## February 11, 2005

### 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 $\left(0,1\right)$ must be mapped to the other space’s $0$ and similarly for $1$. This makes $\left(0,1\right)$ 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 $\mathrm{Ps}\left({C}^{\mathrm{op}},\mathrm{Cat}\right)$ 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 $\mathrm{Spec}\left(C\right)$ to $ℂ$.

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

(1)$H=\left\{f:S\to ℂ\right\}\phantom{\rule{thinmathspace}{0ex}}.$

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 $𝒮$, 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

(2)$ℋ:𝒮\to \mathrm{Hilb}$

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.

Posted at February 11, 2005 3:35 AM UTC

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### 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

(1)$*\to X$

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

(2)$A\cap B\subseteq U⇒\left(A\subseteq U\right)\mathrm{or}\left(B\subseteq U\right)$

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.

Posted by: Kea on February 11, 2005 9:43 PM | Permalink | Reply to this

### things 2-do

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

1-stack = sheaf

2-stack = stack a la Grothendieck

Hm, OK, I wasn’t aware of that. In fact, this shows that we have talked about the same things without noticing: I was talking about 2-sheafs as a way to talk about stacks, while you are referring to sheafs as 1-stacks.

That’s why one needs initial objects now in order to define points.

I guess it is a trivial misunderstanding but I still don’t see what you want to say. An morphism from an initial object can hardly specify a point because by definition there is precisely one morphism only from any initial object to any other one. A similar remark applies to terminal objects.

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).

It was my idea to look at the categorified Gelfand-Naimark theorem as a first step towards categorified NCG. John Baez has not looked at it from this point of view as far as I know, so you are not diagreeing with him but with me.

But since there seems to be nothing more natural than getting 2-NCG from 2-GN I would like to better understand your objection.

So what does it mean to say that 1-Hilbert spaces are no longer 1-categorical?

(Maybe you could repost your respective PF posting here, so that we can get on the same page.)

Alvarez is at Riverside, no?

He was until he received his PhD a few weeks ago, so I haven’t met him here.

He appears to be an expert on the Gelfand-Naimark side of things.

Looks that way. But the paper you pointed out seems to be just about a generalization of the uncategroified thing.

Hence my interest in 2-toposes (and higher).

Hm, I still don’t see which magic precisely you are envoking by referring to 2-toposes here.

So say we want to describe stringy geometry. I have an idea how to do it using something like 2-NCG, which I have tried to indicate. What would 2-toposes have to say here?

Posted by: Urs on February 12, 2005 3:17 AM | Permalink | Reply to this

### Re: Conversation with Kea: Categorified NCG

Maybe this helps:

http://mathworld.wolfram.com/NoncommutativeTopology.html

I’ll get around to saying more when I get time!

Posted by: Kea on February 11, 2005 10:40 PM | Permalink | Reply to this

### Noncommutative Topology

Isn’t the idea of noncommutative topology just that underlying noncommutative geometry but with the Hilbert space representation and the Dirac operator omitted (which define the geometry on top of the topology)?

Posted by: Urs on February 12, 2005 2:58 AM | Permalink | Reply to this

### Re: Conversation with Kea: Categorified NCG

Heh! We are mostly talking about the same stuff after all!

…and morphisms INTO the initial object are ‘dual’ to morphisms OUT of a terminal object.

Objection to the Gelfand-Naimark idea:

Firstly, I was hoping we were (at least partly) talking about what NCG ‘should’ be, as opposed to what Connes et al call NCG.

The way I came to study lattice theory in the first place was in trying to understand the tetracategorical structure behind premonoidal higher categories. For simple physical reasons (see the quark confinement reference cited above) we require the pentagon condition to be broken. This occurs very naturally at the tetracategorical level of trimorphisms between tricategories - such as the well studied tricategories of representation theory. This is very nice, because of course we would like something special to happen in 4D.

Anyway. Things like the Swan theorem (but I’m far from an expert on NCG) can be discussed in a purely topos theoretic framework. For a ring $R$ one finds a space $X$ such that $R$ is recovered from the global sections of a sheaf of rings. That is, one studies a ring in the topos Sh($X$).

