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  "body": "\n## Definition\n\nLet $F$ be a [[functor|28NQ0KRHD3]] from the category of [[sets|0BSJ7RQ9BW]] to itself.\nAn *$F$-coalgebra* is a pair $(X, \\alpha)$ where $X$ is a set and $\\alpha \\colon X \\to F(X)$ is a [[function|0EPAT51DXQ]].\nThe set $X$ is the carrier, or set of states, and $\\alpha$ is the structure map.\nA *homomorphism* of $F$-coalgebras from $(X, \\alpha)$ to $(Y, \\beta)$ is a function $h \\colon X \\to Y$ such that $\\beta \\circ h = F(h) \\circ \\alpha$.\n*Universal coalgebra* is the study of $F$-coalgebras and their homomorphisms, for an arbitrary functor $F$ of this kind.\n\n## Remarks\n\nExample: let $A$ be a set and $F(X) = A \\times X$, with $F(h)(a, x) = (a, h(x))$.\nAn $F$-coalgebra is a set $X$ with a map $\\alpha(x) = (o(x), n(x))$, an output $o(x) \\in A$ and a next state $n(x) \\in X$.\nThe set $A^{\\N}$ of streams with $\\beta(s) = (s_0, (s_1, s_2, \\dots))$ is such a coalgebra.\nFor every coalgebra $(X, \\alpha)$, the map $h(x) = (o(x), o(n(x)), o(n(n(x))), \\dots)$ satisfies $\\beta \\circ h = F(h) \\circ \\alpha$, and it is the only such map, so the streams form a final $F$-coalgebra.\n\nNon-example of a homomorphism: let $A = \\set{a, b}$, let $X = \\set{x}$ with $\\alpha(x) = (a, x)$, and let $Y = \\set{y_1, y_2}$ with $\\beta(y_1) = (a, y_2)$ and $\\beta(y_2) = (b, y_2)$.\nThe map $h(x) = y_1$ keeps the output of $x$, but $\\beta(h(x)) = (a, y_2)$ while $F(h)(\\alpha(x)) = (a, y_1)$, so the square does not commute.\n\nThe same definition, with any category in place of sets, gives the [[F-coalgebra|6BS0300CMV]].\n",
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