Editing Order theory (section)

=== Category theory ===
The visualization of orders with [[Hasse diagram]]s has a straightforward generalization: instead of displaying lesser elements ''below'' greater ones, the direction of the order can also be depicted by giving directions to the edges of a graph. In this way, each order is seen to be equivalent to a [[directed acyclic graph]], where the nodes are the elements of the poset and there is a directed path from ''a'' to ''b'' if and only if ''a'' ≤ ''b''. Dropping the requirement of being acyclic, one can also obtain all preorders.

When equipped with all transitive edges, these graphs in turn are just special [[category theory|categories]], where elements are objects and each set of morphisms between two elements is at most singleton. Functions between orders become functors between categories. Many ideas of order theory are just concepts of category theory in small. For example, an infimum is just a [[product (category theory)|categorical product]]. More generally, one can capture infima and suprema under the abstract notion of a [[limit (category theory)|categorical limit]] (or ''colimit'', respectively). Another place where categorical ideas occur is the concept of a (monotone) [[Galois connection]], which is just the same as a pair of [[adjoint functor]]s.

But category theory also has its impact on order theory on a larger scale. Classes of posets with appropriate functions as discussed above form interesting categories. Often one can also state constructions of orders, like the [[product order]], in terms of categories. Further insights result when categories of orders are found [[equivalence of categories|categorically equivalent]] to other categories, for example of topological spaces. This line of research leads to various ''[[representation theorem]]s'', often collected under the label of [[Stone duality]].