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  "body": "\n## Summary\n\nFor an endofunctor $F$ of a category $\\mathcal{C}$, an $F$-algebra is an object $A$ with a morphism $F A \\to A$.\nFor a suitable functor on sets, $F$-algebras are exactly the algebras of a given signature, so the notion generalizes universal algebra without equations.\nInductive data types are [[initial F-algebras|R76EC56V9R]], and the dual notion, [[F-coalgebra|6BS0300CMV]], models state-based systems.\nUnlike an algebra over a monad, an $F$-algebra need not satisfy any unit or associativity law.\n\n## Explanations\n\n### L1 human\n\nA structure map $F A \\to A$ says how to combine data of the \"shape\" $F$ into a single element of $A$.\nFor $F X = 1 \\sqcup X$, an $F$-algebra is a set with a chosen element and a function from the set to itself.\nThe natural numbers with $0$ and $n \\mapsto n + 1$ are such an algebra.\nFor $F X = 1 \\sqcup X \\times X$, an $F$-algebra is a set with a constant and a binary operation, which is the signature of monoids.\n",
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