6 ms·

Category Theory: Lecture Notes and Online Books

129 points by coalgebraic 5y ago

euroderf 5y ago
N00b question here. Does the mathematical characterization of category theory relate in any way to the bridges that the Langlands program is building ?
- Koshkin 5y ago
  Well, one of the well-known category theorists has said [0], > I’ve never succeeded in understanding the slightest thing about it. [0] https://golem.ph.utexas.edu/category/2010/08/what_is_the_langlands_programm.html https://golem.ph.utexas.edu/category/2010/08/what_is_the_lan...
mbrodersen 5y ago
I have read and understood enough Category Theory to realise that it is kinda fun/interesting if you enjoy abstract math. But it won’t make you a better software (or Haskell) developer. I would instead recommend learning Coq/Lean/Idris/Agda to move up the abstraction level from Haskell, while still learning things that are relevant to developing software.
OracB7 5y ago
I have some interest in the topic (thanks to Arraycast and Conor Hoekstra) and bought to Mock a Mockingbird by Raymond Smullyan, however it is still a bit too involved for my small brain. On the other hand, I am very much enjoying How to Bake Pi by Eugenia Cheng, which is a few levels down in difficulty.
bob_coecke 5y ago
Within the applied category realm where seven sketches lives, there are low-level concrete industrial applications that are becoming increasingly dominant with key industry players in quantum tech. The most known one is ZX-calculus which is explicitly used in things like compiler optimisation, error-correction, translations between computational models etc. This might be the 1st time that CT is playing a central role in a new technology. The reason is the high-level connotation that has been pushed far too long concerning category theory, originating in its abstract mathematical origins. Bob Coecke, Chief Scientist, Quantinuum.
elnatro 5y ago
I find this topic fascinating but hard to start with. Does anyone have any non-Haskell example of category theory? Something that shows why is useful with a use-case.
- Jhsto 5y ago
  This series has some C++ examples alongside Haskell: https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/ https://bartoszmilewski.com/2014/10/28/category-theory-for-p... and acquirable as a PDF at https://github.com/hmemcpy/milewski-ctfp-pdf https://github.com/hmemcpy/milewski-ctfp-pdf
- cashsterling 5y ago
  Cambridge Quantum Computing applies Category Theory to development of quantum computing algorithms. Recent paper by Bob Coecke: https://arxiv.org/abs/2110.05327 https://arxiv.org/abs/2110.05327. Look up more papers by Bob Coecke and co-authors on Arxiv. Cambridge Quantum Computing and Honeywell Quantum Solutions recently merged to form Quantinuum. I work on the hardware side at Quantinuum. I don't actually use or apply Category Theory on a day to day basis but I know it is useful.
- bern4444 5y ago
  I wrote this article[0] as part of my own journey to understand monads and category theory. I think you may find it helpful. It has a practical use case that builds up an intuitive model to a new monad. It goes through a few solutions for a stated problem which solve increasing requirements and eventually coalesces into writing a new monad as a complete solution to resolve all the built up requirements. [0] https://sambernheim.com/blog/building-a-monad https://sambernheim.com/blog/building-a-monad
- vkk8 5y ago
  Seven Sketches in Compositionality by Spivak and Fong (https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf) is supposed to be very practical. It includes many examples, but I'm not sure how useful category theory actually is in the example cases. There is the theory of monotone co-design (https://co-design.science/index.html https://co-design.science/index.html) which is formulated using category theory. It seems pretty practical. Then there's topological data analysis, which is clearly a practical subject, and can be formulated in terms of category theory: https://en.wikipedia.org/wiki/Topological_data_analysis https://en.wikipedia.org/wiki/Topological_data_analysis I'm not very knowledgeful of any of these subjects, but I, like you, got somewhat interested in category theory and tried to find how and where it's used some time ago. These are the main things I found.
ryan93 5y ago
Category theory seems like a "meme" interest among programmers.
- DaiPlusPlus 5y ago
  It is, because modern programming languages today are far more expressive than previous generations of languages are, with powerful type-systems - and languages' type-systems are grounded in category-theory (and group-theory, and type-theory, and set-theory). Modern languages are our tools, and you can't use your tools effectively without understanding them, and you can understand them better by understanding their mathematical underpinnings. ------------- Disregarding the comparatively more esoteric languages like Haskell, consider TypeScript, Rust, Haxe, and to an extent: Kotlin and Scala: you can take a self-taught (i.e. non-CS/SE graduate) programmer and eventually they'll figure their way through, and eventually grok, that languages' type-system - and hey-presto, you've just unwittingly taught someone something an entire course of graduate-level abstract mathematics would do, and been productive in the process.
meiji163 5y ago
The best book I've found for a non-math audience is Fong & Spivak, 7 Sketches in Compositionality.
- JosephHatfield 5y ago
  Thanks for the recommendation. Found it available for free download at David Spivak's homepage on MIT's Math department website
rg111 5y ago
Seven Sketches in Compositionality is easily the best book to get started on Category Theory. Learning something becomes highly fun and enjoyable when the authors/lecturers are deliberately trying to make it so. Most books on anything serves to document something and looks like accumulated notes that are an overt attempt to impress the authors' peers. There is no focus on actually teaching. This book is not like that. This book focuses entirely on teaching, tries deliberately so that you can learn. Some other books that I have come across in my life are: 1. Intro to Electrodynamics by Griffiths 2. High School Physics book by Halliday, Resnick, Walker. 3. ToC book by Sipser. You get the idea.

