{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(\\mathcal{C}, \\otimes, \\mathbb{1})$ be a [[symmetric monoidal category|5S7ER95XAC]], such as sets or topological spaces with the cartesian product.\nAn *operad* $\\mathcal{O}$ in $\\mathcal{C}$ consists of objects $\\mathcal{O}(n)$ for $n \\ge 0$ and the following data.\nThere is a right [[action|0WCY42EV9V]] of the [[symmetric group|0515A6D2Y1]] $\\Sigma_n$ on each $\\mathcal{O}(n)$, and a unit $\\eta \\colon \\mathbb{1} \\to \\mathcal{O}(1)$.\nThere are composition maps, which are associative, unital and equivariant:\n$$\\gamma \\colon \\mathcal{O}(k) \\otimes \\mathcal{O}(j_1) \\otimes \\cdots \\otimes \\mathcal{O}(j_k) \\to \\mathcal{O}(j_1 + \\cdots + j_k).$$\nAn *algebra over $\\mathcal{O}$*, or *$\\mathcal{O}$-algebra*, is an object $A$ of $\\mathcal{C}$ with maps\n$$\\theta_n \\colon \\mathcal{O}(n) \\otimes A^{\\otimes n} \\to A \\qquad (n \\ge 0).$$\nThese maps must satisfy three conditions, written here with elements as for sets or spaces:\n\n1. (associativity) $\\theta_j(\\gamma(c; d_1, \\ldots, d_k); \\mathbf{a}_1, \\ldots, \\mathbf{a}_k) = \\theta_k(c; \\theta_{j_1}(d_1; \\mathbf{a}_1), \\ldots, \\theta_{j_k}(d_k; \\mathbf{a}_k))$, where $j = j_1 + \\cdots + j_k$ and $\\mathbf{a}_i$ is a $j_i$-tuple of elements of $A$;\n2. (unit) $\\theta_1(1; a) = a$, where $1 = \\eta(\\ast) \\in \\mathcal{O}(1)$;\n3. (equivariance) $\\theta_n(c \\cdot \\sigma; a_1, \\ldots, a_n) = \\theta_n(c; a_{\\sigma^{-1}(1)}, \\ldots, a_{\\sigma^{-1}(n)})$ for $\\sigma \\in \\Sigma_n$.\n\nIn a general $\\mathcal{C}$ the three clauses are the commutativity of the corresponding diagrams.\n\n## Remarks\n\nEquivalently, an $\\mathcal{O}$-algebra structure on $A$ is a morphism of operads $\\mathcal{O} \\to \\mathrm{End}_A$, where $\\mathrm{End}_A(n)$ is the internal hom from $A^{\\otimes n}$ to $A$, when $\\mathcal{C}$ is closed.\n\nExample: the commutative operad in sets has $\\mathcal{O}(n) = \\set{\\ast}$ for every $n$.\nAn algebra is a set $A$ with one map $\\theta_n \\colon A^n \\to A$ for each $n$.\nEquivariance makes $\\theta_2(a, b) = \\theta_2(b, a)$, the associativity clause makes $\\theta_2$ associative, and $\\theta_0$ picks a unit.\nSo the algebras are exactly the commutative [[monoids|0N6R902FG1]].\nLikewise the associative operad, with $\\mathcal{O}(n) = \\Sigma_n$, has the monoids as its algebras.\n\nNon-example: take $A = \\Z$ and try to make it an algebra over the commutative operad with $\\theta_2(a, b) = a - b$.\nEquivariance fails for the transposition, since $1 - 0 \\ne 0 - 1$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "ES9YATDWA3",
    "kind": "definition",
    "msc": [
      "18M60",
      "18M75",
      "55P48"
    ],
    "name": "Algebra over an operad",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Operad_algebra"
      }
    ],
    "tag": "H5765CJ7VB",
    "uses": [
      "5S7ER95XAC",
      "0515A6D2Y1",
      "0WCY42EV9V",
      "0N6R902FG1"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/H5765CJ7VB/",
  "links": [
    {
      "field": "defines",
      "target": "ES9YATDWA3"
    },
    {
      "field": "uses",
      "target": "5S7ER95XAC"
    },
    {
      "field": "uses",
      "target": "0515A6D2Y1"
    },
    {
      "field": "uses",
      "target": "0WCY42EV9V"
    },
    {
      "field": "uses",
      "target": "0N6R902FG1"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/H5765CJ7VB.md",
  "msc": [
    "18M60",
    "18M75",
    "55P48"
  ],
  "name": "Algebra over an operad",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/H57/H5765CJ7VB.md",
  "status": null,
  "tag": "H5765CJ7VB",
  "uri": "m2://H5765CJ7VB",
  "url": "https://tamnd.github.io/math2/wiki/algebra-over-an-operad-definition/",
  "writing": {
    "tag": "H5765CJ7VB",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}