Difference between revisions of "Category Theory"
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Category Theory is | 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, | 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= | ||
For starters, the following video series would be great starting points for people who wants to know more about Category Theory. | For starters, the following video series would be great starting points for people who wants to know more about Category Theory. One of the better presentation is by [[Jeremy Mann]], who has a four session series on [[Intro to Category Theory]]. | ||
{{:Intro to 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 [[Category Theory in 30 minutes|30 minutes]]. | To obtain an orientation of Category Theory, the following 3 part Category Theory introductory video series by [[John Peloquin]] can be finished in [[Category Theory in 30 minutes|30 minutes]]. | ||
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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]]. | 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]]. | ||
{{:Table/Declarative vs. Imperative}} | |||
==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<ref>{{:Video/Category Theory For Beginners: Kan Extensions}}</ref> on [[Kan Extension]] by [[Richard Southwell]] is a good start. | |||
==Richard Southwell== | ==Richard Southwell== | ||
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==William Lawvere== | ==William Lawvere== | ||
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=Online Resources on Category Theory= | |||
# [[Jeremy Mann]] has a video series on [[Intro to Category Theory]] | |||
# [[John Peloquin]] also has a video series organized on the page:[[Category Theory by John Peloquin]] | |||
# [[Altexploit]] is a website that specializes in Category Theory and Philosophy | |||
# [[MATH3MA]] is a website by [[Tai-Danae Bradley]] | |||
==Content in Chinese== | |||
[http://swarma.org 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: [https://campus.swarma.org/course/2723 Category Theory Lecture Series #1]. | |||
<noinclude> | |||
=References= | =References= | ||
<references/> | |||
=Related Pages= | |||
[[Category:Physics]] | |||
[[Category:Combinatorial Physics]] | |||
[[Category:Meta mathematics]] | |||
[[Category:Quantum Physics]] | |||
[[Category:Formal Method]] | |||
[[Category:Semantics]] | |||
[[Category:Data Science]] | |||
</noinclude> | |||
Latest revision as of 05:45, 13 January 2024
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. One of the better presentation is by Jeremy Mann, who has a four session series on Intro to 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.
| 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
|https://www.youtube.com/watch?v=ZYGyEPXu8as%7C%7C%7C%7C%7C}}
Online Resources on Category Theory
- Jeremy Mann has a video series on Intro to Category Theory
- John Peloquin also has a video series organized on the page:Category Theory by John Peloquin
- Altexploit is a website that specializes in Category Theory and Philosophy
- MATH3MA is a website by Tai-Danae Bradley
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
- ↑ Awodey, Steve (August 13, 2010). Category Theory (PDF) (2nd ed.). Oxford University Press. ISBN 978-0199237180.
- ↑ 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.
- ↑ 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.
- ↑ Southwell, Richard (Jun 28, 2021). Category Theory For Beginners: Kan Extensions. local page: Richard Southwell.