A finite dimensional vector space over the (two-sided Dedekind) reals in Sh($X$) is the same thing as a finite dimensional vector bundle on $X$. The Swan theorem says that for compact Hausdorff $X$ the category of vector bundles is equivalent to….whatever.

The point is, proofs are ‘intuitionistic’ - ie. they use topos logic. As much as one tries to avoid formulating everything in alternative logics - one seems to come crashing back to the inevitability of it!

Another example: in

Duality for Bounded Lattices G. Allwein C. Hartonas citeseer.ist.psu.edu/allwein93duality.html

the authors show that in order to describe the lattice duality including non-distributive lattices (such as subspaces of a Hilbert space) one MUST drop the axiom of choice. Oh no! No escape! (At some point I’ll drag some interesting PF posts across here, I guess)

I’m enjoying talking to you.

Posted by: Kea on February 12, 2005 5:39 AM | Permalink | Reply to this

### Re: Conversation with Kea: Categorified NCG

Heh! We are mostly talking about the same stuff after all!

Good. Hopefully this happens again!

Firstly, I was hoping we were (at least partly) talking about what NCG ‘should’ be, as opposed to what Connes et al call NCG.

Oh, ok. So while I am trying to get my hands on it you are trying to replace it but something more fleeting.

The way I came to study lattice theory in the first place was in trying to understand the tetracategorical structure behind premonoidal higher categories. For simple physical reasons (see the quark confinement reference cited above) we require the pentagon condition to be broken. This occurs very naturally at the tetracategorical level of trimorphisms between tricategories - such as the well studied tricategories of representation theory.

Let’s see if I am following you: In weak 2-categories composition of 1-morphisms is weakly associative, the failure of strong associativity being measured by the associator which is a 2-morphism that goes between two ways to compose three 1-morphisms.

This associator must satisfy the pentagon identity, i.e. an equation between 2-morphisms (which ensures that it does not matter in which order the associator is used to rebracket expressions of 1-morphsism).

Identities always only need to hold at the top level. So once we go to 3-categories the pentagon identity will no longer be required to be an hnoest identity, but there will be a 3-morphism between its left side and its right side. (The pentagoniator maybe?)

So unless I am mixed up the ‘pentagon condition is broken’ (as you put it) already in 3-categories. Do you agree?

Of course a 4-category is, among other things, a 3-category if you forget about the objects, regard 1-morphsims as the new objects, 2-morphisms as the new 1-morphisms etc.

So since the pentagon identity of 2-morphims is weakened in a 3-category the ‘2-pentagon’ identity of 3-morphisms will be weakened in a 4-category, I think.

Do I understand correctly that you expect this to be related to physics since representation theory (of groups I assume?) gives rise to some 3-categories? I don’t know about that. What do these 3-categories describe?

I won’t find the time to look at that paper on quark confinement that you mentioned at the moment. Could you try to sketch the main ideas so that I can get a feeling for what you are talking about?

This will have a coherence law of its own, i.e. an equation between 3-morphisms.

This is very nice, because of course we would like something special to happen in 4D.

But the pentagon identity (of 2-morphisms!) is already weakened in 3-categories.

Anyway. Things like the Swan theorem (but I’ m far from an expert on NCG) can be discussed in a purely topos theoretic framework. For a ring R one finds a space X such that R is recovered from the global sections of a sheaf of rings. That is, one studies a ring in the topos Sh(X).

Maybe I am beginning to see the point of what you are trying to tell me: In NCG one regards spaces in terms of properties of algebras. You seem to be saying that this is the same as regarding the space not in Set but in some other topos. Is that right?

That sounds interesting. Sorry, that was not obvious to me at all. Can you suggest any reference where I can read more about that?

A finite dimensional vector space over the (two-sided Dedekind) reals in Sh(X) is the same thing as a finite dimensional vector bundle on X. The Swan theorem says that for compact Hausdorff X the category of vector bundles is equivalent to… whatever.

