I'm struggling with the definition of the category of elements--the direction of morphisms. Grothendieck worked with presheaves (C^{op} \to \mathbf{Set}), with a morphism ((a, x) \to (b, y)) being an an arrow (a \to b) in (C). The question is, what is it for co-presheaves? Is it (b \to a)? nLab defines it as (a \to b) and doesn't talk about presheaves. Emily Riehl defines both as (a \to b), which makes one wonder what it is for (𝐶ᵒᵖ)ᵒᵖ→𝐒𝐞𝐭 , not to mention (C^{op}\times C \to \mathbf{Set}).
I've been exploring string theory from a categorical perspective. Since you've mentioned your background in theoretical physics somewhere on your blog, I was hoping you could help clarify something for me. Could you explain in simple terms why the derived category of coherent sheaves on one Calabi–Yau manifold is equivalent "in a certain sense" to the Fukaya category of a completely different Calabi–Yau manifold?
@johncarlosbaez@BartoszMilewski Oh okay, that goes quite into the weeds. I was intrigued by the simple idea that one can consider categories where the objects are D-branes and the morphisms between two branes A and B are states of open strings stretched between A and B. Then, I wanted to follow this idea and its implications in categorical terms, but when I jump around in nLab and papers, I get immediately hit with mathematics beyond my comprehension. What I really wish for is a blog like Bartosz's with easy to understand categorical explanations that also builds intuition about the matter. Thank you for your detailed answer.
@johncarlosbaez To clarify, I do not think that your answer was going into the weeds, but the direction in which you pointed me is (the paper you referred to).
Based on the title "A prehistory of n-categorical physics," is much more what I'm looking for. And if I can find a bird's eye view there, then I'm more than happy, but unfortunately, the URL gives me:
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The requested URL /home/baez/prehistory.pdf was not found on this server.
Physics is tied to mathematics, so we have to assume that, by Goedel, that it must be undecidable. We should be able to come up with an experiment whose outcome cannot be derived, but which Nature "knows" how to answer.
