{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(T, \\eta, \\mu)$ be a monad on a [[category|A2DE66937D]] $\\mathcal{C}$.\nThe *Eilenberg–Moore category* $\\mathcal{C}^T$ has these parts.\n\n1. Its objects are the [[$T$-algebras|QG1ZXRC8EH]] $(A, a)$.\n2. Its morphisms $(A, a) \\to (B, b)$ are the morphisms of $T$-algebras, that is, the $f \\colon A \\to B$ with $f \\circ a = b \\circ T(f)$.\n3. Composition and identities are those of $\\mathcal{C}$.\n\nThe forgetful functor $U^T \\colon \\mathcal{C}^T \\to \\mathcal{C}$ sends $(A, a)$ to $A$.\nThe free functor $F^T \\colon \\mathcal{C} \\to \\mathcal{C}^T$ sends $X$ to $(T(X), \\mu_X)$ and $f$ to $T(f)$.\n\n## Remarks\n\nThe functor $F^T$ is [[left adjoint|TZHF6ZM92S]] to $U^T$, and $U^T F^T = T$.\nThe unit of the adjunction is $\\eta$, and the monad of the adjunction is $(T, \\eta, \\mu)$.\n\nExample: for the list monad on sets, $\\mathcal{C}^T$ is isomorphic to the category of [[monoids|0N6R902FG1]] and monoid homomorphisms.\nA morphism of algebras $f$ satisfies $f(m_1 \\cdots m_n) = f(m_1) \\cdots f(m_n)$, which is exactly the condition to be a monoid homomorphism.\n\nNon-example: the category whose objects are monoids and whose morphisms are all functions between them is not $\\mathcal{C}^T$.\nA function such as $n \\mapsto n + 1$ on $(\\N, +)$ does not satisfy clause 2, since it does not send $0 + 0$ to $1 + 1$.\n\nThe notation $\\mathcal{C}^T$ is standard.\nThe full subcategory of free algebras is equivalent to the Kleisli category of $T$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "FEPRJR3MND",
    "kind": "definition",
    "msc": [
      "18C20"
    ],
    "name": "Eilenberg–Moore category",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Monad_(category_theory)"
      }
    ],
    "tag": "B1TVEMYMC0",
    "uses": [
      "A2DE66937D",
      "QG1ZXRC8EH",
      "TZHF6ZM92S",
      "0N6R902FG1"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/B1TVEMYMC0/",
  "links": [
    {
      "field": "defines",
      "target": "FEPRJR3MND"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "QG1ZXRC8EH"
    },
    {
      "field": "uses",
      "target": "TZHF6ZM92S"
    },
    {
      "field": "uses",
      "target": "0N6R902FG1"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/B1TVEMYMC0.md",
  "msc": [
    "18C20"
  ],
  "name": "Eilenberg–Moore category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/B1T/B1TVEMYMC0.md",
  "status": null,
  "tag": "B1TVEMYMC0",
  "uri": "m2://B1TVEMYMC0",
  "url": "https://tamnd.github.io/math2/wiki/eilenberg-moore-category-definition/",
  "writing": {
    "tag": "B1TVEMYMC0",
    "kind": "definition",
    "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
      }
    ]
  }
}