I think the Serre-Swan theorem is best summarized as saying that Vector bundles over a space are essentially the same as respresentations (finitely generated modules) of the algebra of functions on that space.

That’s nice for NCG generalizations, because they are based on more general algebras and we can always talk about the modules of these algebras and consider them to be the generalized version of a vector bundle over the generalized space ‘described’ by the algebra.

Another example: in

Duality for Bounded Lattices G. Allwein C. Hartonas citeseer.ist.psu.edu/allwein93duality.html

the authors show that in order to describe the lattice duality including non-distributive lattices (such as subspaces of a Hilbert space)

Wait. I think I understood what you mean by lattice. This is supposed to be a category of subspaces, roughly, with objects being subspaces and morphisms being inclusions.

What does it mean for such a lattice to be ‘distributive’?

Posted by: Urs on February 12, 2005 11:24 PM | Permalink | Reply to this

### Re: Conversation with Kea: Categorified NCG

You need projective modules for the Serre-Swan theorem. This is because, as Swan ( I think) showed, given a vector bundle, there always exists another vector bundle such that the direct sum is a trivial bundle. A trivial bundle corresponds to a free module and any direct summand in a free module is a projective module.

Posted by: Aaron on February 12, 2005 11:33 PM | Permalink | Reply to this

### So, where is this going?

Let’s see. Perhaps you could explain a bit more about your spectral triple idea. I guess I should make more of an effort to understand it and not jump to the conclusion that it’s less fundamental than other stuff.

To recap: Hilbert spaces are not fundamental. The complex numbers are not fundamental. Even if we want $ℕ$, such as in the topos $\mathrm{Set}$, we need an axiom of infinity. Then we can define $ℤ$ fairly easily, and then $ℚ$ but by the time we get to the reals things start to get complicated.

The question I have for you is: if we could axiomatise (category theoretically) ALL the features of Hilbert spaces that are essential to physics, why do we need Hilbert spaces? They are but a model for the axioms. Moreover, if we want to do quantum gravity (or NCG) then the axioms of conventional QM will be modified, and it is no longer clear that Hilbert spaces have anything to do with it.

Posted by: Kea on February 12, 2005 9:27 PM | Permalink | Reply to this

### Re: So, where is this going?

Let’s see. Perhaps you could explain a bit more about your spectral triple idea.

Sure. I’ll make it brief though, since I have to hurry, but I’ll try to give a rough idea.

First recap: Particles are points. The configuration spaces of points are 1-spaces. Wavefunctions are (locally at least) functions on these spaces. They form an algebra from whose spectrum the space can be reconstructed. Using that algebra in a spectral triple and adding a Dirac operator to it and representing the generated superalgebra on some Hilbert space yields a supersymmetric quantum mechanics of our particle.

Now categorify: Strings are morphisms between points. The configuration spaces of strings are loop/path spaces. These are naturally 2-spaces, i.e. smooth categories, with the objects being ordinary space and the morphisms being string configurations (paths) in that space. Wavefunctions on these spaces (states of string) are hence 2-maps (smooth functors) (at least locally). They form a 2-algebra…

Wait, do they? See, this is the point where Baez’s HDA II comes into play. It is obvious how the rest of the categorification of the above paragraph should sound, but one has to make it work.

Yes, they do. If correctly set up (actually this has been done in detail only for finite base spaces as far as i am aware) we can continue the above sentence by saying:

… they form a 2 ${H}^{*}$-algebra from whose spectrum the underlying 2-space can be reconstructed.

That’s the categorified Gelfand-Naimark theorem (proven only for the discrete case so far, but one gets the idea).

There is probably some freedom with where you want the 2-states being 2-maps from the configuration 2-space to take values in. In the uncategorified case they take values in $ℂ$ and John Baez argues in his paper that hence the 2-states should take values in $\mathrm{Hilb}$.

I am not sure yet that this is the right target category to describe string states, but we can talk about that in more detail later, maybe.

