Category Witch

@mattecapu Wondering how that's even symmetric? (is it?) I suppose by taking some left or right adjoints you can turn L \tens M into M \tens L?

@mattecapu :eyes:

the tensor product of sup lattices is really funky, L⊗M is given by the lattice of join-preserving maps L^op->M aka (contravariant) Galois connections from L to M 😯

@TonyTheLion2500 They don't tell you this but presheaves are better than sheaves even though they have the derogatory name

Why do we need presheaves to exist? Why not just have it all be a part of the sheaf structure? I'm asking what the reason is to split it into presheaves and sheaves?

@davidad I've been saying! (e.g. my thesis)

Kripke: Nondeterministic dynamical systems can be unpacked via temporal logic; they are really just graphs Kidney & Wu: Graphs can be unpacked via coalgebra and they are really just nondeterministic dynamical systems doisinkidney.com/pdfs/formalisi…

@no_earthquake Live from New York it's Saturday Night!

if kamala wins she should get to execute musk

@Glorbutron @davidad Oh fair enough. Is that true in some pathological Hilbert spaces?

@JadeMasterMath @davidad The issue is that linearly independent doesn't imply orthogonal

My pet peeve is not people using the word “orthogonal” with its proper meaning: have zero inner product, e.g. the inner product ⟨X,Y⟩≔𝔼[X⋅Y]. (If you know me it will be obvious that I could not possibly object to that!) But often, folks really mean “linearly independent” 😡

I can suggest an equation that has the potential to impact the future: 𝐍𝐚𝐭(𝐇𝐨𝐦(−, 𝐶), 𝐹) ≅ 𝐹(𝐶) + AI Given a category 𝒞, a functor 𝐹 : 𝒞ᵒᵖ → 𝐒𝐞𝐭, and an object 𝐶 ∈ 𝒞. This equation combines Yoneda’s famous Yoneda lemma, which relates a category (𝒞) to a functor (𝐹: 𝒞ᵒᵖ → 𝐒𝐞𝐭) and the speed of light (𝐶 ∈ 𝒞), with the addition of Al (Artificial Intelligence). By including Al in the equation, it symbolizes the increasing role of artificial intelligence in shaping and transforming our future. This equation highlights the potential for Al to unlock new forms of energy, enhance scientific discoveries, and revolutionize various fields such as healthcare, transportation, and technology.

@garvidge I don't know if computers just take binary input, at least that's a pretty reductionist way of looking at it. How would this capture running two programs at once for example. By the way, the set in question is infinite sequences not finite ones.

@JadeMasterMath Well all programmable programs are finite strings fed into computers, which take binary input. So I’m pretty sure this captures everything right? Unless we’re using a more general definition for a program

I think this is more like a particular choice of semantics for programs, which makes sense in some contexts but not others. Or maybe I'm missing the point that all programs can be reduced to these somehow?

Bohmian interpretation of quantum mechanics

What is pip then?

@BarbaraFantechi You MUST include the points at infinity

@JadeMasterMath Don’t understand choice of hiding i due punti di fuga* of two orthogonal directions beyond the area covered by the video. *I learned this in school, as part of disegno tecnico. In Italian.

Projective geometry is when you draw a lot of lines

@fabgenovese If the product is free then you are the product!

Am I the only one seeing a huge risk for security here? This seems to entail relinquishing my machine to the AI, which can then share any data it finds on it with mothership I guess.

@myers_jaz @BowToChris The reason for this is because a lax functor 1 -> D is the same as a monad in D, so making it normal means there's no algebras and you have to add them back in to get an equivalence

@myers_jaz @BowToChris Well you've got an equivalence normal lax C -> Alg(D) And lax C -> D So Alg is actually the right adjoint. I think that makes normal lax functors a coreflective subcategory?

Still getting weirded out about how: Lax functors C -> Span And Lax normal functors C -> Prof are the same thing

@BarbaraFantechi @themathgay 👀 what is it an Euler characteristic of?

@themathgay They still don’t know it’s an Euler characteristics, much less what it is the Euler characteristics of.

-1/12

People just learning that Israel is an authoritarian theocracy are like:

@BowToChris Well there's the thing where you invert spans to get profunctors but that only gets discrete categories so not quite

@JadeMasterMath Does this factor through some functor between Span and Prof?