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 to, connection, category and/or theory:
“It may comfort you to know that if your child reaches the age of eleven or twelve and you have a good bond or relationship, no matter how dramatic adolescence becomes, you children will probably turn out all right and want some form of connection to you in adulthood.”
—Charlotte Davis Kasl (20th century)
“The connection between dress and war is not far to seek; your finest clothes are those you wear as soldiers.”
—Virginia Woolf (1882–1941)
“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)
“Lucretius
Sings his great theory of natural origins and of wise conduct; Plato
smiling carves dreams, bright cells
Of incorruptible wax to hive the Greek honey.”
—Robinson Jeffers (1887–1962)