Can you state what's the mechanism that makes it fast? Is this a breakthrough in rendering optimizations or does it introduce smarter data structures? Or is it an accumulation of small optimizations everywhere?
It looks like she wants to talk about symmetric monoidal categories, not just monoidal categories, for that’s where the calculus of string diagrams is actually useful.
Tensor categories is another word for symmetric monoidal categories, indeed it’s less of a mouthful, and perhaps more “intuitive” - it’s conventional to write the symmetric monoidal structure as a tensor product.
Genuinely tried reading the "examples" you wrote in this thread, can't make any sense out of it. Happy to discuss it if you clarify what you mean.
Just to address your original comment in this thread, perhaps it's relevant to note the following. Consider the homotopy theory of the category of nice topological spaces. The full subcategory of topological spaces supported on discrete topological spaces inherits a homotopy theory. This inherited homotopy theory is equivalent to the trivial homotopy theory on the category of sets: where weak equivalences are isomorphisms. This is the sense in which discrete spaces don't have an interesting homotopy theory, at least naively.
(This statement you can precise in your favorite model for the homotopy theory of spaces, via infinity categories, model categories etc.)
The category of sets embeds into the category of topological spaces as a full subcategory, the essential image of which are the discrete spaces. Hence the equivalence I claimed is a precise statement.
As you observe one can't really recover any non-trivial number theoretic things by looking at integers with the discrete topology. The theory of discrete topological spaces is just the theory of sets.
Instead you can look at things like prime ideals of integers localized at some prime, and consider algebro-geometric topologies on that
What a confusing name in an era where homotopy-theoretic methods in arithmetic geometry are flourishing. Looks Hatcher covers non of that and use "topology" to mean "geometrically motivated".
Some context would be nice as this (very nice library) has been around for a while: was there a new release? What are we supposed to be looking for Here?
Ugh, I'm cringing a little that they consider adjunctions 'advanced mathematics'. Adjunctions are ubiquitous even in elementary math. For example, in linear algebra whenever one writes down a matrix to represent a operator in some basis, this is using the tensor-hom adjunctions for modules.
Sorry this is just wrong. Maybe you’re trying to get at the “co/contravariant” properties of tensors, in which case your statement can be more clearly stated as, e.g., the space rank (2, 0), and rank (1,1) vectors, admit different interpretations as internal hom spaces of vector spaces. But in any interpretation of your statement the distinction is never important because all spaces distinguished by this distinction are isomorphic via cononical isomorphisms.
Had to scroll down the page for a while before understanding the point of this: drag-and-drop data transformation pipelines that comes with app integrations at both ends. It’s a great idea!
Not an answer but a general comment on learning languages: instead of setting your goal as “to learn as much of a languageas possible”, instead you can go for “I want to be able to do this class of things that will be made very easy once i know thia language”.
If you do the former it’s very easy to learn something then forget it. Furthermore, echoing other comments, you would lack clear measures of success because it almost makes no sense to ask whether you truly “know” a language.
As someone with a high energy theory background but has only read titles/headlines about this time crystal business, can you give a precise TL;DR on what a time crystal is?
Love the idea - looks like a well-thought-out and much needed abstraction. The way I’d pitch it is that it allows one to create UI in a completely declarative and stateless manner.