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:
“When the apple is ripe it will fall.”
—Irish proverb.
An English equivalent to this might be, “To everything there is a season.”
“All cultural change reduces itself to a difference of categories. All revolutions, whether in the sciences or world history, occur merely because spirit has changed its categories in order to understand and examine what belongs to it, in order to possess and grasp itself in a truer, deeper, more intimate and unified manner.”
—Georg Wilhelm Friedrich Hegel (1770–1831)