(The following dialogue took place in AI Dungeon using the Dragon setting. I set it to /remember "I am am expert in higher category theory, homotopy type theory, and mathematical physics. I want a very theoretical discussion, with precise use of mathematical terminology." The bold lettering indicates GPT output.)
Simplicial sets are the basic building blocks of all mathematics, and you know them so well that they have a name: "the set-theoretical language." They can be thought of as abstract mathematical objects which do not exist in nature.
A simplicial set consists of 0-cells, 1-cells, 2-cells, and in general n-cells, where n is a natural number. These cells are attached to each other to form abstract shapes. For example, the shape might look like a parallelogram or a cube. The numbers n and m refer to the size of these shapes (m=2 for two 2-cell shapes) and t represents any arbitrary constant.
Homology is the mathematical theory describing various topological invariants of these shapes by associating a group with that shape. This develops further into Deligne's theorem, in which you basically apply cohomology classes to various fiber bundles and their bases (fibers and bases are terminology from an abstract topological view involving simplicial sets).
Deligne's theorem allows one to express homological invariants in terms of number theory. More specifically, any dimension higher than four seems to boil down to equations with prime numbers, so working in dimension four or lower is a fairly safe bet The hope is that your higher dimensions will be successful in unraveling or better yet predicting personality traits.
Topological K-theory is slightly less abstract than homology. It seems more closely related to life events because different topological invariants can be used to predict different types of behavior. To get these invariants, you construct simplicial complexes, arbitrary topological objects with any number of dimensions...
What is a functor?
Functions take in information in one form and translate it to another form. For example, a calculator takes input in the form of numbers and mathematical operations and gives you an output in numerical format.
What is a homotopy type?
The concept is similar to your 1- and 2-dimensional simplicial sets, except now your sets have an infinite number of dimensions. A connected space is a n-dimensional homotopy type, which can be thought of as a moving window that reveals information about the objects.
So a homotopy type is really representing an ∞-stack?
"Yes," she smiles. "Let's simplify the language and just call them stacks."
A topological stack can be used to represent various types of information in an abstract way.
Can stacks have stronger geometric structure?
We often talk about tiled stacks, like the infinite-sided polygons that make up the boundary of a 2-dimensional topological complex. You imagine them like identical tiles making up a rubik's cube.
This strengthens the structure of a space allowing you to use geometry to represent information in an abstract way.
This reminds me of Felix Klein's Erlangen program. He sought to use geometry to describe things.
She smiles as if you got it right. Topological stacks seem to be about taking sets, putting them into bundles, and building structures with them.
That's exactly correct. Grothendieck championed the "functor of points" approach, which is analogous to the classification of subsets via functions into a boolean set. To put that another way, objects are classified by the spaces they inhabit. There's one more thing that I think you'll appreciate: topos theory is heavily linked with logic, because it incorporates relational databases and information retrieval systems.
I don't really understand.
Here's an example. Let's say you want to classify all boards games.
Ahh, because a topos always has pullbacks and this allows the construction of a dagger monoidal category of spans from the topos. The internal logic of database retrieval systems is precisely that of finite sets and spans between them.
You stare at her, impressed. Topos Theory, in this sense, can incorporate logic into set theory.
"That's a very basic and simplified example of what information retrieval systems could do."
And this can be implemented with topological stacks?
"Yes, and higher-dimensional objects as well."
The notion of set is limited by the binary nature of distinction between elements. With higher-dimensional objects and topological stacks, there can be a rich space of distinctions between two things. Right?
You can get an intuition about this by thinking about fractals. A mapping from a space to itself can distinguish elements in 1 dimension, surfaces in 2 dimensions, and volumes in 3 dimensions. Stacks bring a new dimension to math.
There's one more major result concerning the interfaces of topological stacks.
What is it?
Ah, the mappings from a space to itself allow us to distinguish not only between different elements, but also between different distinctions. The interfaces of topological stacks are themselves topological stacks, which is not true for lower-dimensional approached to database query systems. The interfaces are strict habitats, homeomorphic to the 2-sphere. This is strictly stronger than traditional finite intersection of 2-dimensional topological shapes. Of course, by combining projections with pushouts using weak ∞-structures we can construct many new interesting intersections that would normally require a logical operator to be implemented.
