Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2506.11240 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911309948256256 |
|---|---|
| author | Keidar, Shai Ragimov, Shaul |
| author_facet | Keidar, Shai Ragimov, Shaul |
| contents | Given a presentably symmetric monoidal $\infty$-category $\mathcal{C}$ and an $\mathbb{E}_{\infty}$-monoid $M$, we introduce and classify twisted graded categories, which generalize the Day convolution structure on $\mathrm{Fun}(M, \mathcal{C})$. These are characterized by a braiding encoded in symmetric group actions on tensor powers, whose character we show depends only on the $\mathbb{T}$-equivariant monoidal dimension. We analyze the $\mathbb{T}$-action on the dimension of invertible objects and identify it with the $\mathbb{T}$-transfer map. Finally, we compute braiding characters in examples arising from higher cyclotomic extensions, such as the $(\mathbb{S}, n+1)$-oriented extension of $\mathrm{Mod}_{En}^{\wedge}$ at all primes and heights, and of the cyclotomic closure of $\mathrm{Vect}^n$ at low heights. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11240 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Twisted Graded Categories Keidar, Shai Ragimov, Shaul Algebraic Topology Category Theory Given a presentably symmetric monoidal $\infty$-category $\mathcal{C}$ and an $\mathbb{E}_{\infty}$-monoid $M$, we introduce and classify twisted graded categories, which generalize the Day convolution structure on $\mathrm{Fun}(M, \mathcal{C})$. These are characterized by a braiding encoded in symmetric group actions on tensor powers, whose character we show depends only on the $\mathbb{T}$-equivariant monoidal dimension. We analyze the $\mathbb{T}$-action on the dimension of invertible objects and identify it with the $\mathbb{T}$-transfer map. Finally, we compute braiding characters in examples arising from higher cyclotomic extensions, such as the $(\mathbb{S}, n+1)$-oriented extension of $\mathrm{Mod}_{En}^{\wedge}$ at all primes and heights, and of the cyclotomic closure of $\mathrm{Vect}^n$ at low heights. |
| title | Twisted Graded Categories |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2506.11240 |