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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nAn endomorphism $e \\colon A \\to A$ is *idempotent* if $e \\circ e = e$.\nThe *Karoubi envelope* $\\mathrm{Kar}(\\mathcal{C})$ is the category with these parts.\n\n1. Its objects are pairs $(A, e)$ with $A$ an object of $\\mathcal{C}$ and $e$ an idempotent endomorphism of $A$.\n2. A morphism $f \\colon (A, e) \\to (B, e')$ is a morphism $f \\colon A \\to B$ of $\\mathcal{C}$ with $e' \\circ f \\circ e = f$.\n3. Composition is that of $\\mathcal{C}$, and the identity of $(A, e)$ is $e$.\n\n## Remarks\n\nThe condition in clause 2 is equivalent to $f \\circ e = f$ and $e' \\circ f = f$.\nThe functor $\\mathcal{C} \\to \\mathrm{Kar}(\\mathcal{C})$ with $A \\mapsto (A, \\id_A)$ is fully faithful.\nEvery idempotent of $\\mathrm{Kar}(\\mathcal{C})$ splits.\nFor example, an idempotent $e$ of $(A, \\id_A)$ is the composite of the [[retraction|DXJH3DGXBC]] $e \\colon (A, \\id_A) \\to (A, e)$ and the section $e \\colon (A, e) \\to (A, \\id_A)$.\nOther names are idempotent completion and Cauchy completion.\n\nExample: let $\\mathcal{C}$ have one object $*$ and morphisms $1$ and $e$, with $e \\circ e = e$.\nThen $\\mathrm{Kar}(\\mathcal{C})$ has the objects $(*, 1)$ and $(*, e)$.\nThe morphisms $(*, e) \\to (*, e)$ are the $f$ with $e f e = f$, and only $f = e$ qualifies, so $(*, e)$ has only its identity $e$.\n\nNon-example: with the same $\\mathcal{C}$, the morphism $1 \\colon (*, e) \\to (*, e)$ is not a morphism of $\\mathrm{Kar}(\\mathcal{C})$.\nIndeed $e \\circ 1 \\circ e = e \\ne 1$, so clause 2 fails.\n",
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    "name": "Karoubi envelope",
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