Is this shape topologically a triangular prism with the ends

>>8464730
>Is this shape topologically ...
solid torus

The surface is topologically a torus, although one might distinguish it instead as a torus equipped with a homotopy cycle (a loop corresponding to the edge of the chess board), which would facilitate further specification of properties related to connectivity. For example, to a bishop, it is topologically a Möbius band. To a rook, it is a torus. To a queen, it's kind of odd, since it can't move diagonally across the edge, but it can move straight over it. Something more complicated than topology would be needed, as there is now a distinction between diagonal paths and straight paths. I won't comment on pawns as it is not clear how they are meant to be oriented.

>>8464730
>n-th dimensional
>redpanels

>>8465197
Is this pasta?

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>>8465603
Did you find it funny and/or shitposty? It's OC, anon. And, I was being serious.

>>8465607

OHP 11/09/16(Wed)12:24:41 No.8464930▶>>8464990

DUDE, FUCKING YES. I had this idea recently! My reasoning came from an observation that entropic data in cohesive homotopy type theory yields a free notion of curvature, compatible with differential cohesion. The reason gravity is incompatible with the other fundamental forces is because it is the emergency of local interactions. This is exciting stuff, thanks for sharing.

You talked about this before. How do you learn more about this shit?

>>8465603
>>8465626
>when your subtle trump post doesn't bait hard enough and you start shitposting

>>8465644
>SJW butthurt tears leaking into other threads
lmao

>>8465644
Not the pasta asker. I wanted to learn more about math.

faggot.

>>8465626
I just browse the hell out of the nLab, read the references provided there (especially Lurie's work), and then fiddle around with ideas until something feels like it's "right." Once you have the resources, the only way to learn the stuff is to get motivated and put in the time. As far as getting to a place where you can produce novel ideas yourself, it just comes with practice and becoming comfortable with the material. For example, when I first learned about adjoint functors, I had little appreciation for them, but as I worked with them, discovered examples, and found use for them in formalizing my own ideas, I came to realize what they are all about and why they are important.

>>8465885
Ive been working on getting my head around some higher level math. Is there a framework to follow? Ive kinda just been bouncing around different fields aimlessly. Im starting to notice that at this level of math intense amounts of prior knowledge are necessary.

Is this a train to follow?

Abstract Algebra->Elliptic Curves->Algebraic Geometry/Topology->Type Theory-> IUT/Cohomology (your stuff I believe) and beyond?

>>8466769
Yeah, I think bouncing around can work. Ot definitely feels less coherent at the start, but then connections form and you realize that you have been moving around the boundary of one big body of information.

A lot of what you want to learn in that train would benefit from, or necessitate, a good foundation of higher category theory and homotopy theory. I would go with

Abstract Algebra -> Commutative Algebra -> Topology -> Category Theory -> Topos/Sheaf Theory -> Algebraic Geometry -> Higher Category Theory -> Elliptic Curves -> Homotopy Type Theory -> IUTT?

I don't know what the prerequisites are for Interuniversal Teichmüller Theory, but it definitely makes use of category theory. As far as cohomology goes, understanding higher categories and homotopy make cohomology a trivial thing to understand (just reduced mapping objects). Once you know higher category theory, Lurie's work is extremely unifying and deep if you want to really understand algebraic geometry.

>>8466792
Oh wow, thank you man. This looks pretty comprehensive.

Im still struggling with trying to wrap my head around abstract algebra.

You mind if I ask you a question. Is saying that a bijection of points creating a line on a graph is analogous to a functor between categories? Or a morphism between groups?

Am I just a brainlet?

>>8466823
The graph of a curve being a bijection onto the domain is not quite analogous to a functor as much as it is a statement about factorization of functions (you can always factor a function into a surjection followed by an injection, surjecting onto the image and then including the image). This shows up in category theory as well, except now you have a ternary factorization system. The graph of a function also has a fairly useful categorification for functors, but it's harder to visualize.

Good luck with your studies, anon! It's a very rewarding journey.

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>>8466840
Ill have to take your word for it, ha.

Thanks again. Ill bug you later in other threads in time. Cheers.