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  "body": "\n## Summary\n\nThe Karoubi envelope of a category $\\mathcal{C}$ is the category obtained by formally adding an image for every [[idempotent|0N5NJZKKN3]] morphism.\nIts objects are pairs of an object and an idempotent endomorphism of it.\nIt is the smallest category containing $\\mathcal{C}$ in which every idempotent splits, and it is also called the idempotent completion.\nFor example, the Karoubi envelope of the category of finitely generated free modules over a ring is the category of finitely generated projective modules.\n\n## Explanations\n\n### L1 human\n\nAn idempotent map $e$ satisfies $e \\circ e = e$, like a projection onto a subspace.\nIn vector spaces such a projection has an image, and $e$ factors through it.\nIn some categories the image is missing, and the Karoubi envelope adds it as the formal object $(A, e)$.\nIn this way projective modules appear as images of projections on free modules.\n",
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