Category Theory

Category theory is about how objects are related to each other. Specifically, a category is a collection of objects and morphisms that go from one object to another, where each object has to have the identity morphism and there is a way to compose two morphisms into a third one in an associative way. ¹

An example of Category Theory can be seen in Set Theory. ²

Reading material:

• Category Theory Illustrated by Jencel Panic ¹
• Category Theory for Programmers by Bartosz Milewski ²
• The Joy of Abstraction by Eugenia Cheng ¹
• Category Theory in Context by Emily Riehl ¹

Dual

An example of dual categories can be seen in the product and sum operations from Set Theory. In each of the product and sum operations, you can consider "imposter" sums and "imposter" products which contain additional information which can be removed with a morphism. ¹

Specifically, you can always create a new imposter sum with a trivial function that goes from A + B -> A + B + X and you can always create a new imposter product that goes from A x B x X -> A x B. You can see by looking at the formulas how the extra information X is on different sides of the function. ¹

Anything that has this kind of relationship is said to be dual to one another. The concepts of product and sum are dual and it's why sums are known as converse product or coproduct. ¹