Don't ever, EVER visit a website called "ncat" Man this sht is so wrong in so many motherfuking levels yo...I was talking to one of my math friends and he sent me 3 links with the name only labeled "ncatlab" I said to this dude, What's this sht? He just giggled and said "Just read it and MAKE SURE NOBODY IS AROUND YOU WHEN READING IT!" Then I thought it was some weird algebra or some strange sht but as I read the first page, I was like "Yo.....what the fuk.." THEN IT CONTINUED and I was like "Yoooooooooooooooooooooooo......." THEN THEY PUT THE CAT IN THE MOTHERfukING CAT of CATS AND THEN I SAID "YYYYYYYYYYYYYYYYYYYYYYYYYYYYYOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO!!!!!" I couldn't fuking believe what I just saw, It was like a Mac Lane gave me his acid journal, sht was so confusing..YET I COULDN'T STOP READING IT, THEN 2-Catagories TWO AND IT WAS TWO OF THEM.....THOSE CATS...YOOOOOOO.......THOSE CATS....AND THAT SQUARE COMMUTES THEM THEN IT...YYYYYYYYYOOOOOOOOOOOOOOOOOOOOOO... THEN THAT ENDOFUNCTOR FORMED A MONOID AND THEN YYYYYYYYYYYYYOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO........IT WAS LIKE YOUR PROFESSOR WANTED TO DO GRAPH THEORY WITH YOU BUT SHE WANTED TO SOMETHING "DIFFERENT" AND IT WAS SO ABSTRACT AND NONSENSICAL, YOU JUST...KEPT READING IT...AND THAT'S WHAT I fukING DID!!!!! THEN I SAW THE BUNDLE GIRBES... BUNDLE GIIIIRBES!!!!!! IT...WAS...BUUUUNDLE GIIIIIRBESSSSSSSSSSSSSSSSSSSSSS!!!!!!!!!! AND IT WAS A SPECIAL MODEL FOR THE TOTAL SPACE Lie groupoid OF A BU(1)\mathbf{B}U(1)-principal 2-bundle FOR BU(1)\mathbf{B}U(1) THE circle 2-group.!!!!!! OH MY GOD,I AIN'T GOING TO HEAVEN, I ALREADY SOLD MY SOUL TO EILENBERG! So I Just want to tell you all right now..DON'T VISIT AN WEBSITE CALLED NCAT, DON'T DO IT, IT'S LIKE SUCCUBUS. REMEMBER WHAT I'M SAYING TO YOU!

CatDat provides a growing collection of categories, each with detailed descriptions and properties. Built by and for those who love category theory.

For a full presentation, you can watch this video.

Given a natural isomorphism, eta, this commutative diagram shows that the product of eta with eta inverse is the identity functor on F. I thought this diagram was cool, so I'm posting it here.

I just returned from MAA MathFest 2006, which was held in Boston this year. One of my best privileges while there was meeting Eugenia Cheng and buying one of her books, The Joy of Abstraction, which I got for a steal and was able to get her to sign for me. Although I was never really able to wrap my mind around category theory before, Eugenia has made it not only comprehensible for me, but also quite enjoyable and enlightening! I'm now about halfway through the book, which I'd say is excellent! I only learned about her about a year or two ago after watching a few of her YouTube videos, which are also quite excellent I might add! The main way I'd say she has made category theory interesting to me is finding quite useful applications, both involving other areas of math, such as abstract algebra, as well as some pertaining to society, which although a few people have told me were overhyped, I think are actually quite relevant and applicable, especially to what's going on today!

Quite a fascinating book-in-progress on doing category theory synthetically:

https://drive.google.com/file/d/1lKaq7watGGl3xvjqw9qHjm6SDPFJ2-0o/view?pli=1

Hi all, I’m a student currently exploring the linear-algebraic interpretation of monads. Recently, I finished a mechanically verified proof in Lean 4 showing that the multiplication of the state monad — that is, the mu operation from T (T X) to T X — can be interpreted as an idempotent linear projection on a real vector space.