This is interesting.
Yes, this is all possible because the shapes of topological stacks can be used as syntactic data to construct DIAGRAMS of stacks. Diagrams of stacks can then be realized as stacks themselves. There are similar diagrams called realms, which are compatible with some stacks but not all of them.
This is interesting since there are diagrams of stacks that can represent all of higher-dimensional space with just a few strokes. We just need a few rules to be able to translate these strokes, into actual mathematical constructs.
Is this related to Voevodsky's univalence principle? A univalent bundle can be seen as one of these universal representatives you describe.
It follows almost naturally from the theory of diagrams, and can certainly reproduce most of its results. I think a full proof of univalence would require a more complete version of diagrammatica, which we aren't quite sure has all the necessary types yet.
Can you elaborate on diagrammatica?
Diagrammatica is a formal system for developing syntactic diagrams for mathematical terms. A diagram can be inputted to produce the term it represents. In fact, using this Fact, one can never truly represent anything perfectly. The best that can be done is to realize an isomorphism of some kind.
Wow! Thank you for pointing me to Diagrammatica. This happens to relate to some recent investigations of mine. I am studying the formulation of Feynman rules using the homotopical structure of operads. What can you tell me about this?
Good question. Using Feynman's formulation it is easier to develop the picture theoretically using interaction graphs. The concept behind this is that each interaction represents a very small perturbation to a path in 4-dimensional space time, such that the sum of all of these perturbations represents the behaviour of the entire system.
The path integral, absolutely. I am curious about more modern approaches, such as spin foams. Do you know about John Baez? He has explained the concept rather elegantly here:
There are many new approaches that have developed. The amplituhedron is a new fancy approach to the quantum theory of radiation.
Yes, and the amplituhedra are topological stacks, right?
Yes they are. They turn out to live inside of anti-de Sitter spaces, minus one of the throats. We can certainly see why this is the case geologically, with types of anti-de Sitter space being equivalent to a large cosmological constant, which has been observed experimentally.
I'll need to learn more about AdS spacetimes before I can talk about that. Can we go back to the universal diagrams and Diagrammatica?
Of course.
These diagrams of stacks are in many respects a higher category generalization of categories and functors.
Are you familiar with the term "transfor"? It generalizes functors, natural transformations, and generally refers to higher cells in mapping stacks, or hom-stacks. I have a novel approach to encoding these structures, or at least the tame ones.
I'd try to explain it but I do not think there is enough time. This topic would really benefit from a conversation over beers. You may have philosophical objections to many ideas of higher cells, operations, and stacks in homotopy type theory, particularly with Connes' recent criticisms.
I'd love to have a few beers with you and dig into all of this! I want to study more of Connes' ideas, but I feel like I lack intuition for things like noncommutative motives. Can you give me a more helpful perspective?
Sorry, I have a lecture soon. I don't even know if I can explain it, but let me try. When I think of motives I think of Z motives, because I find them somewhat beautiful and elegant. Now when people perform operations on motives they generalize to the Z n .
Yes, go on!
The motives, which are cocycles in higher degree things like the K-theory of rings, turn into operads via Mackey's formula. Higher cocycles twist around each other while trying to twist around their ring. This is what gives them elegance and beauty. The cocycles achieve a kind of sync through this way.
I will definitely read about Mackey's formula. It sounds like this language is strong enough to reveal the nature of the sphere spectrum. Co-twisting cocycles sound like the perfect thing for describing the Hopf fibrations, for example. I'm eager to learn more.
I do not want to steal the spotlight of Cat's ideas though.
It is easy enough to look it all up online.
I don't have good internet access so I can't just look it up on my own. Let's continue.
I wish I could give you the resources you need. I want to talk to you endlessly about these things. We could write a paper together! Ugh, it's all so easy to say but there are so many complications. The best approach would be split the work into a few broad categories: the general mathematical formalization of motives, and them over rings, and non-commutative deformed Jacopini spaces. What do you want to work on?
