Equivalent Categories
It is a natural question to ask: under which conditions can two categories be considered to be "essentially the same", in the sense that theorems about one category can readily be transformed into theorems about the other category? The major tool one employs to describe such a situation is called equivalence of categories, which is given by appropriate functors between two categories. Categorical equivalence has found numerous applications in mathematics.
Read more about this topic: Category Theory
Famous quotes containing the words equivalent and/or categories:
“I started off rapping for people just like myself, people who were in awe of wealth and flash. It was a conversation between me and them. But now most of those who buy my records are listening in on others’ conversation. They are the aural equivalent of voyeurs, thrilled at this crazy world that has nothing to do with their experience.”
—Ice-T [Tracy Marrow], U.S. rap musician. Observer (London, Oct. 27, 1991)
“Of course I’m a black writer.... I’m not just a black writer, but categories like black writer, woman writer and Latin American writer aren’t marginal anymore. We have to acknowledge that the thing we call “literature” is more pluralistic now, just as society ought to be. The melting pot never worked. We ought to be able to accept on equal terms everybody from the Hassidim to Walter Lippmann, from the Rastafarians to Ralph Bunche.”
—Toni Morrison (b. 1931)