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
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