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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a category and let $F \\colon \\mathcal{C} \\to \\mathcal{C}$ be a functor.\nAn *initial $F$-algebra* is an [[F-algebra|4RB8834SEJ]] $(I, \\iota)$ that is an [[initial object|EK85VSSRMR]] of the category of $F$-algebras.\nThat is, for every $F$-algebra $(B, \\beta)$ there is exactly one morphism $h \\colon I \\to B$ with $h \\circ \\iota = \\beta \\circ F h$.\n\n## Remarks\n\nExample: on $\\mathbf{Set}$, let $F X = \\set{\\ast} \\sqcup X$, and let $\\iota \\colon F \\N \\to \\N$ send $\\ast$ to $0$ and $n$ to $n + 1$.\nThen $(\\N, \\iota)$ is an initial $F$-algebra.\nGiven $(B, \\beta)$, the condition on $h$ says $h(0) = \\beta(\\ast)$ and $h(n + 1) = \\beta(h(n))$, and these equations define exactly one function $h$ by recursion.\n\nNon-example: $(\\Z, \\iota')$ with $\\iota'(\\ast) = 0$ and $\\iota'(n) = n + 1$ is an $F$-algebra but not an initial one.\nThere is no homomorphism $h$ from it to $(\\N, \\iota)$: it would satisfy $h(0) = 0$ and $h(-1) + 1 = h(0) = 0$, which has no solution in $\\N$.\n\nFor an algebraic signature $\\sigma$ and the functor $F_\\sigma X = \\bigsqcup_n \\sigma_n \\times X^n$, the initial $F_\\sigma$-algebra is the algebra of closed terms of $\\sigma$.\n",
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