In Category Theory
Every poset (and every preorder) may be considered as a category in which every hom-set has at most one element. More explicitly, let hom(x, y) = {(x, y)} if x ≤ y (and otherwise the empty set) and (y, z)(x, y) = (x, z). Posets are equivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if it exists, is an initial object, and the largest element, if it exists, is a terminal object. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphism-closed.
A functor from a poset category (a diagram indexed by a poset category) is a commutative diagram.
Read more about this topic: Partially Ordered Set
Famous quotes containing the words category and/or theory:
“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)
“A theory of the middle class: that it is not to be determined by its financial situation but rather by its relation to government. That is, one could shade down from an actual ruling or governing class to a class hopelessly out of relation to government, thinking of gov’t as beyond its control, of itself as wholly controlled by gov’t. Somewhere in between and in gradations is the group that has the sense that gov’t exists for it, and shapes its consciousness accordingly.”
—Lionel Trilling (1905–1975)