Difference between revisions of "Category Theory"

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Category Theory is the [[abstract algebra]] of [[function]]s<ref>{{:Book/Category Theory}}</ref>. It directly relates to how data and computation can be represented as [[function]]s or [[relation]]s. Due to its generally applicable nature, it is so general that many mathematicians calls it '''[[abstract nonsense]]'''.
Category Theory is the [[abstract algebra]] of [[function]]s<ref>{{:Book/Category Theory}}</ref>. It directly relates to how data and computation can be represented as [[function]]s or [[relation]]s. Due to its generally applicable nature, it is so general that many mathematicians calls it '''[[abstract nonsense]]'''.


The seminal paper, [[Paper/General Theory of Natural Equivalence|General Theory of Natural Equivalence]]<ref>{{:Paper/General Theory of Natural Equivalence}}</ref> that defined the outline of Category Theory was written by [[wikipedia:Saunders MacLane|Saunders MacLane]] and [[wikipedia:Samuel Eilenberg|Samuel Eilenberg]]. Saunders Mac Lane wrote a book<ref>{{:Book/Categories for the Working Mathematician}}</ref> on this topic.
The seminal paper, [[Paper/General Theory of Natural Equivalences|General Theory of Natural Equivalences]]<ref>{{:Paper/General Theory of Natural Equivalences}}</ref> that defined the outline of Category Theory was written by [[wikipedia:Saunders MacLane|Saunders MacLane]] and [[wikipedia:Samuel Eilenberg|Samuel Eilenberg]]. Saunders Mac Lane wrote a book<ref>{{:Book/Categories for the Working Mathematician}}</ref> on this topic.


=Category Theory Online Tutorials=
=Category Theory Online Tutorials=

Revision as of 03:55, 25 February 2022

Category Theory is the abstract algebra of functions[1]. 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, General Theory of Natural Equivalences[2] that defined the outline of Category Theory was written by Saunders MacLane and Samuel Eilenberg. Saunders Mac Lane wrote a book[3] on this topic.

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-long lecture[4] 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.


References

  1. ↑ Awodey, Steve (August 13, 2010). Category Theory (PDF) (2nd ed.). Oxford University Press. ISBN 978-0199237180. 
  2. ↑ Eilenberg, Samuel; Mac Lane, Saunders. "General Theory of Natural Equivalences". Transactions of the American Mathematical Society (Vol. 58, No. 2 (Sep., 1945), ed.). local page: American Mathematical Society: 231-294. 
  3. ↑ Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. 5 (2nd ed.). local page: Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001. 
  4. ↑ Southwell, Richard (Jun 28, 2021). Category Theory For Beginners: Kan Extensions. local page: Richard Southwell. 

Related Pages

Abstract Interpretation, Book/From Frege to Gödel, Category Theory, Video/James Hefford and Vincent Wang: Categories of Semantic Concepts

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