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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a category.\nA *comonad* on $\\mathcal{C}$ is a triple $(G, \\varepsilon, \\delta)$ made of these data.\n\n1. $G \\colon \\mathcal{C} \\to \\mathcal{C}$ is a [[functor|28NQ0KRHD3]].\n2. $\\varepsilon \\colon G \\Rightarrow \\mathrm{Id}_{\\mathcal{C}}$ is a [[natural transformation|KYCPCC57FT]], the counit.\n3. $\\delta \\colon G \\Rightarrow G \\circ G$ is a natural transformation, the comultiplication.\n\nThe data must satisfy two laws.\nThe coassociativity law is $G\\delta \\circ \\delta = \\delta G \\circ \\delta$ as natural transformations $G \\Rightarrow G G G$.\nThe counit laws are $\\varepsilon G \\circ \\delta = \\id_G$ and $G\\varepsilon \\circ \\delta = \\id_G$.\n\n## Remarks\n\nA comonad on $\\mathcal{C}$ is the same as a [[monad|FF6XXA6D1A]] on the opposite category $\\mathcal{C}^{\\mathrm{op}}$.\n\nExample: fix a set $A$ and let $G(X) = A \\times X$ on the category of sets, with $G(f) = \\id_A \\times f$.\nPut $\\varepsilon_X(a, x) = x$ and $\\delta_X(a, x) = (a, (a, x))$.\nBoth sides of the coassociativity law send $(a, x)$ to $(a, (a, (a, x)))$.\nThe counit laws hold since $\\varepsilon_{G(X)}(a, (a, x)) = (a, x)$ and $G(\\varepsilon_X)(a, (a, x)) = (a, x)$.\n\nNon-example: let $A$ have two elements $a_0 \\ne a_1$, and keep $G$ and $\\varepsilon$ but put $\\delta_X(a, x) = (a_0, (a, x))$.\nThen $G(\\varepsilon_X)(\\delta_X(a_1, x)) = (a_0, x) \\ne (a_1, x)$, so a counit law fails.\n\nIf $F \\colon \\mathcal{C} \\to \\mathcal{D}$ is left adjoint to $U \\colon \\mathcal{D} \\to \\mathcal{C}$ with counit $\\varepsilon$ and unit $\\eta$, then $(F U, \\varepsilon, F \\eta U)$ is a comonad on $\\mathcal{D}$.\n",
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    "defines": "97NPWM3S6E",
    "kind": "definition",
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    "name": "Comonad",
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  "markdown": "https://tamnd.github.io/math2/node/S47R8TG0T3.md",
  "msc": [
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  "name": "Comonad",
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