dbranes

Nice to see Jared Kaplan branching out into ML. He did fundamental work on CFTs/bootstrap in physics.

No part of this comment answers my question.

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?

Cool, but this seems to contain very few illustrations.

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 embedding is full & faithful)

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.

Look up ring localization.

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.

Tangent: I like that the logo for the organization 'adjoint' resembles the notation for adjoint functors.