Dominic Verity: "Zen and the art of ∞-categories"

[@Devasantika]

This is the most illuminating talk I've watched this year! Many thanks Dom!

[@JosiahWarren]

the diagrams are epic . Thanks.

[@CyberneticOrganism01]

I have a basic question: you talked about high-dimensional homotopy theory of (arbitrary) spaces. Can this be done using simple category theory, or somehow it requires infinity categories? Or is the main purpose of infinity categories just to deal with that? Thanks for the talk, it's very beginners friendly 😊

[@treborhuang233]

This can be done using what's called "model categories". They are 1-categories with extra structures put on them so we can perform homotopy-related constructions. Each model category actually has an infinity category behind them, and in some sense operations on the model category is really just indirectly describing operations on the underlying infinity category.

[@fbkintanar]

The discussion around 14:00 brings to mind the Measurement Problem in fundamental physics. Perhaps fundamental infinity groupoids are a suitable tool for understanding why quantum-scale measurements are irreversible when everything else about physics and general relativity leads us to expect that dynamical laws allow us to predict the past just like we predict the future. Einstein's famous comment about God not playing dice. A state space of quantum states may not be simply a space of points (fixing a model for our theory of nature to those points) but a space of infinity groupoids, which might force our theories to be more model-agnostic. The problem may be with our default commitment to theories which answer questions on the nose, when shifting away from a classical perspective on nature may require theories that give us answers "up to homotopy" in some way that still needs to be made precise. It seems we are in for a further conceptual and ontological shift, quantum theory 2.0, that needs an framework beyond the Copenhagen interpretation. It may give a better language to talk about quantum computing and quantum information.

[@jsmdnq]

This doesn't make sense though. If space is an infinite set of of groupoids and groupoids is a set of points then space is a set of points(yes more stuff than just points but still points too). I doubt it will ever be possible to have *THE* mathematics of reality. Whatever physics comes up with will just be a model. The idea is to find a model that is complete(can express reality as accurately as we need) and simple enough for us to comprehend. I imagine reality is infinitely more complex than we can even come close to imagining and unfortunately too many confuse a the model as reality itself. This seems what you are getting at except you think there is, at some point, some model that is what reality is. I think the more we use models to learn about reality the more we learn what we don't know and hence seek out better models... this process though will go in indefinitely until humanity becomes extinct or decides their current understanding is good enough and moves on to other things. It may be that the way the universe works, the true underlying "logic" is not computable by the human mind and all we can do is chase our tails by using more and more complex frameworks to try to understand it. The only harm in this is the energy requirements may not be sustainable. This is why science needs to learn to balance it's own progress out with other factors. Science is not independent of, say, economics, health, etc. For example, currently trillions have been stolen by financial terrorists using the financial system and in particular by rigging the stock market and corrupting the government. That $$$ generally goes to wasted efforts or things that decay society such as prostitution, more corruption, vulture capitalism, lawyers, wars, etc. If, say, that $$$ went to education, healthcare, homeless, etc then that would increase science. How many homeless exist, say, that could become physicists if they had a chance and encouragement? Maybe one of them would be the one that would discover something new that would set physics off in a new direction? What about biology? mathematics? genetics? Science doesn't exist in a bubble and everything it does effects the world and vice versa(which should be obvious now).

[@vaibhavsutrave6117]

@@jsmdnq I don't think he was talking about outer space or the space around us, but STATE space -- spaces representing states of (in this case) quantum stuff. I'm not a physicist so I will shut up now. en.wikipedia.org/wiki/State_space_(physics)

[@jsmdnq]

@@vaibhavsutrave6117 In physics though the "state space" will eventually correspond to physical points(that is the point of physics as it maps math to the real world which means "physical points"). So whatever representation one uses, whatever works best, will still have to be mapped to physical reality, ultimately, to have any use in physics. So one can use whatever layer of abstraction one can handle as long as it provides some benefit but one will never be free from having to map down to physical points(which currently are "Euclidean" in nature). What physics really is concerned about is how reality is structured. What structure is the "substance" of reality so that one can represent it correctly and hence optimally. I doubt we will ever have the tools to do that correctly so all we can do is approximate and find models that work well for the things we need them to work well at.... which is ultimately what we do.

[@vaibhavsutrave6117]

@@jsmdnq Wasn't one of the ideas of QM was that position and velocity were somehow tied together, and the more you know about one the less you know about the other? That is, that if you try to say something definitively about where in space a particle is sitting, you are basically sacrificing knowing anything about its velocity. The best we can do is this kind of fuzzy sort of statistical distribution of possible states? This is stuff I learned a decade ago as an undergrad, so excuse me if I'm wrong.

[@jsmdnq]

@@vaibhavsutrave6117 Yes, position and velocity are not independent. It is true of all operators that do not commute. [X,Y] != 0. xd/dx(f) != d/dx(xf) But ultimately it has to do with the integral and Fourier series and with the Heisenberg uncertainty principle. But these are issues with the mathematics, not with reality. The mathematical physics is not the reality. The map is not the territory. It may be that reality is like that or it may be just that our model is like that and it is most likely the latter. We use the math we have discovered/created to model reality... reality doesn't have to conform to our model. What we do is simply try various models and the ones that work best is what is used. QM is based on probability theory and so is very good in exactly the sense that probability is good at modeling problems. It doesn't' mean that reality is probabilistic. The universe may be on a level of complexity that humans can never understand or create a true mathematical model for. At best we can only create better and better approximations. Maybe one day an entirely new way of doing "mathematics" will be found that then can truly model reality... it is unlikely that it will ever happen or has happened.