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  "body": "\n## Summary\n\nA quotient category is obtained from a [[category|A2DE66937D]] by declaring some parallel morphisms equal.\nThe identification must be an [[equivalence relation|0G8VEZ3WTK]] on each set of morphisms that is compatible with composition, called a congruence.\nQuotient categories are how one builds homotopy categories, where homotopic maps become equal.\n\n## Explanations\n\n### L1 human\n\nKeep all the objects, but glue some arrows together.\nFor this to make sense, gluing $f$ to $f'$ must also glue $g \\circ f$ to $g \\circ f'$, so that composition still works on the glued arrows.\nThe main example is the homotopy category of topological spaces, where two continuous maps become the same arrow when one can be deformed into the other.\nFor a small example, a monoid is a category with one object, and a quotient of that category is a quotient monoid.\nGluing integers of the same parity turns $\\Z$ under addition into $\\Z/2\\Z$.\n",
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