactory abstract framework to study monoidal structures in the setting of perverse sheaves. |
| title | Tensor structures on fibered categories |
| topic | Category Theory Algebraic Geometry 18D30, 18M05 |
| url | https://arxiv.org/abs/2401.13491 |