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  "body": "\n## Summary\n\nAn initial algebra of an endofunctor $F$ is an [[F-algebra|4RB8834SEJ]] from which there is exactly one homomorphism to every other $F$-algebra.\nIt is unique up to unique isomorphism, and by [[Lambek's lemma|S9FVESJ0CS]] its structure map is an isomorphism, so it is a fixed point of $F$.\nInitial algebras give a uniform account of inductive definitions and of definitions by recursion.\nThe unique homomorphism out of the initial algebra is the function defined by structural recursion.\nFor a signature, the initial algebra is the term algebra, which is the basis of [[initial algebra semantics|HVF403PQ4H]] for abstract data types.\n\n## Explanations\n\n### L1 human\n\nThe natural numbers are built from $0$ by repeatedly adding $1$, and nothing else.\nThat is why a function on them can be defined by giving its value at $0$ and a rule for passing from $n$ to $n + 1$.\nInitiality is the abstract form of this fact.\nLists, finite trees and terms are initial algebras of other functors, and the same recursion principle holds for them.\n",
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