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
strict categoryproset
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 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.

Revised on May 29, 2012 22:04:00 by Andrew Stacey (