Categories for order theorists

Preorders can only describe the direction that objects that move in, they cannot explain the dynamic forces responsible for the motion of objects. Categories are just one way to explain motion in a preorder, defined by axiomatizing moves from one point to another as morphisms. Monoid actions are another way to explain motion in a preorder.

The order-theoretic perspective on categories requires further examination beyond what was provided by my first post. In particular, we need order-theoretic foundations for functors. This is provided by a homomorphism theorem which relates functors to subalgebra and congruence lattices. This leads to the following more complete account of the order theoretic foundations of category theory.

1. Order-theoretic introduction of categories
2. Lattices of subcategories
3. Lattices of congruences of categories
4. Theory of functors
5. Natural transformations

Notes:

[1] The idea of recasting "structure preserving maps" as structure including maps provides a very direct accounting of how most categories are extensions of preorders. Monotone maps, continuous maps, relation homomorphisms, etc all correspond to structural inclusions in corresponding partial orders. This is formalized by hom class comparisons.

[2] There is a lot more that can be done to put category theory on a good order-theoretic footing. There are also many partial orders related to categories that are worthy of further examination. Everything is a work in progress.

[3] In these posts I mostly deal with locally small categories as structured sets. There are also large categories which can be treated as structured proper classes. The differences can be worked out in an axiomatic set theoretic account of categories. The important point is that categories can be treated as a sort of structured collection.