Wolfram Physics Project: Relations to Category Theory

Find Stephen's notebook for this session here: www.wolframcloud.com/obj/wolframphysics/WorkingMat…

I woke up to this video playing I have no idea how I came here or what this is, I’m truely confused

Fell asleep watching real civil engineer play poly bridge 3 and woke to a physics lesson.

I love falling asleep to whatever YouTube video I'm watching so I can wake up to smell the Category Theory.

Oh dear - this is about the best example of "smart people aren't necessarily great explainers"... IDK, maybe they didn't have time to prepare, that would explain it. But... come on, there are such great ways for intuition building towards category theory, none of them is explored for at least the first hour - though Tali Baynon does a pretty good job later on. Here's how I start when I explain category theory: 1. For the longest time, most mathematics we did was un-formalized in the sense that it had no canonical axiomatic foundation (euclidean geometry being the exception) 2. In the mid- to late 19th century, philosophical logic and mathematical thought came together and laid the foundation of everything for the next 100 years: predicate logic, propositional logic, boolean algebra, set theory, algebraic topology - and among them, set theory with propositional logic was thought to have the power to be able to serve as the language for grounding, formalizing - axiomatizing all of mathematics. 3. When we describe abstract mathematical structures in terms of sets, the foundational notions are of "inner structure" of the sets - and we represent everything as such inner structure of sets (including functions, relations, elements, unions, intersections, complements etc.) with the help of the "is element" diadic predicate. 4. This leads to loads of operations with powersets and inclusions and in general, there are many different encodings of structures in the language of sets (e.g. the natural number 2 might be represented by {∅,{∅}}, or by {1,1}, or by {0,0}, or by {0,{0}} ... and so on... the relational structure is not very intelligible when "flattened" into a set-theoretical representation. 5. What category fundamentally does differently is this: it talks only about objects and arrows/morphisms between them. The objects can certainly have inner structure (in that morphisms e.g. in the category of sets can be (non)injective and/or (non)surjective functions) - but category theory never looks "inside" the objects! They are black boxes - and all the internal structure is represented by conditions on the arrows/morphisms that go in and out! That's the fundamental idea - and the reason why category-theory (including topos-theory and homotopy type theory) is the language of structuralism: Things are defined in terms of their relations to themselves and to other things. It's the power of this concept that lies at the heart of everything. 6. An example: there is the notion of a categorical product. It is defined as an object AxB with two morphisms left:AxB -> A and right:AxB -> B such that any other object which has such morphisms to A and B factors uniquely through the categorical product, i.e. there is unique (up to unique isomorphism) morphism g from the other candidate Y with its candidate projections Yleft: Y->A and Yright: Y->B to the unique (up to unique isomorphism) actual categorical product AxB: g: Y->AxB such that left(g(Y)) = Yleft and right(g(Y) = Yright. This defined the "universal construction" of the categorical product: an object with morphisms to its left and right parts such that any other object with such morphisms factors through the actual product (in the above way). 7. Notice that at no point in the above description of the product have we needed to draw our objects as anything other than labelled black boxes - and being a product is certainly about inner structure! But it's the ways in which inner structure is revealed in or rather defined through how the object can relate to other things. 8. Now we can place constraints (are the morphisms epic, monic, both, none, what constraints are on the dual category etc.) and look for models. For example, in the category of Sets, the categorical product exactly describes the cartesian product - in this way we abstract and generalize notions of structure to make them universally discoverable and applicable. That is the magic of category theory.

this is the deepest rabbit hole youtube's taken me to and i am genuinely afraid

By the way, being able to be a fly on the wall during conversations like this--its supremely awesome. I've struggled to understand sheaf cohomology since first reading Frankel. Hearing you guys discuss this and walk thru the concepts really opens up this stuff in my head while I'm listening to you.

woke up to this and i just can't sleep through it. where did you bring me youtube

Thank you Prof. Wolfram to clarify the cat. theory - decompose the abtractions into concret explanation. Save lots of time to decode them.

I fell asleep on this tab and woke up to this stream. Apparently I've been watching Sam O'Nella reactions for the past 5 hours.

out of category theory comes the principle of irreducible confusability

I loved this because I really struggle with Category Theory. I am always behind by months as I study to understand this breakthrough Wofram Theory! Exciting and I predict noble prizes in the future !!!! I got so much from this! Thank you to all the guests !!! Thank you Stephen Wolfram!!!!

I like the linguistic side the most from category theory, all those specific and absolutely exact terms for every possible abstract thing, like learning a new language with the maximum possible expressivness.

so glad i found this through autoplay

The ..."morphisms between morphisms between ... " property of infinity Cats looks very close to what we think of when we are looking at infinitely differentiable manifolds, aka the smoothness property: between any two points there is another point such that there is a way to go from a ball of one point to the ball of any other point.

You end this on exactly what I've been wondering - which is how to express the symmetries of q.f.t. as a group, using category theory, and possibly Grothendieck equivalence to encode the group into the rule... i.e. what does a particle look like in rulial space? I think what it looks like is is an exceptional group as per Lisi, that emerges as you drive up the scale from hyperedge to electron size, where the exceptional group is showing you the stable vibrational modes in the spacelike graph.

I fell asleep watching Vsauce and now I am here

A powerful set of ideas about Category Theory... I think this is a important video!!!!

Take a look at Tim Maudlin's Theory of Linear Structures (book). He does exactly what was explained ~2:30 for sieves and open sets. He truncates conventional topology at the 0D point-set axioms, because they do not seem obvious or physical. He retains line elements as connectivity (for points that don't 'exist' :) then shows that the lines must be directed to derive the discrete equivalent of topology (e.g. open/closed). This obviously leads into one of your other sessions about rebuilding calculus over discrete structures. Tim goes on to discuss applications to physics. Perhaps arrange a live session with him! By all means start discussing the topology stuff, but he can also (perhaps mostly) contribute to the philosophical implications of your work. P.S. Echoes here of Rovelli's Relational Quantum Mechanics which I like to call the Zero Worlds Interpretation, because there are relations but no relata (in Mermin's words), i.e. edges but no nodes : ) P.P.S. Also, not by coincidence, in Rovelli's LQG the lowest dimensional spatial operator is area! but the area appears on the incident edges. So neither the nodes nor the edges 'exist' spatially but there are area quanta with a spectrum, which presumably have normals in some limit. There are also volume quanta/spectrum. The outcome originates from one of Penrose's many amazing insights, that spin may be fundamental, not space, not time, not spacetime.

Hi all! With all respect, Don’t ask why im here randomly 3 yrs later of this being published but I do believe there should be a partnership dictionary/re-writing of these terms used, even tho people working on this for ages. Its extremely confusing. Hopefully not changing any of the discussed subject matters. Thanks and All the best to all.