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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a category with [[zero morphisms|YF9F9EG6M3]], and let $f \\colon X \\to Y$ be a morphism of $\\mathcal{C}$.\nA *kernel* of $f$ is an object $K$ with a morphism $k \\colon K \\to X$ such that:\n\n1. $f \\circ k = 0$;\n2. for every morphism $u \\colon T \\to X$ with $f \\circ u = 0$ there is a unique morphism $v \\colon T \\to K$ with $u = k \\circ v$.\n\n## Remarks\n\nEquivalently, $k$ is an [[equalizer|CTJ5AD8ST5]] of $f$ and $0 \\colon X \\to Y$.\nA kernel is unique up to a unique isomorphism, and is written $\\ker f \\to X$.\nEvery kernel is a [[monomorphism|Y2NP1JFR86]], by the uniqueness in clause 2.\n\nExample: in the category of [[modules|0NZ8JA25WK]] over a ring, the inclusion $k \\colon \\set{x \\in X : f(x) = 0} \\to X$ of the [[kernel of the homomorphism|0W88CEGQP1]] is a kernel.\nIf $f \\circ u = 0$, then $u$ takes values in this submodule, so $u$ factors through $k$, and uniquely since $k$ is injective.\n\nNon-example: for the reduction $f \\colon \\Z \\to \\Z/2$ of abelian groups, the inclusion $k \\colon 4\\Z \\to \\Z$ satisfies clause 1 but not clause 2.\nThe inclusion $u \\colon 2\\Z \\to \\Z$ satisfies $f \\circ u = 0$ but does not factor through $4\\Z$, since $2 \\notin 4\\Z$.\n",
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