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  "body": "\n## Definition\n\nLet $(T, \\eta, \\mu)$ be a [[monad|FF6XXA6D1A]] on a category $\\mathcal{C}$.\nA *$T$-algebra* is a pair $(A, a)$ of an object $A$ of $\\mathcal{C}$ and a morphism $a \\colon T(A) \\to A$ with these two properties.\n\n1. Unit law: $a \\circ \\eta_A = \\id_A$.\n2. Associativity law: $a \\circ T(a) = a \\circ \\mu_A$ as morphisms $T(T(A)) \\to A$.\n\nA *morphism of $T$-algebras* $f \\colon (A, a) \\to (B, b)$ is a morphism $f \\colon A \\to B$ of $\\mathcal{C}$ with $f \\circ a = b \\circ T(f)$.\n\n## Remarks\n\nExample: let $T$ be the list monad on sets, so $T(X)$ is the [[free monoid|CV013KHVJG]] of finite words in $X$.\nFor a [[monoid|0N6R902FG1]] $M$, let $a$ send a word $(m_1, \\dots, m_n)$ to the product $m_1 m_2 \\cdots m_n$ in $M$, with the empty word sent to the identity.\nThe unit law holds since a one-letter word $m$ evaluates to $m$.\nThe associativity law holds since multiplying the blocks of a word of words and then multiplying the results gives the product of the concatenated word.\nEvery $T$-algebra arises in this way, so $T$-algebras are the same as monoids.\n\nNon-example: for the same monad, let $a \\colon T(\\N) \\to \\N$ send a word to its length.\nThen $a(\\eta_{\\N}(5)) = 1 \\ne 5$, so the unit law fails.\n\nFor every object $X$, the pair $(T(X), \\mu_X)$ is a $T$-algebra, the free $T$-algebra on $X$.\n",
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