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  "body": "\n## Summary\n\nThe equalizer of two parallel morphisms $f, g \\colon X \\to Y$ in a [[category|A2DE66937D]] is a universal morphism $e \\colon E \\to X$ with $f \\circ e = g \\circ e$.\nIt is the [[limit|CHS5AFJGWE]] of the diagram formed by $f$ and $g$.\nIn sets it is the subset where $f$ and $g$ agree, and in abelian groups it is the kernel of $f - g$.\nEvery equalizer is a monomorphism, and products together with equalizers build all limits.\n\n## Explanations\n\n### L1 human\n\nGiven two functions $f$ and $g$ from $X$ to $Y$, ask where they give the same answer.\nThe answer is the set $E = \\set{x \\in X : f(x) = g(x)}$, which sits inside $X$.\nFor example, for $f(x) = x^2$ and $g(x) = x$ on the real numbers, the equalizer is $\\{0, 1\\}$, the set of solutions of $x^2 = x$.\nThe categorical equalizer describes this \"solution set\" through maps only: anything mapping into $X$ on which $f$ and $g$ agree must map into $E$.\n",
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