Category Theory's Notion of Order, Illustrated

It treats order as a binary relation on a set, and organizes what kind of structure arises when that relation satisfies reflexivity, transitivity, antisymmetry, and totality.

A total order is a structure in which every two elements are comparable, and a poset is this structure minus totality. In a poset, not all elements are comparable with each other, but the structure can be read through chain, maximum element/minimum element, join, meet, and Hasse diagram.

As examples, it cites color mixing, divisibility, and set inclusion. Color mixing and the inclusion order form a lattice, and in the divisibility order, join = least common multiple and meet = greatest common divisor.

Through Birkhoff's representation theorem, it explains that a distributive lattice is isomorphic to the inclusion order of its own join-irreducible elements. In other words, even though a lattice may look like an abstract structure, it can ultimately be represented as the inclusion order of smaller generating elements.

An order isomorphism is an invertible function between underlying sets that also preserves order. This is illustrated through an example in which the divisibility order of numbers and the inclusion order of prime powers correspond to the same structure.

Finally, it covers preorder. A preorder has only reflexivity and transitivity, and by grouping elements connected to each other in both directions into equivalence classes, a poset can be constructed on top of it.

From this perspective, a preorder can be read directly as a category. Each object has an identity morphism, and the transitivity of the relation corresponds to composition. In particular, any total order can be viewed as a thin category in which there is at most one morphism between any two objects, and in this case coproduct corresponds to join and product corresponds to meet.