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  "body": "\n## Summary\n\nIn abstract homotopy theory, the weak equivalences of a category are the morphisms that one wants to treat as if they were isomorphisms.\nThey form one of the three classes of a model category, and the homotopy category is obtained by inverting them.\nThe motivating examples are the [[weak homotopy equivalences|4Q3WY1RX91]] of topological spaces and the [[quasi-isomorphisms|C9JEC2WQKV]] of chain complexes.\nA weak equivalence need not have an inverse morphism, which is why the word weak is used.\n\n## Explanations\n\n### L1 human\n\nTwo spaces can have the same holes in every dimension without there being a nice map back and forth between them.\nA weak equivalence is a map that sees no difference between its source and target from the chosen point of view, for example in all homotopy groups.\nHomotopy theory then declares such a map to be as good as an isomorphism.\nFor chain complexes, a map that induces isomorphisms on all homology groups is a weak equivalence, even when it is far from invertible.\nAn example is the map from the complex $\\Z \\xrightarrow{2} \\Z$, placed in degrees one and zero, to the complex with $\\Z/2$ in degree zero and $0$ elsewhere.\n",
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