nLab order theory

Idea

Order theory, the study of orders, is one of the main branches of mathematics, but it often gets short shrift. Many (but not all) of the ideas of category theory that don't apply to groupoids or sets already apply to posets. (Of course, order theory also has features of interest for its own sake.)

As low-dimensional higher category theory

In the context of (n,r)-category theory, ordinary category theory is the study of $(1,1)$-categories. Following the spirit of negative thinking, we find that one dimension below this is not just set theory as the study of $(0,0)$-categories, but a richer theory, that of $(0,1)$-categories. These are posets: sets with order. Thus order theory is the study of $(0,1)$-categories.

A comparative discussion of order theory as compared to category theory is at category theory vs order theory.