Anyway, it is clear how the categorification of my first paragraph should end: We want to have a 2-Dirac operator acting on these 2-states. This will involve the supercharge on the worldsheet as its arrow part, which is nothing but a generalized Dirac operator on path space. Then the whole thing should be turned into a 2-spectral triple and we hope that this 2-spetral triple encodes superstring quantum mechanics (i.e. the background fields that the string propapagtes in) just like ordinary spectral triples encode supersymmetric quantum mechanics.

I have more details here than I mentioned so far, but that’s the rough idea.

Just yesterday I exchanged emails with Nils Baas, who is workingon trying to categorify point physics in terms of K-theory in order to get string physics in terms of elliptic cohomology and mentioned aspects of this program to him. He said that he is currently working with A. Stacey on working out 2-differential geometry and 2-Dirac operators.

I am glad to see that I am not the only one thinking in this direction. Currently I am thinking (among other things) about how to correctly define the exterior 2-bundle over a given 2-space. Superstring states should be sections of such an exterior 2-bundle.

What kept confusing me for a while is that, as it turns out, the categorification of a vector bundle is not necessarily a 2-vector (2-)bundle (as Baas, Dundas and Rognes implicitly assumed in their paper) but can for instance be a loop space over a vector bundle or things like that.

Posted by: Urs on February 12, 2005 11:58 PM | Permalink | Reply to this

### How can a lattice be distributive?

Let’s start with sets and subsets. There are two operations on subsets. $\wedge$ is intersection and $\vee$ is union. $\wedge$ is distributive over $\vee$. For any topos lattices are distributive.

Now consider linear subspaces. Set theoretic union $U\vee V$ must be replaced by the smallest subspace containing $U$ and $V$. Check for yourself that this breaks distributivity.

Posted by: Kea on February 14, 2005 12:27 AM | Permalink | Reply to this

### References for topos theory

“Toposes, Triples and Theories” Michael Barr and Charles Wells, link

“Elementary Categories, Elementary Toposes” C. McLarty, Oxford Science (1992)

“Topoi” R. Goldblatt, North-Holland (1984)

Posted by: Kea on February 14, 2005 12:35 AM | Permalink | Reply to this

### Re: References for topos theory

Sorry - the link to Barr and Wells is
http://www.cwru.edu/artsci/math/wells/pub/ttt.html

Posted by: Kea on February 14, 2005 12:42 AM | Permalink | Reply to this

### …

Urs: In NCG one regards spaces in terms of properties of algebras. You seem to be saying that this is the same as regarding the space not in Set but in some other topos. Is that right?

No. Ordinary toposes are not good enough to do NCG. But NCG appears to be very ‘topos theoretic’. Some people study quantales, where one allows $\wedge$ to be non-commutative. I prefer the work of Street, Stubbe, Walters et al. (which will take me a long time to understand!) on alternative characterisations of ${\mathrm{Sh}}_{J}\left(C\right)$ (which is sheaves on a site with respect to a Grothendieck topology) because this is more category theoretic and will help us understand tetracats.

About breaking pentagons: yes, one can do this in 3D if one means the tricategorical (see Gordon, Power and Street) TD7 (a modification) which fills in the cube, five sides of which are the Mac Lane pentagon. But if, in addition, one wants freedom on the sixth face, then one is moving up to tetracategories.

By the way - there aren’t really any references for this. I’m just an old grad student with lots of crazy ideas that I’m being strongly discouraged from discussing!

EXCERPT FROM PF:

The objects of a tricategory T are labelled $p$, $q$ etc. For each pair of objects $p$ and $q$ there is a bicategory $T\left(p,q\right)$. The internalisation of weak associativity is a pseudonatural transformation $a$. The parity cube is labelled by bracketed words on four objects, which may be loosely thought of as the bicategories of ‘particles’ a la Heisenberg.

The Mac Lane pentagon lives on 5 sides of the cube. The new top face of the cube is the premonoidality deformation (of Joyce), which closes the horn. This square appears as a piece of data for a trimorphism, that is a map between tricategories.

The tetracategorical breaking of hexagons amounts to the breaking of topological invariance as described by Pachner moves for 4D spin foams. This view thus sort of explains why four dimensional gravity is not topological in the usual sense of the word.

