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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $F \\colon \\mathcal{C} \\to \\mathcal{C}$ be a [[functor|28NQ0KRHD3]].\nAn *$F$-algebra* is a pair $(A, \\alpha)$ of an object $A$ of $\\mathcal{C}$ and a morphism $\\alpha \\colon F A \\to A$.\nA *homomorphism* of $F$-algebras $(A, \\alpha) \\to (B, \\beta)$ is a morphism $h \\colon A \\to B$ with $h \\circ \\alpha = \\beta \\circ F h$.\nWith these morphisms, $F$-algebras form a category.\n\n## Remarks\n\nExample: on $\\mathbf{Set}$, let $F X = \\set{\\ast} \\sqcup X$.\nThe natural numbers with $\\alpha(\\ast) = 0$ and $\\alpha(n) = n + 1$ form an $F$-algebra $(\\N, \\alpha)$.\nMore generally, for an algebraic signature $\\sigma$ with $\\sigma_n$ the set of $n$-ary symbols, the $F_\\sigma$-algebras for $F_\\sigma X = \\bigsqcup_n \\sigma_n \\times X^n$ are exactly the $\\sigma$-algebras, and their homomorphisms are the usual ones.\n\nNon-example: a morphism $\\gamma \\colon A \\to F A$ in the other direction does not make $A$ an $F$-algebra; such a pair $(A, \\gamma)$ is an [[F-coalgebra|6BS0300CMV]].\nFor $F X = \\set{\\ast} \\sqcup X$ on $\\mathbf{Set}$, a coalgebra describes a system in which each state either stops or has one next state.\n\nIf $(T, \\eta, \\mu)$ is a monad, every algebra over the monad is a $T$-algebra in this sense, but not conversely, since no unit or associativity law is required here.\n",
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