In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.
This article gives a general treatment of universal properties. To understand the concept, it is useful to study several examples first, of which there are many: all free objects, direct product and direct sum, free group, free lattice, Grothendieck group, product topology, Stone–Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.
Read more about Universal Property: Motivation, Formal Definition, Duality, Examples, History
Famous quotes containing the words universal and/or property:
“It is long ere we discover how rich we are. Our history, we are sure, is quite tame: we have nothing to write, nothing to infer. But our wiser years still run back to the despised recollections of childhood, and always we are fishing up some wonderful article out of that pond; until, by and by, we begin to suspect that the biography of the one foolish person we know is, in reality, nothing less than the miniature paraphrase of the hundred volumes of the Universal History.”
—Ralph Waldo Emerson (1803–1882)
“It is clearly better that property should be private, but the use of it common; and the special business of the legislator is to create in men this benevolent disposition.”
—Aristotle (384–322 B.C.)