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  "body": "\n## Summary\n\nA pseudo-element of an object $A$ in an [[abelian category|Y9CZ3MTNF1]] is an equivalence class of morphisms with codomain $A$.\nTwo such morphisms are identified when they agree after composing with suitable [[epimorphisms|RVXDSY04KG]].\nPseudo-elements behave enough like elements of a module to allow diagram chasing inside any abelian category.\nWith them, one proves diagram lemmas such as the [[snake lemma|Q8AYGD93CW]] without an embedding theorem.\n\n## Explanations\n\n### L1 human\n\nIn a category of modules one proves diagram lemmas by following elements around the diagram.\nA general abelian category has objects without elements, so one uses morphisms $x \\colon X \\to A$ as \"generalized elements\" instead.\nTwo generalized elements are treated as the same if they cover each other after pulling back along surjections.\nFor example, in abelian groups the maps $\\Z \\to \\Z$ given by $n \\mapsto n$ and $n \\mapsto -n$ give the same pseudo-element of $\\Z$.\nSo pseudo-elements are coarser than elements, but they detect zero, monomorphisms, epimorphisms and exactness.\n",
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