Constructing monoidal structures on fibered categories via factorizations
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929625129549824 |
|---|---|
| author | Terenzi, Luca |
| author_facet | Terenzi, Luca |
| contents | Let $\mathcal{S}$ be a small category, and suppose that we are given two (non-full) subcategories $\mathcal{S}^{sm}$ and $\mathcal{S}^{cl}$ that generate all morphisms of $\mathcal{S}$ under composition in the same way as morphisms of quasi-projective algebraic varieties are generated by smooth morphisms and closed immersions. We show that a monoidal structure on a given $\mathcal{S}$-fibered category is completely determined by its restrictions to $\mathcal{S}^{sm}$ and $\mathcal{S}^{cl}$; in fact, any such pair of monoidal structures satisfying a natural coherence condition uniquely determines a monoidal structure over $\mathcal{S}$. The same principle applies to morphisms of $\mathcal{S}$-fibered categories and monoidality thereof. Under further assumptions on the subcategories $\mathcal{S}^{sm}$ and $\mathcal{S}^{cl}$, and with suitable restrictions on the $\mathcal{S}$-fibered categories and morphisms involved, we provide a variant of the above factorization method in which inverse images under closed immersion are partially replaced by the corresponding direct images: the latter variant is more adapted to the setting of perverse sheaves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13489 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructing monoidal structures on fibered categories via factorizations Terenzi, Luca Category Theory Algebraic Geometry 18D30, 18M05 Let $\mathcal{S}$ be a small category, and suppose that we are given two (non-full) subcategories $\mathcal{S}^{sm}$ and $\mathcal{S}^{cl}$ that generate all morphisms of $\mathcal{S}$ under composition in the same way as morphisms of quasi-projective algebraic varieties are generated by smooth morphisms and closed immersions. We show that a monoidal structure on a given $\mathcal{S}$-fibered category is completely determined by its restrictions to $\mathcal{S}^{sm}$ and $\mathcal{S}^{cl}$; in fact, any such pair of monoidal structures satisfying a natural coherence condition uniquely determines a monoidal structure over $\mathcal{S}$. The same principle applies to morphisms of $\mathcal{S}$-fibered categories and monoidality thereof. Under further assumptions on the subcategories $\mathcal{S}^{sm}$ and $\mathcal{S}^{cl}$, and with suitable restrictions on the $\mathcal{S}$-fibered categories and morphisms involved, we provide a variant of the above factorization method in which inverse images under closed immersion are partially replaced by the corresponding direct images: the latter variant is more adapted to the setting of perverse sheaves. |
| title | Constructing monoidal structures on fibered categories via factorizations |
| topic | Category Theory Algebraic Geometry 18D30, 18M05 |
| url | https://arxiv.org/abs/2401.13489 |