I have been trying to figure out this topic for a while (they don't call it "Abstract Nonsense" for nothing...) and especially if it can help me make sense of the properties and ordering of hyperbolic 3-manifolds generated by links... this was a very smooth introduction to a lot of things I missed, but I wish there was more depth!! Do you have any other videos/accessible sources on the topic?
@VisualMath
Жыл бұрын
Yes, sorry, the video itself is too short to go into details. I once tried to write down lecture notes on this stuff (braided categories abstractly and friends). I do not claim that they are readable, but I park them here: www.dtubbenhauer.com/qinvariants.pdf
@freeformcreations
Жыл бұрын
@@VisualMath This is an excellent expose! Again all what you show us in this field pointing to new programming languages with available variable domains in Categories. The problem is that the professional mathematicians do not understand software/programming languages and therefore reluctant to even look into that direction. Your exposes on these subjects are our guiding lights to better and freer thinking.
@VisualMath
Жыл бұрын
@@freeformcreations I am not 100% on board. Left aside whether the file is good or not 🙂: we clearly have a long way to go, but times change and quite a few professional mathematicians are really good at programming languages.
@spogel9981
9 ай бұрын
Wow, thanks to you, I got further inside into CATs.👍 At 4:59 I am not sure if I got it. The standard 3dim vectorspace over R with the vector cross product belongs not to braided CAT, but with the condition x cross y is equivalent to y cross x, it is. Right?
@VisualMath
9 ай бұрын
Yes, but then the cross product would just be the flip map 😅
@darashayda1
2 жыл бұрын
Could we compose BRAIDE WORDS comprised of Obj which might have repeated objects e.g. you showed us xyz, but could have a braided word i.e. xyztx where in that instance of the word we might or might not braid x the same way given the neighbours. If so, is this braided word a representative of a braided word in some linear braided group?
@VisualMath
2 жыл бұрын
I think I do not understand the question, sorry. Maybe it helps to stress that the x, y, z are arbitrary objects with x=y=z being an allowed option.
@darashayda1
2 жыл бұрын
@@VisualMath x,y,z,u,v,w... where we have more than 3 objects, and some might be equal to each other. Is this allowed in this braiding scheme?
@VisualMath
2 жыл бұрын
Ah, yes that is allowed: note that e.g. x is arbitrary, so it could be x=yz or any finite combination of objects.
@darashayda1
2 жыл бұрын
@@VisualMath Then is the word xyz...,assuming finite number of objects used, a braid word in a braid group of some kind? Or this braiding concept has nothing to do with braiding groups.
@VisualMath
2 жыл бұрын
Ah, I see where the confusion comes from. You are on the wrong level: category theory is about arrows ;-) In other words, the braid group action happens on morphism, not on objects. The best statement to expect on objects is that they (meaning isomorphism classes) form a commutative monoid.
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