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)
“The theory [before the twentieth century] ... was that all the jobs in the world belonged by right to men, and that only men were by nature entitled to wages. If a woman earned money, outside domestic service, it was because some misfortune had deprived her of masculine protection.”
—Rheta Childe Dorr (1866–1948)