So what’s the point of higher category theory, and weak n-categories and (weak?) infinity categories? What do I know so far?
Well category theory is a concise way of describing a bunch of seemingly different mathematical ideas. A category has:
-objects (these could be sets, or numbers, or dots on a paper)
-morphisms, (these could be functions between sets, or arrows between dots on a paper)
-composition of morphisms (do one function, then do another. this gives you a third function)
-identity morphisms (when you compose this with another morphism, you just get that other morphism itself, like multiplying by 1 or adding 0)
-strict associativity of composition of morphisms (like (a+b)+c = a + (b+c))
And this framework describes a lot of the stuff I would want to do math on: sets and groups and fields and topological spaces. It means I can talk about all these things at once and I LOVE talking about a lot of things at once. Yay, category theory is great.
And it gets even better: you can define functors, which are like functions between categories. And then you can make a category whose objects are categories and whose morphisms are functors! This is bliss.
But then, you realize you can define these things called natural transformations, which are—wait for it—like functions between functors. So you can define a category that has functors as its objects and natural transformations as its morphisms! So now you have this thing called a 2-category, where you sort of have two categories stacked on top of each other with the morphisms of one category forming the objects of the next category. And you can stack these as high as you want. 3-categories, n-categories. This, truly, is bliss.
BUT there was a problem all along that I was hiding from you: some things that I would want to do math on do not form a category in any nice, sensible way. I need to study this part more to get clear on exactly why, but I think it has something do do with considering things as equivalent even though they don’t have an isomorphism between them. It definitely has something to do with associativity breaking down.
So sometimes instead of (a+b) + c and a + (b+c) being exactly equal, you get that those two things are relatedin some nice way without being the exact same object. You can describe this relationship by a morphism, or a morphism between morphisms, or a morphism between morphisms between morphisms.
But somehow then your associativity laws for those higher-dimensional relationships are also never strict, so you keep needing to jump up a dimension, and in fact you need to know that you’re able to jump up to every dimension and never stop.
Aaaand you end up with something called an infinity category. And it’s pretty cool. And I need to do more reading to pin down these last 4 paragraphs.