{
  "id": "2407.11627",
  "version": "v1",
  "published": "2024-07-16T11:44:13.000Z",
  "updated": "2024-07-16T11:44:13.000Z",
  "title": "Filtering the linearization of the category of surjections",
  "authors": [
    "Geoffrey Powell"
  ],
  "comment": "14 pages. Comments welcome",
  "categories": [
    "math.RT"
  ],
  "abstract": "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. This yields the `primitive' subcategory $ k \\mathbf{FS}^0 \\subset k \\mathbf{FS}$ that is of independent interest. 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 $k \\mathbf{FS}^0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-16T11:44:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite sets",
      "surjections",
      "linearization",
      "filtration",
      "exploiting additional structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}