Yes and no. The fundamental ideas, once they start to sink in, show clear parallels between vastly different fields of analytic thought. The more you understand the framework though, the more its limitations can be concerning. Dependence on the axiom of choice, for example, and the naturality of choice in the field itself, leads some to speculate that if contradiction can be chosen true, the theory's implementation (with the vast majority of categorical proofs appealing to choice) is completely broken.

It's overwhelming at times how applicable category theory is from the right perspective, but underwhelming how its implementation in set theory can be expected to pan out.

tl;dr: category theory is dope but aoc is sus

In "laymans" terms, it gives you a) an environment for ML models to verify their proofs and b) and rich space for ML to study relations between different fields of mathematics, logic, philosophy, ethics, and everything else by default at that point.

propositions are implementations of types so just being able to say when propositions are equivalent in like a rigorous way is good for science anyways.

Resources are quite scarce I'm afraid. Emily Riehl and company are working on (inf,1) categories to establish homotopy between derived functors, for applications in univalent foundations. For a computer algebra system or proof assistant, type equivalence is required to abstract away implementation details. To actually compute homotopy equivalence, it's better to compute cohomology equivalence, but simplicial cohomology is often too expensive to compute. So it's an open problem whether we can optimize a derived homology functor between a derived (enriched) functor and an abelian (enriched) category (which still lacks proper definition afaik). But its a goal I heard at a HoTT talk once to get non-simplicial cohomology of types instead of computing homotopy (which is computationally impossible at scale). Feel free to steal and spread this idea but it's kinda original and speculative.

tl;dr application is computing homotopy equivalence of types at a reasonable expense

Persistent homology for real data is developing plenty of classical techniques, but we probably need a very good LLM with some very good HoTT to derive non-simplicial (co)homology for some given category -> some given abelian category

CT is kinda good for ML if you have a complex topology of solution spaces. When programmers try to implement categories from a naive view, instead of applying sophisticated categorical constraints on their models, I definitely feel a sort of way about it. With that said, LLMs with analytic modules should be able to do categorical constructions in the not too distant future, which will be nice as hell. Optimizing functors might be a thing someday too, but its definitely not there mathematically yet

Im bullish on deriving (co)homologies using ML but it will be some time before we get there i think.

Well youre missing a whole lot of externalities in your thesis, like assuming it will be mostly plastic, which is not feasible if oil production plummets due to global warming, or if we all die because they dont stop massive oil demand.

If the time horizon is 30 years until mass production, then titanium may be cheaper than plastic if we can effectively mine the moon by then. If it's 10 years, then the AI probably isnt ready yet.

We would also need reliable hydraulics that arent dependent on fossil fuels, and those parts cant be plastic anyways. If you want mass production of robots ASAP then we need to produce a lot more oil and that will probably kill us js.

Depends how much you value the IP behind the AI. Tesla's self driving feature (which is hella broken and doesnt get turned on just because you pay) is >10k.

So far Boston Dynamics has burned >2b for about 20 working robots which is about 100m a peice so far. They are also hella broken.

Dont let your dreams be memes.

When you buy a call option, you give somebody money for them to have to deliver something for a price in the future, but where you don't have to take it if you change your mind. A put option is the same except you give someone money so they will be obliged to receive something for a price in the future. They say an option that you buy is "the right but not the obligation to buy or sell something at a fixed date in the future for a fixed price." An option that you sell is the obligation to fulfill that contract.

Why would someone do this? Well, say you will receive 2000 tons of gold next month because you found el dorado or something, but you don't have anywhere to put it. You can pay people today to receive gold at the current price next month, regardless of what the price is next month. If the price of gold goes up, you can just sell the gold you received for the market price, but if the price goes down, you can sell it for last month's price.

It's an insurance contract that dates back to Babylon. Farmers could negotiate a delivery price in summer for their harvest in the fall. That is, pay a little bit of gold in the summer to guarantee they could receive much more gold in the fall. Babylonian traders usually could then export the received goods at a higher price, so they were just looking for income on both ends of the trade.