• gerryflap@feddit.nl
    18·
    1 day ago

    Okay, so:

    1. 🍇 = 1, because 1 + 1 + 1 = 3
    2. Therefore, 🍪 = 2, because 2 - 1 - 1 = 0
    3. 🥪 is the set of all integer numbers Z, as defined
    4. I am not sure about 🍔, but I assume that it’s the set of integers with all even numbers removed, therefore it’s the set of all odd numbers.

    Beyond that starts the nonsense for me. I’m very curious whether that stuff actually checks out. Some of the terms I remember from group theory, but other stuff seems incorrect to my (limited) knowledge.

    • The second definition of 🍕 seems to contain redundant information, as far as I can see " --> " defines a morphism, so why does the predicate “φ is a morphism” matter?

    • The first definition of 🍕 with the contravariant thing also doesn’t parse for me, what does that “-” mean in the function arguments?

    • In the definition of 🌭, what is the n (or the P)? ChatGPT started yapping about real projective space, but I’m not sure if that’s correct.

    If there’s an actual mathematician here who knows then I’d love to know the answer. I’ve kinda been nerd sniped by this question but I don’t possess the knowledge to fully get this one

    • Kogasa@programming.dev
      4·
      18 hours ago

      🍕(–, B) : C -> Set denotes the contravariant hom functor, normally written Hom(–, B). In this case, C is a category, and B is a fixed object in that category. The – can be replaced by either an object or morphism of C, and that defines a map from C to Set.

      For any given object X in C, the hom-set Hom(X, C) is the set of morphisms X -> B in C. For a morphism f : X -> Y in C, the Set morphism Hom(f, B) : Hom(Y, B) -> Hom(X, B) is defined by sending each g : Y -> B to gf : X -> B. This is the mapping C -> Set defined by Hom(–, C), and it’s a (contravariant) functor because it respects composition: if h : X -> Y and f : Y -> Z then fh : X -> Z and Hom(fh, C) = Hom(h, C)Hom(f, C) sends g : Z -> B to gfh : X -> B.

      –

      P^(n)® AKA RP^n is the n-dimensional real projective space.

      –

      The caveat “phi is a morphism” is probably just to clarify that we’re talking about “all morphisms X -> Y [in a given category]” and not simply all functions or something.

      –

      For more context, the derived functor of Hom(–, B) is called the Ext functor, and the exactness of that sequence (if the typo were fixed) is the statement of the universal coefficient theorem (for cohomology): https://en.wikipedia.org/wiki/Universal_coefficient_theorem The solution to this problem is the “Example: mod 2 cohomology of the real projective space” on that page. It’s (Z/2Z)[x] / <x^(n+1)> or 🍔[x]/<x^(n+1)>, i.e. the ring of polynomials of degree n or less with coefficients in 🍔 = Z/2Z, meaning coefficients of 0 or 1.

      • gerryflap@feddit.nl
        3·
        16 hours ago

        Okay I have some reading to do haha. Thanks for the explanation!

        As a programmer (who also did quite some math) it never ceases to amaze me how often math just uses single character variable/function names that apparently have a specific meaning. For instance the P^(n)® thingy. Without knowing this specific notation, one might easily assume it meant something else like power sets. Even within the niche I’m more familiar with (machine learning) there was plenty of that stuff going around.

        Then again, this meme has an incentive to make it harder, it wouldn’t be funny if it explained symbols.

        • Kogasa@programming.dev
          3·
          15 hours ago

          Math builds up so much context that it’s hard to avoid the use of shorthand and reused names for things. Every math book and paper will start with definitions. So it’s not really on you for not recognizing it here

    • MBM@lemmings.world
      5·
      23 hours ago

      🍔 is the set of integers modulo 2 (more literally, if two integers differ by an even integer you consider them the same). I can write out more in a bit.

      Edit: this previous post has some good comments, and you can find some of the notation and the answer on the wikipedia page for cohomology ring (they use F2 for integers mod 2 and RPn instead of Pn(R)). I don’t know enough algebraic topology to actually know why that’s the answer but I can at least answer these:

      The first definition of 🍕 with the contravariant thing also doesn’t parse for me, what does that “-” mean in the function arguments?

      I assume it’s shorthand for saying that if you define f(x) = 🍕(x, B) then f : C --> Set is contravariant.

      In the definition of 🌭, what is the n (or the P)? ChatGPT started yapping about real projective space, but I’m not sure if that’s correct.

      It’s not the notation I’m used to (I’d also think of power sets first), but I think it’s n-dimensional real projective space.

      • gerryflap@feddit.nl
        2·
        16 hours ago

        Ah thanks for the info! Together with the other in-depth comment this is painting a good picture of what’s happening. Though I have some terms to study before I’ll get it.

    • Zenoctate@lemmy.worldOP
      2·
      1 day ago

      All I can say is that P(ℝ) refers to a power set of ℝ (all rational numbers). Although I don’t know what n stands for in Pⁿ(ℝ)

      Basically P(A), where A = {1,2,3}, equal {Φ,1,2,3,(1,2),(2,3),(1,3),(1,2,3)}

      • gerryflap@feddit.nl
        1·
        1 day ago

        Yeah this was a possibility I was thinking as well. The superscript n could just be n recursive applications, but then n is still not defined. It’s one of the things that makes me thing that it’s just nonsense. Also, how do you do math on Lemmy? Can you just use LaTeX math syntax or did you copy those symbols?