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  "body": "\n## Summary\n\nThe Kleisli category of a [[monad|FF6XXA6D1A]] $T$ on a category $\\mathcal{C}$ has the same objects as $\\mathcal{C}$.\nA morphism from $X$ to $Y$ in it is a morphism $X \\to T Y$ of $\\mathcal{C}$.\nComposition uses the multiplication of the monad to flatten $T T$ to $T$.\nIt is equivalent to the full subcategory of free algebras in the [[Eilenberg–Moore category|FEPRJR3MND]].\nAmong all adjunctions that induce the monad, it gives the initial one.\nIn computer science, Kleisli morphisms model programs with side effects, such as partiality, nondeterminism or state.\n\n## Explanations\n\n### L1 human\n\nFor the powerset monad, a morphism from $X$ to $Y$ in the Kleisli category is a map that sends each element of $X$ to a set of elements of $Y$.\nSuch a map is the same as a relation between $X$ and $Y$.\nComposing two of them means following all possible paths, which is the usual composition of relations.\nSo the Kleisli category of the powerset monad is the category of sets and relations.\n",
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