Category Of Preordered Sets
The category Ord has preordered sets as objects and monotonic functions as morphisms. This is a category because the composition of two monotonic functions is monotonic and the identity map is monotonic.
The monomorphisms in Ord are the injective monotonic functions.
The empty set (considered as a preordered set) is the initial object of Ord; any singleton preordered set is a terminal object. There are thus no zero objects in Ord.
The product in Ord is given by the product order on the cartesian product.
We have a forgetful functor Ord → Set which assigns to each preordered set the underlying set, and to each monotonic function the underlying function. This functor is faithful, and therefore Ord is a concrete category. This functor has a left adjoint (sending every set to that set equipped with the equality relation) and a right adjoint (sending every set to that set equipped with the total relation).
Read more about Category Of Preordered Sets: 2-category Structure
Famous quotes containing the words category and/or sets:
“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)
“Nothing sets a person up more than having something turn out just the way it’s supposed to be, like falling into a Swiss snowdrift and seeing a big dog come up with a little cask of brandy round its neck.”
—Claud Cockburn (1904–1981)