{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "220PPH6N7Q"
    },
    {
      "field": "uses",
      "target": "3TGTNRBKW0"
    },
    {
      "field": "uses",
      "target": "ESRDNKF6WJ"
    }
  ],
  "body": "\n## Summary\n\nSweedler notation is a shorthand for the comultiplication of a [[coalgebra|A06RA8KB0T]]: one writes $\\Delta(c) = \\sum c_{(1)} \\otimes c_{(2)}$, or even $c_{(1)} \\otimes c_{(2)}$, without naming the terms of the sum.\nCoassociativity lets one write iterated coproducts as $c_{(1)} \\otimes c_{(2)} \\otimes c_{(3)}$ without brackets.\nIt is named after Moss Sweedler and is the standard notation for computations with [[Hopf algebras|7XFYP0N84H]] and [[comodules|7WEM2E6RTV]].\n\n## Explanations\n\n### L1 human\n\nThe coproduct of an element is a sum of pairs, but the pairs are rarely worth naming one by one.\nSweedler notation writes a typical pair as $c_{(1)} \\otimes c_{(2)}$ and lets the summation stay implicit.\nWith it, the counit law reads $\\varepsilon(c_{(1)}) c_{(2)} = c$, and the antipode law reads $S(c_{(1)}) c_{(2)} = \\varepsilon(c) 1$.\nThe symbols $c_{(1)}$ and $c_{(2)}$ are not elements on their own; they only make sense inside expressions that are linear in each of them.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "16T15",
      "16T05"
    ],
    "name": "Sweedler notation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Coalgebra"
      }
    ],
    "tag": "NC26CE0Y8B"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/NC26CE0Y8B/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/NC26CE0Y8B.md",
  "msc": [
    "16T15",
    "16T05"
  ],
  "name": "Sweedler notation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NC2/NC26CE0Y8B.md",
  "status": null,
  "tag": "NC26CE0Y8B",
  "uri": "m2://NC26CE0Y8B",
  "url": "https://tamnd.github.io/math2/wiki/sweedler-notation/",
  "writing": {
    "tag": "NC26CE0Y8B",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}