The n-Category Café

With all the integers $[n]_0$ equal to 1, and hence all the factorials, it’s hard to see where the excitement comes from.

If you’ve got quantum groups $GL_q(2, k)$ acting on quantum planes, perhaps we could see there how something interesting happens as $q \to 0$. Or is it only the $q = 0$ limit of deformations of universal enveloping algebras which are interesting?

Urs has written eloquently on this ‘one step further up’ […]. But I still feel it’s mysterious.

I also still feel it is mysterious – but to me it now feels much less mysterious than it did a while ago. In other words, I got the feeling that I am understanding at least some basics underlying it.

Let me maybe recall the latest version of how I think about what is going on:

That Shift in Dimension™ which we encounter all over the place, in different guises, is essentially the result of a kind of obstruction problem which is very fundamental:

The obstruction to some $n$-thing being flat is itself an $(n+1)$-thing: its curvature.

And the point is: here I am using the word “obstruction” in a very technical, very systematic sense.

This is what I think obstruction theory is about, and the way it ought to be conceived:

it is about having an exact sequence of $n$-things in some context $$A \to B \to C$$ and having a map into $C$ $$\array{ A &\to& B &\to& C \\ &&&& \uparrow \\ &&&& P }$$ and asking whether this map can be lifted to a map to $B$ $$\array{ A &\to& B &\to& C \\ &&&{}_{dl}\nwarrow& \uparrow \\ &&&& P } \,.$$

Here “dl” is for “desired lift”.

What is the systematic way to ask for the obstruction for this lift to exist? The systematic way uses weak cokernels – and that’s where the shift in dimension appears.

And there is a nice story to be told as the formalism proceeds: it’s a story of weakness and of equivalence, and of things holding almost, up to something else. So therefore it is a story in the world of $n$-categories.

The thing is this: $n$-categories offer a nice way to say

“We can try to lift the map, but we might make a mistake thereby. However, the kinds of mistakes we can possibly make are rather well behaved.”

All this is encoded in the following technical statement:

regard the above diagram of $n$-things as a diagram of $(n+1)$-things. Then the desired lift always exists in a weak sense. And then the obstruction problem amounts to constructing the weak lift and checking if it does restrict to a strict lift.

So it’s really a big tautology, in a sense, but formalized nicely enough to be actually useful.

So this is how it works: we form the cokernel of $A \to B$ in the context of $(n+1)$-things. That’s the weak cokernel of $A \to B$ (or “homotopy quotient”, if you like). Since its definition turns out to be somewhat tautologous, also its denotation is somewhat tautologous: I call this weak cokernel itself $(A \to B)$. So

$$A \to B \to (A \to B)$$

is an exact sequence now. And since the previous sequence was exact, too, we get a universal morphism from $(A \to B)$ onto $C$. And in nice cases this is actually an equivalence, in a suitable way. So we get $$\array{ &&&& (A\to B) \\ &&&\nearrow & \uparrow^{al} \\ A &\to& B &\to& C \\ &&&{}_{dl}\nwarrow& \uparrow \\ &&&& P } \,.$$

Here “al” is for “attempted lift”. This attempt always exists, but it need not be successful!

But we can systematically measure the failure of it being succesful: that’s simply what the map to the cokernel of $B \hookrightarrow (A \to B)$ projects out. In simple situations this is simply $A[1]$ ($A$ shifted in degree by one, what I used to write as $\Sigma A$) $$\array{ &&&& (A\to B) &\stackrel{fl}{\to}& A[1] \\ &&&{}^i\nearrow & \uparrow^{al} \\ A &\to& B &\to& C \\ &&&{}_{dl}\nwarrow& \uparrow \\ &&&& P } \,.$$

Here “fl” is for “failed lift”.

Then if we are in a situation where each map is the kernel of its cokernel, by the universal property of cokernels we find that the desired lift dl exists if and only if the composite denoted “obs” in $$\array{ &&&& (A\to B) &\stackrel{fl}{\to}& A[1] \\ &&&{}^i\nearrow & \uparrow^{al} \\ A &\to& B &\to& C && \nearrow_{obs} \\ &&&{}_{dl}\nwarrow& \uparrow & \nearrow \\ &&&& P }$$ is trivializable (equivalent to the trivial such map).

