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  "body": "\n## Summary\n\nLocalizing a [[category|A2DE66937D]] at a class of morphisms means adding formal inverses so that these morphisms become isomorphisms, in the most economical way.\nThe result is a new category with a functor from the old one, characterized by a universal property.\nHomotopy categories and derived categories are localizations: one inverts weak equivalences or quasi-isomorphisms.\nThe construction generalizes the localization of a commutative ring, where chosen elements are made invertible.\n\n## Explanations\n\n### L1 human\n\nSometimes two objects should count as the same because a chosen map connects them, even though the map has no inverse.\nLocalization forces such maps to have inverses by allowing formal backward steps in chains of maps.\nA morphism of the new category is a zigzag of old morphisms, some of which point the wrong way and are chosen ones.\nFor example, in the category with two objects $0$ and $1$ and one arrow $0 \\to 1$, inverting the arrow makes $0$ and $1$ isomorphic.\n",
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