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
Famous quotes containing the words categorical and/or definition:
“We do the same thing to parents that we do to children. We insist that they are some kind of categorical abstraction because they produced a child. They were people before that, and they’re still people in all other areas of their lives. But when it comes to the state of parenthood they are abruptly heir to a whole collection of virtues and feelings that are assigned to them with a fine arbitrary disregard for individuality.”
—Leontine Young (20th century)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)