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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $A$ be an object of $\\mathcal{A}$.\nConsider the [[morphisms|859F0TFVA3]] with codomain $A$.\nTwo such morphisms $x \\colon X \\to A$ and $y \\colon Y \\to A$ are *equivalent*, written $x \\sim y$, if there are an object $Z$ and [[epimorphisms|RVXDSY04KG]] $u \\colon Z \\to X$ and $v \\colon Z \\to Y$ with $x \\circ u = y \\circ v$.\nThis is an equivalence relation.\nA *pseudo-element* of $A$ is an equivalence class for $\\sim$; one writes $x \\in_p A$.\nFor a morphism $f \\colon A \\to B$ and $x \\in_p A$, the pseudo-element $f(x) \\in_p B$ is the class of $f \\circ x$.\nThe *zero pseudo-element* of $A$ is the class of a [[zero morphism|YF9F9EG6M3]] $0 \\to A$.\n\n## Remarks\n\nTransitivity of $\\sim$ uses that pullbacks of epimorphisms are epimorphisms in an abelian category.\nA morphism $x$ is equivalent to a zero morphism exactly when $x = 0$.\nThe basic rules are these.\nA morphism $f \\colon A \\to B$ is a monomorphism if and only if $f(x) = 0$ implies $x = 0$.\nIt is an epimorphism if and only if every $y \\in_p B$ is $f(x)$ for some $x \\in_p A$.\nA sequence $A \\xrightarrow{f} B \\xrightarrow{g} C$ is exact if and only if $g \\circ f = 0$ and every $y \\in_p B$ with $g(y) = 0$ is $f(x)$ for some $x \\in_p A$.\n\nExample: in the category of abelian groups, let $x, y \\colon \\Z \\to \\Z$ be $x(n) = n$ and $y(n) = -n$.\nTake $Z = \\Z$, $u = \\id$ and $v = y$, both epimorphisms; then $x \\circ u = y \\circ v$, so $x \\sim y$.\nMore generally, two morphisms into an abelian group $A$ are equivalent exactly when they have the same image, so the pseudo-elements of $A$ correspond to the subgroups of $A$.\n\nNon-example: the morphisms $x(n) = n$ and $w(n) = 2n$ from $\\Z$ to $\\Z$ are not equivalent.\nIf $x \\circ u = w \\circ v$ with $u, v$ epimorphisms, then $x$ and $w$ have the same image, but these images are $\\Z$ and $2\\Z$.\n",
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