Category Theory

Category Theory is a foundational representation of mathematics. It directly relates to how data and computation can be represented as functions or relations. Due to its generally applicable nature, it is so general that many mathematicians calls it abstract nonsense.

The seminal paper, A General Theory of Natural Equivalence that defined the outline of Category Theory was written by Saunders MacLane and Samuel Eilenberg.

Category Theory Online Tutorials

For starters, the following video series would be great starting points for people who wants to know more about Category Theory.

To obtain an orientation of Category Theory, the following 3 part Category Theory introductory video series by John Peloquin can be finished in 30 minutes.

An ideal starting point for learning Category Theory is to know its nature of being a pure declarative style of reasoning, in contrast to the imperative style of reasoning. It can also be understood as a global vs. local approach to prescribe concepts. Bartosz Milewski has a great video on this subject, please see Video/Declarative vs Imperative Approach.

Declarative vs. Imperative Reasoning

Concepts\Programming Style	Imperative	Declarative
Mathematical Semantics	Algorithmic Sequence	Category Theory
Scopes	Local	Global
Scientific Doctrines	Classical Physics	Quantum Physics
Scientific Doctrines	Action-Reaction	Stationary Action Principle
Analytical Modeling	Newtonian Mechanics	Lagrangian Mechanics
Infrastructure Automation	Ansible	Terraform

Category Theory is about composition

Compositionality is the property that provides universality in Category Theory's expressiveness. In broad stroke, there are many ways of composing functions and objects. The two stands out most are: Horizontal composition and Vertical composition.

Category Theory should start with Kan Extension

Instead of introducing Category Theory from the definitions of function, functor, and natural transformation, one should start teaching or learning Kan Extension. A 5 hour lecture on Kan Extension by Richard Southwell is a good start.

Richard Southwell

Richard Southwell has a youtube channel on many subjects about math, particularly having a long series on Category Theory.

Topos Institute

Topos Institute, founded by David Spivak and Brendan Fong, also has a youtube channel on Category Theory.

Topos institute publishes its lectures on Youtube, for example:

Toby St Clere Smithe has a talk at Topos Institute on: Compositional Active Inference: A “Process Theory” for Finding Right Abstractions

William Lawvere

{{#ev:youtube|https://www.youtube.com/watch?v=ZYGyEPXu8as%7C%7C%7C%7C%7C}}

Content in Chinese

SWARMA.ORG] is a science club in China, which offer high quality intellectual content on many subject matters. One of the lecture series is on Category Theory, and can be found on this link: Category Theory Lecture Series #1.