129 points by coalgebraic 5y ago

N00b question here. Does the mathematical characterization of category theory relate in any way to the bridges that the Langlands program is building ?

Koshkin 5y ago
Well, one of the well-known category theorists has said [0], > I’ve never succeeded in understanding the slightest thing about it. [0] https://golem.ph.utexas.edu/category/2010/08/what_is_the_langlands_programm.html https://golem.ph.utexas.edu/category/2010/08/what_is_the_lan...

mbrodersen 5y ago

I have read and understood enough Category Theory to realise that it is kinda fun/interesting if you enjoy abstract math. But it won’t make you a better software (or Haskell) developer. I would instead recommend learning Coq/Lean/Idris/Agda to move up the abstraction level from Haskell, while still learning things that are relevant to developing software.

OracB7 5y ago

I have some interest in the topic (thanks to Arraycast and Conor Hoekstra) and bought to Mock a Mockingbird by Raymond Smullyan, however it is still a bit too involved for my small brain. On the other hand, I am very much enjoying How to Bake Pi by Eugenia Cheng, which is a few levels down in difficulty.

bob_coecke 5y ago

Within the applied category realm where seven sketches lives, there are low-level concrete industrial applications that are becoming increasingly dominant with key industry players in quantum tech. The most known one is ZX-calculus which is explicitly used in things like compiler optimisation, error-correction, translations between computational models etc. This might be the 1st time that CT is playing a central role in a new technology. The reason is the high-level connotation that has been pushed far too long concerning category theory, originating in its abstract mathematical origins. Bob Coecke, Chief Scientist, Quantinuum.

elnatro 5y ago

I find this topic fascinating but hard to start with. Does anyone have any non-Haskell example of category theory? Something that shows why is useful with a use-case.

Jhsto 5y ago
This series has some C++ examples alongside Haskell: https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/ https://bartoszmilewski.com/2014/10/28/category-theory-for-p... and acquirable as a PDF at https://github.com/hmemcpy/milewski-ctfp-pdf https://github.com/hmemcpy/milewski-ctfp-pdf
cashsterling 5y ago
Cambridge Quantum Computing applies Category Theory to development of quantum computing algorithms. Recent paper by Bob Coecke: https://arxiv.org/abs/2110.05327 https://arxiv.org/abs/2110.05327. Look up more papers by Bob Coecke and co-authors on Arxiv. Cambridge Quantum Computing and Honeywell Quantum Solutions recently merged to form Quantinuum. I work on the hardware side at Quantinuum. I don't actually use or apply Category Theory on a day to day basis but I know it is useful.
bern4444 5y ago
I wrote this article[0] as part of my own journey to understand monads and category theory. I think you may find it helpful. It has a practical use case that builds up an intuitive model to a new monad. It goes through a few solutions for a stated problem which solve increasing requirements and eventually coalesces into writing a new monad as a complete solution to resolve all the built up requirements. [0] https://sambernheim.com/blog/building-a-monad https://sambernheim.com/blog/building-a-monad
vkk8 5y ago
Seven Sketches in Compositionality by Spivak and Fong (https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf) is supposed to be very practical. It includes many examples, but I'm not sure how useful category theory actually is in the example cases. There is the theory of monotone co-design (https://co-design.science/index.html https://co-design.science/index.html) which is formulated using category theory. It seems pretty practical. Then there's topological data analysis, which is clearly a practical subject, and can be formulated in terms of category theory: https://en.wikipedia.org/wiki/Topological_data_analysis https://en.wikipedia.org/wiki/Topological_data_analysis I'm not very knowledgeful of any of these subjects, but I, like you, got somewhat interested in category theory and tried to find how and where it's used some time ago. These are the main things I found.

ryan93 5y ago

Category theory seems like a "meme" interest among programmers.

DaiPlusPlus 5y ago
It is, because modern programming languages today are far more expressive than previous generations of languages are, with powerful type-systems - and languages' type-systems are grounded in category-theory (and group-theory, and type-theory, and set-theory). Modern languages are our tools, and you can't use your tools effectively without understanding them, and you can understand them better by understanding their mathematical underpinnings. ------------- Disregarding the comparatively more esoteric languages like Haskell, consider TypeScript, Rust, Haxe, and to an extent: Kotlin and Scala: you can take a self-taught (i.e. non-CS/SE graduate) programmer and eventually they'll figure their way through, and eventually grok, that languages' type-system - and hey-presto, you've just unwittingly taught someone something an entire course of graduate-level abstract mathematics would do, and been productive in the process.

meiji163 5y ago

The best book I've found for a non-math audience is Fong & Spivak, 7 Sketches in Compositionality.

JosephHatfield 5y ago
Thanks for the recommendation. Found it available for free download at David Spivak's homepage on MIT's Math department website

rg111 5y ago

Seven Sketches in Compositionality is easily the best book to get started on Category Theory. Learning something becomes highly fun and enjoyable when the authors/lecturers are deliberately trying to make it so. Most books on anything serves to document something and looks like accumulated notes that are an overt attempt to impress the authors' peers. There is no focus on actually teaching. This book is not like that. This book focuses entirely on teaching, tries deliberately so that you can learn. Some other books that I have come across in my life are: 1. Intro to Electrodynamics by Griffiths 2. High School Physics book by Halliday, Resnick, Walker. 3. ToC book by Sipser. You get the idea.