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If X is a scheme and Y→X is its normalization, then the morphism Y→X has property P and any other morphism Z→X with property P factors uniquely through Y. universal-property ag.algebraic-geometry normalization ac.commutative-algebra 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 According to Yoneda’s lemma, this property determines the space Zup to homotopy equivalence. is a universal element for the (contravariant) functor which sends an object $ C $ 777.8 777.8 777.8 777.8 777.8 777.8 1333.3 1333.3 500 500 946.7 902.2 666.7 777.8 >> /FirstChar 33 (The Universal Property of the Quotient Topology) Let X Universal mapping properties are extremely efficient ways of proving things without having to descend to the level of elements (not that the latter is a bad thing.) 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