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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $(T, \\eta, \\mu)$ be a [[monad|FF6XXA6D1A]] on $\\mathcal{C}$, with unit $\\eta \\colon \\mathrm{Id} \\to T$ and multiplication $\\mu \\colon T T \\to T$.\nThe *Kleisli category* $\\mathcal{C}_T$ is the category given as follows.\n\n1. Its objects are the objects of $\\mathcal{C}$.\n2. A morphism $X \\to Y$ in $\\mathcal{C}_T$ is a morphism $f \\colon X \\to T Y$ in $\\mathcal{C}$.\n3. The composite of $f \\colon X \\to T Y$ and $g \\colon Y \\to T Z$ is $g \\circ_T f = \\mu_Z \\circ T g \\circ f \\colon X \\to T Z$.\n4. The identity on $X$ is $\\eta_X \\colon X \\to T X$.\n\n## Remarks\n\nThe monad laws $\\mu \\circ T \\eta = \\mathrm{id} = \\mu \\circ \\eta T$ and $\\mu \\circ T \\mu = \\mu \\circ \\mu T$ give the identity laws and the associativity of $\\circ_T$.\nThe functor $\\mathcal{C} \\to \\mathcal{C}_T$, $f \\mapsto \\eta \\circ f$, has a right adjoint $\\mathcal{C}_T \\to \\mathcal{C}$ that sends $X$ to $T X$ and $f$ to $\\mu \\circ T f$, and this adjunction induces $T$.\n\nExample: let $T$ be the monad $X \\mapsto X \\sqcup \\set{\\ast}$ on $\\mathbf{Set}$, with $\\eta_X$ the inclusion and $\\mu_X$ sending both added points to $\\ast$.\nA Kleisli morphism $X \\to Y \\sqcup \\set{\\ast}$ is a partial function from $X$ to $Y$, undefined where the value is $\\ast$, and $\\circ_T$ is composition of partial functions.\nSo $\\mathbf{Set}_T$ is isomorphic to the category of sets and partial functions.\n\nNon-example: composing $f \\colon X \\to T Y$ and $g \\colon Y \\to T Z$ by $T g \\circ f$ alone gives a morphism $X \\to T T Z$, not $X \\to T Z$.\nWithout the multiplication $\\mu$, clause 3 does not produce a morphism of the right type.\n\nFor the monad of a variety on $\\mathbf{Set}$, the Kleisli category is equivalent to the category of free algebras of the variety and the homomorphisms between them.\n",
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