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ADITTYA CHAUDHURI (Oct 10 2023 at 14:17):Are there situations where higher differential geometric objects/ideas (like Lie groupoids, differentiable stacks, higher bundles,..etc) have been used in dealing with applied category theory problems? In almost, all the references I encountered, I found their uses only in Physics. I am currently a researcher in higher gauge theory and related ares and soon to submit my PhD thesis in Mathematics. I am very curious to know whether ideas from these or related fields have found uses in areas other than Physics. In particular, I am more curious about their potential uses in dealing problems in network theory, systems theory, etc.
John Baez (Oct 10 2023 at 15:18):One reason I quit working on these things is that they are not currently very useful in applied category theory, except for physics. That may change someday, and I hope it does, but right now other aspects of category theory are proving more useful.
ADITTYA CHAUDHURI (Oct 10 2023 at 16:10):John Baez said:
One reason I quit working on these things is that they are not currently very useful in applied category theory, except for physics. That may change someday, and I hope it does, but right now other aspects of category theory are proving more useful.
Ok, I got your point Sir.
John Baez (Oct 10 2023 at 16:38):One thing I recommend is looking at the work of Eugene Lerman. Some of his work is about networked systems, like "Dynamics on networks I. Combinatorial categories of modular continuous-time systems", "Modular dynamical systems on networks" and "Algebras of open dynamical systems on the operad of wiring diagrams". And some of his work involves Lie groupoids, etc.
ADITTYA CHAUDHURI (Oct 10 2023 at 17:04):John Baez said:
One thing I recommend is looking at the work of Eugene Lerman. Some of his work is about networked systems, like "Dynamics on networks I. Combinatorial categories of modular continuous-time systems", "Modular dynamical systems on networks" and "Algebras of open dynamical systems on the operad of wiring diagrams". And some of his work involves Lie groupoids, etc.
Thank you Sir.
ADITTYA CHAUDHURI (Oct 10 2023 at 17:09):One thing, I am curious to know. How a "smooth setting" can be used in Applied category theory? I mean, what is the meaning of "smoothness" in Applied category theory problems?
John Baez (Oct 10 2023 at 18:45):Differential equations - both ODE and PDE - are extremely important in applied math, and applied math often uses configuration and phase spaces that are manifolds (e.g. in robotics, the space of positions of your robot).
Jean-Baptiste Vienney (Oct 10 2023 at 20:34):There is also smoothness in computer science. It's composed of "languages": differential lambda calculus and differential linear logic, and their categorical models, ie. the mathematical structures they are speaking about: differential categories, cartesian differential categories and tangent bundle categories.
It's a quite active field.
John Baez (Oct 10 2023 at 21:17):It would be interesting to blend ideas from higher differential geometry (like the ones @ADITTYA CHAUDHURI mentioned in the beginning of this discussion) with ideas from the differential lambda calculus. After all, theories in the lambda calculus give 2-categories of types, terms and rewrites, and the something similar must be true of the differential lambda calculus.
Jean-Baptiste Vienney (Oct 10 2023 at 21:38):I don't remember who but some people are working on replacing equality by rewriting to obtain 2-categorical models of differential lambda calculus etc... But I don't know if the higher-order categorical structure in higher differential geometry is the same one ie. rewriting and rewriting between rewriting etc... ?
Jean-Baptiste Vienney (Oct 10 2023 at 21:41):Mhh, probably it's the same. Anyway, personally, I already have much stuff to work on on the 1-categorical side. But I think you're right: probably lot of work to do here. Especially using the ideas from mathematicians that computer scientist don't know and translating them in logic etc...
ADITTYA CHAUDHURI (Oct 11 2023 at 06:50):John Baez said:
It would be interesting to blend ideas from higher differential geometry (like the ones ADITTYA CHAUDHURI mentioned in the beginning of this discussion) with ideas from the differential lambda calculus. After all, theories in the lambda calculus give 2-categories of types, terms and rewrites, and the something similar must be true of the differential lambda calculus.
Thank you Sir. I actually don't know much about differential lambda calculus. Can you please suggest a starting point (some references) to start looking in that direction ?
John Baez (Oct 11 2023 at 08:06):There are other people here who would know the best introductory references. As for myself, I would just google "differential lambda calculus".
ADITTYA CHAUDHURI (Oct 11 2023 at 08:20):Ok Sir.
Jules Hedges (Oct 11 2023 at 11:34):This thread is off-topic for #general: off-topic because it's on-topic!
Jules Hedges (Oct 11 2023 at 11:38):ADITTYA CHAUDHURI said:
Are there situations where higher differential geometric objects/ideas (like Lie groupoids, differentiable stacks, higher bundles,..etc) have been used in dealing with applied category theory problems?
I also didn't hear about anybody using these kinds of things, but not-higher differential geometric ideas are starting to become quite common in applied category theory approaches to things like general systems theory, dynamical systems and machine learning... so I'd bet there's potential for the more advanced stuff to start being used at some point in the future
Emilio Minichiello (Oct 11 2023 at 11:42):Hey Adittya. I've been studying higher differential geometry (HDG) for my PhD and am now finishing up. I've been trying to transition from HDG to applied category theory. I think that if you are looking for applications of HDG, then applied topology is a better place to start, especially topological data analysis. Here are some papers that I find really interesting and have lots of overlap with (at least the sheafy part of) HDG: Sheaves, Cosheaves and Applications; A Sheaf-Theoretic Construction of Shape Space; A derived isometry theorem for constructible sheaves on ℝ. There is also geometric deep learning, which I don't know a ton about, but it seems like its getting close to ideas of HDG. I have found that while there aren't a ton of direct applications of HDG to applied category theory, the tools one learns from studying higher differential geometry can be really helpful and applicable. Take this paper by David Spivak for instance, it uses the idea of lifting problems, which are ubiquitous in HDG and homotopy theory in general, to study databases.
