I’m very interested in connecting constructor theory to category theory and, perhaps, assembly theory on the other side. I’ve been looking for a sanity check and found this video very helpful.

Really cool to hear system dynamics and category theory being uttered in the same breath. It is like a supremely-surprising collaboration.

This is the first time I understand what monads are all about. Thank you so much!

This is great.

I kinda like this professor. But she should have picked some better subject to talk about. I mean, Yoneda Lemma for the Category of Matrices is for college students to lecture, not professors.

Ciara should stick to computer science and stay away from juvenile protest movements. Even better - leave the UK.

Day 4 is the best day, thank you Actegories -- how is this only 1.2K views!

14:50 "You can think of the spreading of the posterior as a learning experience" .... said Oscar Wilde to his toy-boy ;)

oldie but goodie tutorial

Awesome talk, thanks!

🐈‍⬛

Beautiful slides. Is there any place I can download them?

Promo-SM 😡

By the quiet pond, A duck quacks and spreads its wings, Ripples fade to still.

😅

Starts at 3:44

I'm sorry - is this Bryce's handwriting, or is this a typeface I'm unfamiliar w/? If the former is the case then I am just in awe!

let's gooooo

🪩🔌

Wow amazing. 👍

Love it!!!!!!!!!!!!!!!

Cool

starts at 8:17

Great lecture by Profesor Emily Riehl. Thank you very much. 🙏

Bartosch Milevski already did right to left, bottom to top, lmao

What is the application of this talk? Any thought?

Nice presentation. It was a very intuitive example, illuminating the Yoneda lemma

Starts at 36:40

Unfortunately all this is pointless until the social structure of humanity is fixed. Financial terrorism reigns due to the greed and flaws in the financial system that allows sociopaths to gain power. Sociopaths will only care about category theory if it can make them money... and so like all things thing, money will ruin category theory and math in general as it is hijacked to be used to extract wealth from society which will then produce fewer mathematicians. As you can see from this presentation, there are already mathematicians selling out to the MIC for a buck.... the same MIC that will eventually cause a nuclear war. Then what is the purpose? Just to pay the bills? It's like "How can we use category theory to design more efficient bombs?", "How can category theory be used to manage our AI robotic force to more efficiently control the slaves?", etc. It's coming. Advancements in our intelligence will always be diverted by the financial terrorists psychopaths for their own ends.

Very clear! (Provided one knows what a natural transformation is) I’d seen parts of some of these ideas before, but this video really made it clear to me

Basically if AB for matrices A and B is their product which we write .(A,B) then if C is some column operation we have .(A,C(B)) = C(.(A,B)). By the Yoneda lemma we have ABC_I where C_I is a special matrix that represents our column operation which we can equivalently do by multiplying on the right by it. This acts, in some sense, like homomorphism: f(AB) = f(A)f(B) = ABF where F is a representation of f. Here f(A) = A if f is matrix multiplication. It's not quite the same as a homomorphism because if f(A) = A then we end up with the identity. It is effectively f(A,B) = Af(B) = ABF as the position of the arguments are treated differently. That is, column operations "commute" through to the second argument of matrix multiplication. I'm sure in the mirror/opposite g(XY) = g(X)Y = GXY when we are talking about row transformations although maybe there is some transposes(OP's) thrown it there to make things work out.... well, lets see f^T((XY)^T) = f((XY)^T)^T = f(Y^TX^T)^T = (Y^Tf(X^T))^T = (Y^TX^TF)^T = F^TX^TY^T = GXY = g(XY) by setting g = f^T, G = F^T, X = B^T, and Y = A^T then we get a row version. E.g., if A = mxn, B = nxk and f(AB) = mxk then X = kxn, Y = nxm and g(XY) = kxm.

Beautiful! But what is the Yoneda Lemma of Yoneda Lemmas? After all, functors are functions towo!

So the Yoneda lemma is just a high level view of a "change of basis"... a sort of "change of functors"? It's expressed categorically but shows up everywhere. It seems quite powerful(because of the simple answer it gives) then as it gives us a way to "change perspectives" in a universal way.

Could one in any way say that naturality *IS* the thing that mathematics tries to pin down? I mean, it's so ubiquitous in math surely it is not coincidental. Given that we can probably safely say there is some deeper connection that is independent of humans(surely if it was just "choice" or "preference" then it wouldn't be so common) then it could be more of a "law" of the universe or at least some type of structure the universe itself contains.

