{
  "id": "2407.11623",
  "version": "v1",
  "published": "2024-07-16T11:38:03.000Z",
  "updated": "2024-07-16T11:38:03.000Z",
  "title": "Functors on the category of finite sets revisited",
  "authors": [
    "Geoffrey Powell"
  ],
  "comment": "27 pages. Comments welcome",
  "categories": [
    "math.RT",
    "math.AT"
  ],
  "abstract": "We study the structure of the category of representations of $\\mathbf{FA}$, the category of finite sets and all maps, mostly working over a field of characteristic zero. This category is not semi-simple and exhibits interesting features. We first construct the simple representations, recovering the classification given by Wiltshire-Gordon. The construction given here also yields explicit descriptions of the indecomposable projectives. These results are used to give a convenient set of projective generators of the category of representations of $\\mathbf{FA}$ and hence a Morita equivalence result. This is used to explain how to calculate the multiplicities of the composition factors of an arbitrary object, based only on its underlying $\\mathbf{FB}$-representation, where $\\mathbf{FB}$ is the category of finite sets and bijections. This is applied to show how to calculate the morphism spaces between projectives in our chosen set of generators, as well as for a closely related family of objects (the significance of which can be shown by relative nonhomogeneous Koszul duality theory).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-16T11:38:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite sets",
      "relative nonhomogeneous koszul duality theory",
      "yields explicit descriptions",
      "morita equivalence result",
      "first construct"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}