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}).
@BartoszMilewski There are two possible conventions and both work (this happens a lot in math, specially in category theory with all the op's —and things get worse in 2-category theory which has op's and co's which you can pick independently!). You can use a→b which produces a category with a functor E→C which is a discrete opfibration, or you can use b→a to get a category with a functor E→C^op, which is a discrete fibration. Neither is better or worse than the other. I usually choose depending on whether fibration or opfibration is more natural for what I want to later.
@BartoszMilewski - it's just a convention; either convention is possible. So in what sense are you struggling with it? Are you struggling to decide what you like best? You don't really need to pick a favorite.
This reminds me that two different communities write the specialization order for points x and y of a topological space as x ≤ y and y ≤ x respectively.
It is just a matter of convention.
But it does make papers from the other community very hard to read, as we need to flip the order every time in our heads to match our intuition and previous knowledge.
@johncarlosbaez It's actually a serious problem. How do I define the category of elements for a profunctor: both arrows in the same direction, or twisted?
@BartoszMilewski - they both exist, neither is "right" by decree of god, so use the one that works for what you're doing... which may change next week.
@johncarlosbaez I thought that once you define a thing called the "category of elements" for (C \to Set), then it should work the same if I replace (C) with (C^{op}) or with (C \times C) or (C^{op} \times C) and so on. All other definition in nLab (or in math, in general) worked this way.
@BartoszMilewski@johncarlosbaez It does work for all those other cases too, but there is also a second possible definition which also works for all those cases. 🤷🏽
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?
@madnight My knowledge of string theory is totally disjoint from my knowledge of category theory, both being rather shallow at this point. Maybe @johncarlosbaez can help.
@madnight@BartoszMilewski - I doubt anyone can explain in simple terms why the derived category of coherent sheaves on one Calabi–Yau manifold should be equivalent to the Fukaya category of some other Calabi–Yau manifold.
This is an unproved conjecture due to Kontsevich called "homological mirror symmetry". You can read an elementary introduction to it on Wikipedia:
but one thing you won't find is anything about why it should be true! They say there was a year-long program on it at the Institute for Advanced Studies, but "only in a few examples have mathematicians been able to verify the conjecture."
gives a bit of an explanation: physicists believe for every 3d complex Calabi-Yau variety (X) there are associated two field theories, and at least some X have a "mirror partner" (\hat{X}) such that the first field theory built using (X) is equivalent to the second one build using (\hat{X}). But this has not been proved - in fact, these field theories have only been fully constructed in some cases!
I would hope some good string theorists could tell a good simplified story about why they believe this stuff, but I haven't seen it.
When you get mathematicians involved in a difficult unsolved problem like this, things tend to become technical. This supposedly introductory paper:
@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.
@madnight - Actually I was staying well away from the weeds, giving the view from miles up.
There are categories like what you said, but formalizing them requires formalizing some particular string theory well enough to know what these 'states of open strings' are: there should be a vector space of such states. In mirror symmetry people are studying two different string theories, the "A-model" and the "B-model", both of which are quite technical, and claiming that sometimes the A-model on one space is equivalent to the B-model in some other space. People got excited because some physicists used this conjecture to do some astounding computations that mathematicians had been struggling to do for decades.
If you want something much more easy to get into, I immodestly recommend my paper "A prehistory of n-categorical physics":
@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.
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