Categorical Definition
Let A be a category. A localization is an idempotent and coaugmented functor. A coaugmented functor is a pair (L,l) where L:A → A is an endofunctor and l:Id → L is a natural transformation from the identity functor to L (called the coaugmentation). A coaugmented functor is idempotent if, for every X, both maps L(lX),lL(X):L(X) → LL(X) are isomorphisms. It can be proven that in this case, both maps are equal.
Read more about this topic: Localization Of A Category
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