Bartosz MilewskiI'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 (š¶įµįµ)įµįµāššš , not to mention \(C^{op}\times C \to \mathbf{Set}\).
I posted this question to MathOverflow. https://mathoverflow.net/questions/470999/morphisms-in-the-category-of-elements
Fabian Beuke@BartoszMilewski Hi Bartosz,
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 CalabiāYau manifold is equivalent "in a certain sense" to the Fukaya category of a completely different CalabiāYau manifold?
John Carlos Baez@madnight @BartoszMilewski - I doubt anyone can explain in simple terms why the derived category of coherent sheaves on one CalabiāYau manifold should be equivalent to the Fukaya category of some other CalabiāYau manifold.
This is an unproved conjecture due to Kontsevich called "homological mirror symmetry". You can read an elementary introduction to it on Wikipedia:
https://en.wikipedia.org/wiki/Homological_mirror_symmetry
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."
The nLab article
https://ncatlab.org/nlab/show/mirror+symmetry
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:
https://arxiv.org/abs/0801.2014
says
"Part of the difficulty in dealing with homological mirror symmetry is the breadth of knowledge required for a proper formulation."
It doesn't give a good story about why the homological mirror symmetry conjecture should be true.
If you want to study this stuff, learn lots and lots of math first.
@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:
Not Found
The requested URL /home/baez/prehistory.pdf was not found on this server.
A Prehistory of n-Categorical Physics
This paper traces the growing role of categories and n-categories in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts which manifest themselves in Feynman diagrams, spin networks, string theory, loop quantum gravity, and topological quantum field theory. Our chronology ends around 2000, with just a taste of later developments such as open-closed topological string theory, the categorification of quantum groups, Khovanov homology, and Lurie's work on the classification of topological quantum field theories.
Giulio@BartoszMilewski @madnight @johncarlosbaez
Thanks for sharing!
P.S.
A few people eventually moved from #twitter to #mastodon.Suddenly, the old #buzzwords (*) of #stringtheory and #categorytheory pop up in the new, #distributed #fediverse.
(*) The above link is an example of a "history" of #physics that mentions #CarloRovelli but not #EnricoFermi.
@giuliohome @BartoszMilewski @madnight - it's a "prehistory of n-categorical physics" and Fermi didn't do much for that unless you get into the modified symmetric monoidal category used for fermions, which I didn't cover.
@johncarlosbaez @BartoszMilewski @madnight
Recalling real-world #history, #Calabi left Italy around the same time as #Fermi, for the same reason. (Nowadays the problem is that people tend to forget and repeat the errors of the past.)
Of the next generation, Tullio #Regge was more of an eclectic dreamer and a mathematician than a physicist like Fermi, which aligns him more closely with the abstract nature of n-category theory.
@madnight - whoops, I should change "prehistory.pdf" to "history.pdf" in the URL I gave... but Bartosz gave a better link: the arXiv is always better.
Okay, I see what you mean about "weeds".