So just to setup some linear algebra maps to her notation and CT: For any pxq matrix with p rows and q cols we can effectively associate in the category of natural numbers maps from one natural number to another that represent matrices of the corresponding p=domain and q=codomain. That is, an arrow from p to q in the category of natural numbers is really any matrix with size pxq. In reality it is not the category of natural numbers but the category of matrices. Here the objects are the natural numbers and the morphisms/arrows are transformations, linear transformations, or equivalently matrices. Hence the homset(p,q) is a set of morphisms. In fact, it is the set of all pxq matrices over whatever field the matrix category is over. So the homset(p,q) itself is generally a "set of matrices". Composing arrows or homsets correspond to matrix multiplication(at least that is one way we get a valid composition structure to turn Mat_F in to a category). So what she is effectively describing categorically is everything you learn in Linear Algebra but by putting it in an "abstract" framework. Everyone that knows LA knows that you can multiply matrices and that the sizes have to "match", that matrix multiplication is associative, and usually that matrices and linear transformations are "isomorphic". This is just translating that language in to the language of category theory as arrows, categories, homsets, and functors. h_k(n), the k-column functor is a functor from Mat->Set that maps the objects to objects but fixes exactly what is mapped so we are only dealing with some fixed-k column matrices. This can be considered a contravariant functor homset(_,k) where _ is our variation parameter and corresponds to n. Effectively h_k(n) = homset(n,k) (but this homset is not in Mat but "across Mat"(in a category that contains Mat and Set as objects)). That is, look at the category Mat and go to the k object. Then an arrow from n to k(or in her OP-notation k to n) is an nxk matrix. There are a number of them for each n(one for matrix for each n->k) and for each object p in the category there is an arrow from p to k(or in her OP-notation k to p). h_k(n) is essentially looking at a sub-category of Mat with k fixed (it has just as many objects as Mat but only arrows in to k so k acts like a terminal object). h_k is just a very long winded way to think of _xk matrices(all matrices with k -columns such as 1xk, 2xk, 3xk, 4xk, etc). Of course n is a variable here too but k is fixed over in. E.g., this is a given k, ("given n do stuff but pretend k is fixed"). Around 18:46 she states h_k(m) <- h_k(n) which is just the statement that the inner dimensions must agree for matrix multiplication/our composition rule. Hence by defining the functor h_k one can define the "inner dimensions must agree" rule quite succinctly. But of course h_k is defined in categorical terms so it's true for any abstract structure that behaves in a similar way. So, in fact, as you were learning linear algebra and all that you were learning the categorical ideas without them being expressed/translated in to category theory. The idea how is that, well, this basic idea of having to have inner dimensions agree for composition has been shown to be a functor... and hence can, once one knows more category theory, be treated more "abstractly" and seen how it interacts with other categorical concepts and structures. (e.g., one can then ask about natural transformations between h_k(n) and another functor between Mat and Set.) The "gradation" is just a fancy way to say that there is an integer(or possibly other) parameter, in this case k, that sorta is "fixed" but isn't fixed. h_1(n) is a "grade", h_2(n) is a another "grade", h_3(n) is another "grade", etc and the gradation is due to the ordering k inherits from the fact that the integers has an ordering. E.g., say one has a real valued sequence f_n(x) = x^n. Then one can think of each f_n as being a "grade" and each grade consisting of a bunch of real numbers. so the 1st grade is f_1 = {x | x \in R), f_2 = {x^2 | x \in R}, etc. The natural transformation on h_k just shows that the "math" is independent of k... or "parallel" or "analogous". E.g., you can have an nx3 and 3xm and the same "logic" applies as if you are using a nx2 and 2xm matrix... or nx100 or 100xm. Yes something specific changes but overall the same structure exists between them, that of the definition of the matrix product with the only thing changing column size(nothing else). So the idea is that because k-column functor is natural (that is, sort of "isomorphic" to the j-column functor(or any q-column functor) there is really nothing new between the two structures(changing k from 34 to 1929 doesn't change anything structurally except a transposition of structure. This will just turn the logic from sum_{j=1..34,i=1..n} ... to sum_{j=1..1929,i=1..n} ... where ... is exactly the same mathematical formula). Bear in mind that a commuting diagram is that two things compute: f(g(_)) = g(f(_)). It should be understood from this that most people use natural transformations all the time. It is so ubiquitous that no one really realizes it.It's sort of like atoms. Almost everything is made up of atoms but virtually no one can see them. Generally speaking there usually are more ways than one to skin a category and what is important is that those "different" ways are really the "same" way(same but different).

CompositionOrder(X) = DiagramOrder(X^(OP)) = DiagramOrder(X)^(OP) Life would be very simple if everyone used the same number of OP's!!! Duality, what a joke!

When I was in the womb I always used the matrix multiplication operation on two matrices A:nxm and B:mxk by selecting random points from each with a uniform weighted probability of n/(n+k) for A and k/(n+k) for B and placing them randomly within a C:nxk matrix. Now I know why I always got the wrong answer.

It's quite funny that when I took linear algebra in college it took me about 1-3 minutes to understand that matrix multiplication had a rule on it about how row dimension and column dimension had to "align up" properly. But to understand it from the categorical perspective it takes about 20-30 minutes. In "basic" math it was like about 3 sentences and 5 examples. In "advanced" math it's like 2000 sentences and 0 examples. I get that the human brain thinks naturally in terms of functions, functions, and natural transformations but it seems the more I become aware just how fundamental of concepts they are the more things seem to become complex-ified. It's quite cool seeing the world as one huge category of vector spaces... but I get tired of having to transform my kinematical movements in to a vector space to fix my coffee now. When category theorists say "I know category is a bit abstract" they are absolutely lying! It's so abstract that it is the most concrete thing possible.

My problem with Emily is that she "harps" way to much on what people may or may not understand rather than skipping over having to put up warnings at every turn. She should just state things as if everyone was essentially on her level. I have a feeling she's spend way to much of her life trying to explain things to people who don't know even the basics and so is constantly in the mindset that she is explaining things to novices. I believe this is the wrong approach.

"I'm gunna help you answer your own question" is savagely efficient black-belt mathematics. 🙂🙂

Pretty amazing talk. Thank you!

Amazing video!

This helped me understand snake equations. Thank you

This is an awesome example, and an awesome explanation.

I was a TA in 1978 for Prof. Linton when he taught introductory mathematics at Wesleyan University in Middletown, CT. No specific memories other than I enjoyed working for him.

Thank you. That was helpful to me. I’m curious about CT. I bought the Mac Lane book but soon got lost.

Amazing talk on an amazing topic! Thanks!

Hello

I love how she describes the 'procedure called abuse of notations' at around 16:00.

Is it often the case that natural transformations form a vector space?