Filtering the linearization of the category of surjections
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866918259804078080 |
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| author | Powell, Geoffrey |
| author_facet | Powell, Geoffrey |
| contents | A filtration of the morphisms of the $k$-linearization $k \mathbf{FS}$ of the category $\mathbf{FS}$ of finite sets and surjections is constructed using a natural $k \mathbf{FI}^{op}$-module structure induced by restriction, where $\mathbf{FI}$ is the category of finite sets and injections. In particular, this yields the `primitive' subcategory $ k \mathbf{FS}^0 \subset k \mathbf{FS}$ that is of independent interest; for example, the category of $k \mathbf{FS}^0$-modules is closely related to the category of $k \mathbf{FA}$-modules, where $\mathbf{FA}$ is the category of finite sets and all maps.
Working over a field of characteristic zero, the subquotients of this filtration are identified as bimodules over $k \mathbf{FB}$, where $\mathbf{FB}$ is the category of finite sets and bijections, also exhibiting and exploiting additional structure. In particular, this describes the underlying $k \mathbf{FB}$-bimodule of $k \mathbf{FS}^0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11627 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Filtering the linearization of the category of surjections Powell, Geoffrey Representation Theory 18A25 A filtration of the morphisms of the $k$-linearization $k \mathbf{FS}$ of the category $\mathbf{FS}$ of finite sets and surjections is constructed using a natural $k \mathbf{FI}^{op}$-module structure induced by restriction, where $\mathbf{FI}$ is the category of finite sets and injections. In particular, this yields the `primitive' subcategory $ k \mathbf{FS}^0 \subset k \mathbf{FS}$ that is of independent interest; for example, the category of $k \mathbf{FS}^0$-modules is closely related to the category of $k \mathbf{FA}$-modules, where $\mathbf{FA}$ is the category of finite sets and all maps. Working over a field of characteristic zero, the subquotients of this filtration are identified as bimodules over $k \mathbf{FB}$, where $\mathbf{FB}$ is the category of finite sets and bijections, also exhibiting and exploiting additional structure. In particular, this describes the underlying $k \mathbf{FB}$-bimodule of $k \mathbf{FS}^0$. |
| title | Filtering the linearization of the category of surjections |
| topic | Representation Theory 18A25 |
| url | https://arxiv.org/abs/2407.11627 |