Galois Connection - Connection To Category Theory

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 xy. 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:

    The virtue of art lies in detachment, in sequestering one object from the embarrassing variety. Until one thing comes out from the connection of things, there can be enjoyment, contemplation, but no thought.
    Ralph Waldo Emerson (1803–1882)

    I see no reason for calling my work poetry except that there is no other category in which to put it.
    Marianne Moore (1887–1972)

    The theory of the Communists may be summed up in the single sentence: Abolition of private property.
    Karl Marx (1818–1883)