Some useful REFERENCES on Stacks:

A. Grothendieck “Pursuing Stacks” available at http://www.math.jussieu.fr/~leila/mathtexts.php

Ross Street “Categorical and Combinatorial Aspects of Descent Theory” available at www.maths.mq.edu.au/~street/DescFlds.pdf

Posted by: Kea on February 14, 2005 1:58 AM | Permalink | Reply to this

### Re: Baas and Stacey

Oh, that’s interesting. I see that Baas is right into categories (I’ll meet him in July in Australia I guess) and Stacey is a young guy.

I wouldn’t be surprised if there were a LOT more people out there thinking along these lines.

Posted by: Kea on February 14, 2005 2:16 AM | Permalink | Reply to this

### Re: Baas and Stacey

(I’ll meet him in July in Australia I guess)

Is there some conference going on in July in Australia where you expect to meet him?

Posted by: Urs on February 17, 2005 4:52 AM | Permalink | Reply to this

### Re: Baas and Stacey

Posted by: Kea on February 20, 2005 3:34 AM | Permalink | Reply to this

### Re: Baas and Stacey

Ah, right. We talked about that conference. Let’s see if Nils Baas has worked out 2-differential geometry and 2-Dirac operators by then.

Posted by: Urs on February 22, 2005 5:10 PM | Permalink | Reply to this

### 2-NCG: an example (discrete strings)

Above I gave some hints on what I consider categorified NCG should be. I have thought more about it since then and there seems to be a nice illuminating example, being the categorification of the discrete NCG that I had discussed with Eric Forgy last year.

More thinking is required, but I would like to sketch some basic ideas:

Recall that the ordinary Gelfand-Naimark theorem says that every commutative ${H}^{*}$-algebra $H$ (usually this is formulated in terms of ${ℂ}^{*}$-algebras, but I follows HDA2 here) is isomorphic to a commutative ${H}^{*}$-algebra of functions from a set $\mathrm{Spec}\left(H\right)$ to $ℂ$.

This gets categorified by first sending

(1)$\mathrm{Set}\to \mathrm{Cat}$
(2)$ℂ\to \mathrm{Hilb}$
(3)${H}^{*}-\mathrm{algabra}\to 2-{H}^{*}-\mathrm{algebra}$
(4)$\text{functions from sets to}ℂ\to \text{functors from categories to}\mathrm{Hilb}$

and then discussing how to reconstruct a groupoid from the 2-${H}^{*}$-algebra of functors into $\mathrm{Hilb}$ from it.

I don’t feel like spelling all the details out right now, please see HDA2.

Actually, there only the special case where the underlying discrete 2-space is a groupoid is discussed. Without much ado I’ll assume in the following that one does not run into major difficulties by assuming that the entire theory can be generalized to base 2-spaces which are not necessarily groupoids. In these cases a functor into $\mathrm{Hilb}$ will assign finite Hilbert spaces ${ℂ}^{N}$ to objects and (not necessarily unitary) linear operators ${ℂ}^{N}\to {ℂ}^{{N}^{\prime }}$ to morphisms.

To fill the formalism a little more with life I’ll adopt the following point of view:

Given a base space $ℳ$ which is a groupoid, the category $\mathrm{Rep}\left(ℳ\right)$ of functors

(5)$ℳ\to \mathrm{Hilb}$

can be regarded as the category of discrete $U\left(N\right)$-connections on $ℳ$ for arbitrary $N$.

Each such functor assigns ${C}^{N}$ to each vertex and an element of $U\left(N\right)$ to each morphism. That’s a connection in $U\left(N\right)$.

I will eventually think of the 2-space $ℳ$ as a configuration space of string. The string states which assign elements of $U\left(N\right)$ to each piece of string are precisely the boundary states for $N$ coinciding D9-branes with the given gauge field turned on (see the references given in hep-th/0408161).

Hence I’ll think of the ${ℂ}^{N}$ assigned to each vertex as inidciating the number of D-branes at that point and of the linear operators assigned to the edges as describing how strings stretch between the D-branes at two shuch vertices.

