### Conversation with Kea: Categorified NCG

#### Posted by Urs Schreiber

Over on sci.physics.strings Kea and I seem to have found a lot of very interesting things to talk about, related to NCG, categories and - I hope - categorified NCG, but it becomes quite of a stretch to consider this discussion on-topic for sps. Therefore I’ll reply to Kea’s latest post here at the Coffee Table.

I’ll start by replying to some things Kea said and then talk about what looks like a promising approach to categorifying NCG to me.

It seems that I wrote this:

I guess the point is that once you realize that category theory is the language in which god wrote math it becomes clear that at the heart of it one is dealing with omega-categories

To which Kea replied:

Great! Do many String theorists think this way?

I doubt it!

In the category of topological spaces a point is specified by a morphism from the one point space, which is an initial object.

Wait, the one point space is not an initial object, since there are many morphisms from it to any other space. (?)

A sufficiently general type of distributive lattice with 0 and 1 is called a frame (see Mac Lane and Moerdijk). In the category of frames the initial object is the 2 point lattice (0,1) […]

Ok, so I guess a morphism of frames must respect the partial ordering and hence the $0$ of $(\mathrm{0,1})$ must be mapped to the other space’s $0$ and similarly for $1$. This makes $(\mathrm{0,1})$ initial.

[…] so one defines a ‘point’ of a generalised space to be a morphism into this object (remember the contravariance).

Wait, I don’t understand what you are saying here. How does that define a point?

But by this definition a space might not have any points at all! A space is said to be ‘geometric’ if for any two objects of the lattice there exists a point (morphism) p such that p^{-1} distinguishes the objects.

What do you mean by ‘distinguishes’?

Back to Stone. The category of sober spaces is equivalent to the category of generalised spaces which are ‘geometric’. This may be viewed as a duality in which the two point space plays a special ‘self-dual’ role (it’s called a schizophrenic object). Another example of these so-called Stone dualities is Pontrjagin duality, for which U(1) is the schizophrenic object.

Sorry, you have lost me here.

So…what about NCG? Well, this is the question, isn’t it?

Yes, that’s the question that interest me!

We need 2-toposes.

Why?

Let’s see: I know what a topos it, namely a category which has the ‘essential’ properties of Set. So I can guess what a 2-Topos should be: A 2-categorry with the ‘essential’ properties of Cat?

Hm, in which sense is that be necessary for talking about NCG?

This is my pet fundamental thing! To Ross Street a 2-topos involves 2-stacks, which are, first of all, pseudofunctors from a site C into Cat.

This sounds like the property of a 1-stack, not a 2-stack.

The stack condition is a descent diagram, and the inclusion of Stack(C) into $\mathrm{Ps}({C}^{\mathrm{op}},\mathrm{Cat})$ is a nice biadjunction.

I am assuming that $\mathrm{Ps}$ stands for ‘pseudofunctor’.

Gerbes, as you say, are related to this.

Yes, gerbes *are* stacks ‘in’ groupoids (meaning stacks that send source objects to categories which are groupoids) which are ‘non-empty’ and ‘transitive’ in some sense. (Kind of amazing actually that people figured out that it is this definition of gerbe that one should be interested in.)

But the lattice theory is more fundamental. The logic of a topos depends on it. Topos (1-stack) lattices are always distributive. Quantum lattices are not. But quantum lattices are well understood, and a proper understanding of 2-toposes means getting the lattice theory right. I guess this is what I’ve been trying to say!

Maybe you could just say it yet once more? :-) Above I have indicated some questions I have which apparently I need to know the anwer to in order to see where you are headed. Seems that you are kind of saying that ordinary geometry lives in 1-toposes while ‘quantum’ geometry should live in 2-toposes? I need more details to understand this!

Ok, now let me make some remarks about how I imagine categorified NCG should be approached.

I had some vague ideas about this before. Yesterday I mentioned these to John Baez (whom I am currently vising at UC Riverside in California) and he pointed out to me what seems to be an important step in the right direction:

So concentrate for simplicity on the case where we are dealing with spaces which are finite sets of points. Then the basic observation of NCG says that all information about this set is encoded in the algebra of functions over it.

More precisely, there is the Gelfand-Naimark theorem which says that every commutative ${C}^{ast}$-algebra $C$ is isomorphic to a ${C}^{ast}$-algebra of *functions* from some set $\mathrm{Spec}(C)$ to $\u2102$.

For a finite set $S$ we can easily turn this algebra of functions into a Hilbert space in which case we call it an ${H}^{ast}$-algebra. This gives us essentially two of three ingredients in a spectral triple, namely the algebra and the Hilbert space. So the crucial item is the Hilbert space

What John Baez pointed out to me is that there is a very nice categorification of this construction that comes equipped with an even nicer categorified version of the above theorem. This is the content of

J. Baez: Higher Dimensional Algebra II. 2-Hilbert Spaces (1997)

The idea is based on taking the category $\mathrm{Hilb}$ as a categorification of the complex numbers. Then we want to equip the set $S$ with some morphisms and we assume these to be invertible. This is then a discrete version of a *2-space* $\mathcal{S}$, namely a space which is a configuration space of strings, if you wish. The above function then becomes a functor and the space of these functors

turns out to form no longer a Hilbert space but a 2-Hilbert space in a very nice way. Even better, the above theorem generalizes and now tells us that these ‘2-spaces’ are essentially the same as these 2-Hilbert spaces of functors.

In fact, in the above paper this theorem is stated for the graded case, which is what we need to define a spectral triple anyway.

Now, all this is discussed much better in several old TWFs of course, but I had missed these and only now appreciate the impact of this.

Namely I believe this is useful for describing strings in terms of categorified spectral triples. In fact I have a vague idea how using boundary states we can find a very natural realization of the above 2-spectral triple involving 2-Hilbert spaces, which I’ll try to work out in more detail. Of course the above gives only two items in the 2-spectral triple, the missing one is the categorified Dirac operator. This certainly will be something involving the string supercharge as its arrow part. We’ll see.

In any case, if you can see a relation of all this to what you have been talking about, please let me know.

## Re: Conversation with Kea: Categorified NCG

Sorry - I meant terminal object for the one point space.

Regarding terminology: it may not be standard, but the Aussie Cat theorists say

1-stack = sheaf

2-stack = stack a la Grothendieck

and so on, because one ought to standardise higher categorical indexing.

For a topological space $X$, the open sets form a lattice with $0=$ the empty set and $1=X$. A locale is just a generalisation of such a lattice, ie. one which is complete and distributive. Hence a point for a locale is a morphism

Frames are dual (opposite category) to locales. That’s why one needs initial objects now in order to define points.

From Mac Lane and Moerdijk - the definition of sober is that, for any proper open subset $U$ of the lattice $X$ satisfying

there is a unique point $x$ in the space such that $U=X-\overline{x}$ (where bar means closure).

If a space is sober then there is a homeomorphism between the space and the points of a locale.

Elsewhere (on PF) I mentioned Hilbert space lattices, which are no longer distributive because set theoretic union is replaced by the appropriate subspace construction.

Here I’m afraid I must disagree with John Baez on what is the way to approach NCG. Lattice theory tells us that even 1-Hilbert spaces are no longer 1-categorical. Hence my interest in 2-toposes (and higher).

Alvarez is at Riverside, no? See http://arxiv.org/PS_cache/math/pdf/0402/0402150.pdf He appears to be an expert on the Gelfand-Naimark side of things.