Connection To Category Theory
Every partially ordered set can be viewed as a category in a natural way: there is a unique morphism from x to y if and only if x ≤ y. A Galois connection is then nothing but a pair of adjoint functors between two categories that arise from partially ordered sets. In this context, the upper adjoint is the right adjoint while the lower adjoint is the left adjoint. However, this terminology is avoided for Galois connections, since there was a time when posets were transformed into categories in a dual fashion, i.e. with arrows pointing in the opposite direction. This led to a complementary notation concerning left and right adjoints, which today is ambiguous.
Read more about this topic: Galois Connection
Famous quotes containing the words connection, category and/or theory:
“Accept the place the divine providence has found for you, the society of your contemporaries, the connection of events.”
—Ralph Waldo Emerson (1803–1882)
“The truth is, no matter how trying they become, babies two and under don’t have the ability to make moral choices, so they can’t be “bad.” That category only exists in the adult mind.”
—Anne Cassidy (20th century)
“The great tragedy of science—the slaying of a beautiful theory by an ugly fact.”
—Thomas Henry Huxley (1825–1895)