Here “obs” is for, guess what, “obstruction”.

It is this kind of reasoning that makes an additional categorical dimension appear out of nowhere:

since we are bound to be making mistakes in the $n$ dimensions that we started with, we device an ingenious method to export all these mistakes into an additional dimension, to get rid of them.

It’s some very elementary principle of category theory at work here, really, which says something like: don’t take quotients, take weak quotients if possible.

While I think it provides a huge reduction of my internal entropy, you might still feel the above diagrammatics is too baroque to serve as a good explanation for That Shift in Dimension™.

I promise that feeling will disappear after some time, but then there is still room to wonder why this kind of shift appears to be so crucial throughout all of quantum theory.

And I think this also has a good answer:

Quantum theory is, I claim, about colimits of transport $n$-functors. But actually not of the transport $n$-functors themselves, but of the $(n+1)$-functors which obstruct flat $n$-functors to exist.

There is really a pretty vast generalization of

- the fundamental theorem of calculus

- Stokes’ law

hiding here, encompassing most of the occurences of the word “curvature” as it appears in the physical literature.

So I am saying: if you are willing to accept that the notion of “gauge theory” is at all relevant, then from that it follows that you are bound to run into the mysterious Shift in Dimension™ from time to time:

the extra dimension is the home for the obstructions to flatness.

Khovanov homology still feels like a ‘trick’

Now this I don’t understand. Maybe I’ve just grown up too close to it, but Khovanov homology never seemed like a trick to me, and especially not once it was broken down into the combinatorics for links.

The bracket skein relation shoves one way of splitting the crossing up in degree and the other one down, providing an algebraic separation. What Khovanov homology does is turn this algebraic degree separation into a “firmer” homological degree separation, and reads the crossing as a “path” from one splitting to the other rather than just a relation between them. Equations turn into processes, just as they should.

Or maybe there’s something else – something more general – that you’re referring to as Khovanov homology?

John Armstrong wrote:

John Baez wrote:

Khovanov homology still feels like a ‘trick’.

Now this I don’t understand. Maybe I’ve just grown up too close to it, but Khovanov homology never seemed like a trick to me, and especially not once it was broken down into the combinatorics for links.

As you clearly know, it’s often handy to step back from something you’re familiar with, view it from a different angle, and ask yourself if you really understand it — while applying very demanding standards on what counts as ‘really understanding’.

For me, I don’t even need to step back from Khovanov homology, since I was never all that close to it. So I can skip that step.

For me the story goes like this: people discovered knot invariants like the Jones polynomial, which were mysterious at first, but then they ‘understood’ them in terms of quantum groups. Every simple Lie group has a god-given 1-parameter deformation in the world of quantum groups. The category of representations of a Lie group is symmetric monoidal, but the category of representations of a quantum group is only braided monoidal. So, we get a nice invariant of braids — and ultimately knots — from any representation of a simple Lie group (which gives a rep of the corresponding quantum group). This invariant depends on the parameter $q$ that shows up in the deformation game. It becomes dull as $q \to 1$, which is the limit where our quantum group reduces to a plain old group.

Fine and dandy. But then people start realizing that this whole story should categorify! Maybe for any representation of a simple Lie group we should get a braided monoidal 2-category and an invariant of 2-tangles!

The sensible way to study this is to go back to the basics and categorify the whole theory of simple Lie groups — or for technical reasons, maybe simple Lie algebras — and their deformations.

At its best, categorifying something means understanding it better, by seeing where the equations come from. So, we should take pretty much all the basic equations in the theory of simple Lie algebras and see where they come from!

This sounds really cool, since simple Lie algebras are a venerable subject that people have thought about for over a century. Digging beneath the surface of simple Lie theory and getting a whole deeper level of understanding… that should be really interesting!

And in fact it is. But, most of what I read about Khovanov homology and its relatives does not give me the feeling that I’m digging beneath the surface of simple Lie theory and understanding it all better. Instead, I’m just getting recipes for computing invariants.

So, from that viewpoint, what I see so far seems more like a ‘trick’.