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.
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