ADITTYA CHAUDHURI (Oct 11 2023 at 12:18):Emilio Minichiello said:
Hey Adittya. I've been studying higher differential geometry (HDG) for my PhD and am now finishing up. I've been trying to transition from HDG to applied category theory. I think that if you are looking for applications of HDG, then applied topology is a better place to start, especially topological data analysis. Here are some papers that I find really interesting and have lots of overlap with (at least the sheafy part of) HDG: Sheaves, Cosheaves and Applications; A Sheaf-Theoretic Construction of Shape Space; A derived isometry theorem for constructible sheaves on ℝ. There is also geometric deep learning, which I don't know a ton about, but it seems like its getting close to ideas of HDG. I have found that while there aren't a ton of direct applications of HDG to applied category theory, the tools one learns from studying higher differential geometry can be really helpful and applicable. Take this paper by David Spivak for instance, it uses the idea of lifting problems, which are ubiquitous in HDG and homotopy theory in general, to study databases.
Thanks a lot Emilio for the references and the suggestions! I am also currently finishing up my PhD work in higher gauge theory. Personally, I do not want to completely transition to ACT at this moment, but want to start working on it in parallel with HDG. In particular, the best scenario for me will be applying the ideas of HDG to the problems of Applied category theory. I will definitely read the papers you suggested.
ADITTYA CHAUDHURI (Oct 11 2023 at 12:23):Jules Hedges said:
ADITTYA CHAUDHURI said:
Are there situations where higher differential geometric objects/ideas (like Lie groupoids, differentiable stacks, higher bundles,..etc) have been used in dealing with applied category theory problems?
I also didn't hear about anybody using these kinds of things, but not-higher differential geometric ideas are starting to become quite common in applied category theory approaches to things like general systems theory, dynamical systems and machine learning... so I'd bet there's potential for the more advanced stuff to start being used at some point in the future
Thank you Sir. Professor @John Baez was telling (earlier in this thread) about the potentiality of blending ideas of differential lambda calculus with HDG.
ADITTYA CHAUDHURI (Oct 11 2023 at 12:33):Jules Hedges said:
This thread is off-topic for #general: off-topic because it's on-topic!
At the beginning I thought, my question was not concrete, so asked in #general: off-topic :) to be safe.
Jean-Baptiste Vienney (Oct 11 2023 at 14:17):@ADITTYA CHAUDHURI . You could look at these papers:
which introduce the three main structures in the world of differential categories.
And on the logical side:
I think the category-theoretic papers would be more interesting for you unless you already know about linear logic or lambda calculus. But as to the category-theoretic side, I think the best ressources to learn should be some slides from @JS PL (he/him) . We should ask him.
Unfortunately there are still not good introductory papers to learn this stuff and none of this papers is a very easy read. (Especially on the logical side. Even the paper "An introduction to Differential Linear Logic" fails to introduce the appropriate sequent calculus unless you already know it. It was noted by some people on MathOverlow. )
Jean-Baptiste Vienney (Oct 11 2023 at 14:30):But fortunately JS knows almost everything on these subjects and is on this Zulip :)
Jean-Baptiste Vienney (Oct 11 2023 at 14:33):My intuition is that a line of attack could be that you learn about tangent (bundle) categories and try to see if you can find some relation with higher bundle stuff
Jean-Baptiste Vienney (Oct 11 2023 at 14:37):A more precise idea: I can read on the nlab that a Lie groupoid X is an internal groupoid in the category Diff of smooth manifolds". Maybe you could consider internal groupoids in a tangent (bundle) category to obtain Lie groupoids in a generalized sense and see what propertie of Lie groupoids are still verified by these generalized Lie groupoids.
Indeed, the most canonical tangent bundle categories is the category of smooth manifolds and so you can just replace this category by any tangent category and still write the definition of a Lie groupoid (I think).
Jean-Baptiste Vienney (Oct 11 2023 at 14:44):But well, it seems that some people are already working on this: The Algebroid of a Groupoid in a Tangent Category
Jean-Baptiste Vienney (Oct 11 2023 at 14:45):Maybe you should contact this guy Matthew Burke if you want to find some concrete project in this line of ideas.
ADITTYA CHAUDHURI (Oct 11 2023 at 15:32):@Jean-Baptiste Vienney Thanks a lot!! I think, first I will read about tangent bundle categories.
Jean-Baptiste Vienney (Oct 11 2023 at 15:43):In addition to the above paper, maybe you can find these slides helpful:
Introduction to tangent categories
Jean-Baptiste Vienney (Oct 11 2023 at 15:46):In fact, you can also watch the associated talk on the website of the 2021 conference "Tangent categories and applications" as well a lot of other talks on this area
ADITTYA CHAUDHURI (Oct 11 2023 at 15:52):@Jean-Baptiste Vienney Thanks a lot!
JS PL (he/him) (Oct 11 2023 at 20:18):Yes always happy to talk about differential categories and tangent categories.
ADITTYA CHAUDHURI (Oct 12 2023 at 04:13):JS PL (he/him) said:
Yes always happy to talk about differential categories and tangent categories.
Thank you Sir.