{
  "annotation_id": "e6a0d3dd-9273-4c9e-ab9f-ac14946096d6",
  "date_created": "2026-03-02T18:02:05.749000Z",
  "date_modified": "2026-03-02T18:02:05.749000Z",
  "file_hash": "640f84feed10b6ffa914abdbbd8f32be632d21010b52c1b6375c312c9d27f735",
  "private": false,
  "record": {
    "abstract": "We review the problem of finding a general framework within which one can\nconstruct quantum theories of non-standard models for space, or space-time. The\nstarting point is the observation that entities of this type can typically be\nregarded as objects in a category whose arrows are structure-preserving maps.\nThis motivates investigating the general problem of quantising a system whose\n`configuration space\u0027 (or history-theory analogue) is the set of objects\n$\\Ob\\Q$ in a category $\\Q$.\n  We develop a scheme based on constructing an analogue of the group that is\nused in the canonical quantisation of a system whose configuration space is a\nmanifold $Q\\simeq G/H$, where $G$ and $H$ are Lie groups. In particular, we\nchoose as the analogue of $G$ the monoid of `arrow fields\u0027 on $\\Q$. Physically,\nthis means that an arrow between two objects in the category is viewed as an\nanalogue of momentum. After finding the `category quantisation monoid\u0027, we show\nhow suitable representations can be constructed using a bundle (or, more\nprecisely, presheaf) of Hilbert spaces over $\\Ob\\Q$. For the example of a\ncategory of finite sets, we construct an explicit representation structure of\nthis type.",
    "arxiv_id": "quant-ph/0401175",
    "authors": [
      "C J Isham"
    ],
    "categories": [
      "quant-ph"
    ],
    "doi": "10.1007/s10701-004-1944-3",
    "title": "Quantising on a category",
    "url": "https://arxiv.org/abs/quant-ph/0401175"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "18e34983-34fd-46b0-9e0e-12efd2439a4f",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}