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I'm interested in category theory, and have been tantalized by the remark (such as seen here) that a TQFT is just a functor from a particular category of cobordisms into the category of Hilbert spaces.
But even more interesting, is that we can take a 2-category of cobordisms, and then a functor into the category of 2-Hilbert spaces! And so on to an n-category of cobordisms and n-Hilbert spaces.
So I've been trying to figure out two things: what this "means", and what they heck a 2-Hilbert space is, and why it's useful.
I think I've finally gotten the point, and was hoping for feedback if I'm somewhat close, or if I'm way off the mark.
I'll be working on an n dimensional manifold.
In the most boring case, we simply have n dimensional submanifolds, and each of them have values living in the complex numbers. (i.e. we have a function (0-functor) from the set of n-manifolds to the set of complex numbers)
I suppose that homotopic submanifolds should get the same number? Or at least that if two submanifolds are homotopic, they either both have zero value, or both have nonzero value?
In the more interesting case, we have (n-1)-dimensional submanifolds, and n-dimensional cobordisms to move from one submanifold to another.
Here, to each (n-1)-dimensional submanifold, we assign values in a Hilbert space, and to each cobordism, we assign a linear transformation.
Why do we do this? So that if we take a cobordism from M to itself, the cobordism gives us a linear (unitary?) operator T! So, if the value of M is v, we can put this all together and get the complex number <v, Mv>.
I suppose there should again be some conditions on everything. e.g.
If I took a state over a loop, then transport that loop around the torus, I get a complex number as output.
If I take that state over the loop, then transport it to a state over another loop, and then transport that loop around the torus, I should get the same complex number.
(And I guess that's clear because if I follow it up by the inverse of the original transport, I should be back where I started?)
And now, I think I finally get the point of the next step!
Now, I'm working with the (n-2)-dimensional submanifolds. I have the (n-1)-dimensional cobordisms between those, and also the n-dimensional "cobordisms" between those!
Now, I map this over into the category of 2-Hilbert spaces. So, the (n-2)-dimensional submanifolds take values in 2-Hilbert spaces.
Why do we do this? Well, if I take an (n-2)-dimensional submanifold, and I use a cobordism to transport it around back to itself, I can now take an inner product which takes values in a Hilbert space!
And then when I transport that cobordism around back to itself, I can take an inner product to get a complex number!
The whole point of the 2-Hilbert space, here, is that it has an inner-product which is Hilbert space-valued! And I want those because those have a complex-valued inner-product, which gives me the number I really wanted to see.
And now, the iteration of this idea becomes clear.
So am I close?
But even more interesting, is that we can take a 2-category of cobordisms, and then a functor into the category of 2-Hilbert spaces! And so on to an n-category of cobordisms and n-Hilbert spaces.
So I've been trying to figure out two things: what this "means", and what they heck a 2-Hilbert space is, and why it's useful.
I think I've finally gotten the point, and was hoping for feedback if I'm somewhat close, or if I'm way off the mark.
I'll be working on an n dimensional manifold.
In the most boring case, we simply have n dimensional submanifolds, and each of them have values living in the complex numbers. (i.e. we have a function (0-functor) from the set of n-manifolds to the set of complex numbers)
I suppose that homotopic submanifolds should get the same number? Or at least that if two submanifolds are homotopic, they either both have zero value, or both have nonzero value?
In the more interesting case, we have (n-1)-dimensional submanifolds, and n-dimensional cobordisms to move from one submanifold to another.
Here, to each (n-1)-dimensional submanifold, we assign values in a Hilbert space, and to each cobordism, we assign a linear transformation.
Why do we do this? So that if we take a cobordism from M to itself, the cobordism gives us a linear (unitary?) operator T! So, if the value of M is v, we can put this all together and get the complex number <v, Mv>.
I suppose there should again be some conditions on everything. e.g.
If I took a state over a loop, then transport that loop around the torus, I get a complex number as output.
If I take that state over the loop, then transport it to a state over another loop, and then transport that loop around the torus, I should get the same complex number.
(And I guess that's clear because if I follow it up by the inverse of the original transport, I should be back where I started?)
And now, I think I finally get the point of the next step!
Now, I'm working with the (n-2)-dimensional submanifolds. I have the (n-1)-dimensional cobordisms between those, and also the n-dimensional "cobordisms" between those!
Now, I map this over into the category of 2-Hilbert spaces. So, the (n-2)-dimensional submanifolds take values in 2-Hilbert spaces.
Why do we do this? Well, if I take an (n-2)-dimensional submanifold, and I use a cobordism to transport it around back to itself, I can now take an inner product which takes values in a Hilbert space!
And then when I transport that cobordism around back to itself, I can take an inner product to get a complex number!
The whole point of the 2-Hilbert space, here, is that it has an inner-product which is Hilbert space-valued! And I want those because those have a complex-valued inner-product, which gives me the number I really wanted to see.
And now, the iteration of this idea becomes clear.
So am I close?