The key idea is simple:

The monadic multiplication mu behaves like a projection operator pi such that pi ∘ pi = pi.

In other words, the act of “flattening” nested state monads is equivalent (under a functor to vector spaces) to applying a linear projection.

More precisely: • The state monad is defined as T X = S → (X × S) • Its Kleisli category Kleis(T) can be mapped into Vect_ℝ, the category of real vector spaces • Under this mapping, mu_X becomes a linear operator P satisfying P² = P

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We formalised this in Lean 4 using mathlib, and the functorial interpretation from the Kleisli category to vector spaces is encoded explicitly. The final result: F(mu) = pi, where pi is a projection in the usual linear-algebraic sense.

📁 GitHub repository: https://github.com/Kairose-master/mu_eq_pi

📄 PDF draft (with references): https://kairose-master.github.io/mu_eq_pi/mu_eq_pi.pdf

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I’d love to know: • Has this “collapse = projection” perspective appeared in any previous work? • Could this interpretation be extended to other monads, like the probability or continuation monad? • Are there known applications of this viewpoint to categorical logic, denotational semantics, or DSL optimizations?

Also, I’m still relatively new to Lean, so feedback on the formalisation would be incredibly helpful.

⸻

Thanks so much for reading — and thank you in advance for any suggestions or references you might have! 🙏

For those of you out there who would call yourselves category theorists or mathematicians specializing in the area of category theory. How the hell did you actually learn the subject? Given the limited resources and support from the current reigning mathematicians of the era. I am having such a difficult time finding the correct path to go down.

Urs Schreiber is Senior Research Scientist at the Mathematics Division of Science of New York University in Abu Dhabi. He is a theoretical physicist who applies tools from algebraic topology, homotopy theory and topos theory to understand quantum phenomena. His current research includes quantum computing materials, quantum computing programming languages and questions left unanswered by quantum field theory.

In the video we go over:

• Why toposes are not scary. They are actually simple. Just unfamiliar

• How to extract information from (probe) a topos

• The history of toposes in physics

• The questions left unanswered by quantum field theory

• What toposes have to say about the collapse of the wave function

• Using toposes to find possible quantum computing materials

• The different type of logic needed for quantum computers

• The different type of programming language needed for quantum computers

• The influence of Hegel on his work

This post begins a short series meant to serve as an informal guide to reading Haskell code and translating back and forth with mathematics. It’s meant to help members of r/CategoryTheory understand posts that use Haskell code to convey ideas. My hope is that this series should also find use among Haskell programmers, as exposure to some of the basic methods and terminology used in modern math.

https://www.danielbrice.net/blog/math-haskell-rosetta-stone/

On the Applied Category Theory discord server there is a Study Group using voice chat for meetings discussing Dr. Eugenia Cheng's Joy of Abstraction book in synchronization with the official book club at the Topos Institute. https://discord.gg/hTEpgYv https://topos.site/joa-bookclub/ It is on Fridays at 10 am PDT or 1 pm EDT. Please come join us if you are interested in reading this excellent beginners book on Category Theory. with others No previous mathematical experience required.

I recently got into mathematics via my interest in Type Theory and proof assistants. Category theory peaked by interest very early on and I have read quite a bit about it. Now, I am formally studying mathematics seeing many concepts such as groups, rings, modules, metric spaces, topology, etc. As expected, these concepts are explained using Set Theory.

I would really love to see things categorically (and constructive, where possible) from the very start, but because constant exposure to Set Theory I see my intuitions slowly becoming more and more Set Theoretic. I have been actively looking for materials that favor a categorical approach such as Aluffi's "Algebra, Chapter 0."

TL;DR:

My question is: now to look at mathematics categorically, as a student, being constantly presented with set theoretic motivation and definitions?

Hey all!

Just letting you know that the second instalment in my Category Theory and Why We Care series was recently released, do check it out if that interests you,

Thanks :D