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
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“We have had many harbingers and forerunners; but of a purely spiritual life, history has afforded no example. I mean we have yet no man who has leaned entirely on his character, and eaten angels’ food; who, trusting to his sentiments, found life made of miracles; who, working for universal aims, found himself fed, he knew not how; clothed, sheltered, and weaponed, he knew not how, and yet it was done by his own hands.”
—Ralph Waldo Emerson (1803–1882)
“Let the amelioration in our laws of property proceed from the concession of the rich, not from the grasping of the poor. Let us understand that the equitable rule is, that no one should take more than his share, let him be ever so rich.”
—Ralph Waldo Emerson (1803–1882)