nLab category theory vs order theory


Consider the following table. The basic thesis of this page is that, if you are having trouble understanding an item in the left column, then first you should understand the item in the right column. (Each item on the right is a decategorification, in one way or another, of the item on the left.)

Category theoryOrder theory
skeletal categoryposet
functormonotone function
(dual) adjunctionGalois connection
monadMoore closure
cartesian closed pretoposHeyting algebra
Grothendieck toposlocale

In the first several rows, we trust that people know about the order-theoretic concept before we teach them about category theory, so we can and do use these rows pedagogically. But after that, we cannot trust that people know about the order-theoretic concepts, so we teach the category-theoretic concepts without them —but the order-theoretic concepts are still simpler, so maybe we should teach (or learn) about them first. This is the approach taken, for example, in Paul Taylor's book Practical Foundations.

This is an example of negative thinking, but perhaps not more negative than normal. It is common to go down as far as (0,0)(0,0)-categories, but not so common to move to (0,1)(0,1)-categories before pushing on to (1,1)(1,1)-categories.

See also at relation between preorders and (0,1)-categories.


Last revised on July 23, 2022 at 09:19:27. See the history of this page for a list of all contributions to it.