{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "DGZ8RS5KMZ"
    },
    {
      "field": "uses",
      "target": "DWZA03W93P"
    }
  ],
  "body": "\n## Summary\n\nA Lawvere theory is a category with finite products whose objects are the natural numbers, where $n$ is the $n$-th power of $1$.\nIts morphisms $n \\to 1$ play the role of the $n$-ary operations of an algebraic theory.\nThey are closed under composition and identified modulo the equations of the theory.\nModels are finite-product-preserving functors to $\\mathbf{Set}$, and the models of the theory of a [[variety|AJNZJD8442]] form a category equivalent to the variety.\nLawvere theories give a presentation-free description of varieties, so that different choices of operations and axioms for the same variety give the same theory.\n\n## Explanations\n\n### L1 human\n\nThe theory of groups can be written with multiplication, inverse and identity, or with division alone, and the two descriptions look different.\nA Lawvere theory removes this choice.\nIt collects all operations that can be built from the basic ones, such as $(x, y) \\mapsto x y^{-1} x$, and identifies two of them when they agree in every group.\nThese operations, with their composition, form a category, and a group is then a way to interpret this category by sets and functions.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18C10",
      "08C05"
    ],
    "name": "Lawvere theory",
    "same_as": {
      "wikidata": "Q6505076",
      "wikipedia": "Lawvere theory"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lawvere_theory"
      }
    ],
    "specializes": [
      "A2DE66937D"
    ],
    "tag": "NTJ94H2CNS"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/NTJ94H2CNS/",
  "links": [
    {
      "field": "specializes",
      "target": "A2DE66937D"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/NTJ94H2CNS.md",
  "msc": [
    "18C10",
    "08C05"
  ],
  "name": "Lawvere theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NTJ/NTJ94H2CNS.md",
  "status": null,
  "tag": "NTJ94H2CNS",
  "uri": "m2://NTJ94H2CNS",
  "url": "https://tamnd.github.io/math2/wiki/lawvere-theory/",
  "writing": {
    "tag": "NTJ94H2CNS",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}