So, in this language, the functor

(6)$ℳ\to \mathrm{Hilb}$

for the general non-unitary case assigns a number $N$ of D(-1)-branes to each vertex and a $n×m$ matrix to each edge

(7)${ℂ}^{m}\to {ℂ}^{n}$

describing how strings stretch between these stacks of D-instantons.

Anyway, this is just a picture I offer which I think makes good sense to be worked out in more detail. But if you don’t buy into this D-brane picture just consider it fancy language.

Now the the 2-NCG. The very point of $\mathrm{Rep}\left(ℳ\right)$ is that it is a 2-${H}^{*}$-category, which in particular means that its objects (functors $ℳ\to \mathrm{Hilb}$) form an algebra and can hence be multiplied and added and indeed form a categorified Hilbert space.

Clearly, this serves as the categorification of the bosonic function algbra that enters a commutative spectral triple.

Such an algebra is turned into a graded algebra with a graded nilpotent operator $d$ acting on it by forming its universal abstract differential calculus obtained by generating the universal algebra generated by the elements ${a}_{o}\left(d{a}_{1}\right)\dots \left(d{a}_{p}\right)$ under a product which makes $d$ nilpotent and graded Leibnitz and restricts on $A$ to the original algebra product. By throwing out certain elements consistently one gets any abstract differential calculus from this universal one.

So in particular consider the algebra of the point map of the functors $ℳ\to \mathrm{Hilb}$. They are essentially maps from vertices into the integers. Let ${e}_{i}$ be the function which assigns 1 to the $i$-th vertex and 0 to every other. Discrete differential 1-forms over this algebra are linear combinations of

(8)${e}_{i}\left(d{e}_{j}\right)\phantom{\rule{thinmathspace}{0ex}}.$

This also encodes the presence of an edge $\left(i\to j\right)\in ℳ$.

We can do a similar thing to the algebra of functors which restricts to this operation under the source and target map. Let ${E}_{\mathrm{ij}}$ be the functor which assigns $1:{ℂ}^{1}\to {ℂ}^{1}$ to $\left(i\to j\right)\in ℳ$ and ${ℂ}^{0}$ to every other vertex and the trivial map to every other vertex.

Then ${E}_{\mathrm{ij}}\left(d{E}_{\mathrm{kl}}\right)$ denotes a discrete 1-form on discrete path space which encodes a surface element stretching between $i\to j$ and $k\to l$.

Let us now notationally identify this 1-form with the surface element that it is non-vanishing on. Similarly let us identify the the functor ${E}_{\mathrm{ij}}$ notationally with $i\to j$ in the following. Then just like passing to the differential calculus on functions enlarged the set $S$ of points to the set of elementary discrete differential forms, the above process enlarges the category of the base 2-space to that of elementar discrete edge differential forms.

In order to make elementary discrete edge forms into functors we make the following natural definitions:

We declare the source of this edge 1-form to be ${e}_{i}\left(d{e}_{k}\right)$ and the target to be ${e}_{j}\left(d{e}_{l}\right)$.

(9)$s\left({E}_{\mathrm{ij}}\left(d{E}_{\mathrm{kl}}\right)\right)={e}_{i}\left(d{e}_{k}\right)$
(10)$t\left({E}_{\mathrm{ij}}\left(d{E}_{\mathrm{kl}}\right)\right)={e}_{j}\left(d{e}_{l}\right)$

Furthermore we define composition of such elementary discrete edge forms by functoriality of $d$:

(11)$\left({E}_{\mathrm{ij}}\left(d{E}_{\mathrm{kl}}\right)\right)\circ \left({E}_{\mathrm{jm}}\left(d{E}_{\mathrm{ln}}\right)\right)=\left({E}_{\mathrm{ij}}\circ {E}_{\mathrm{jm}}\right)\left(d\left({E}_{\mathrm{ij}}\circ {E}_{\mathrm{jm}}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

One can draw some pretty pictures illustrating this, but I won’t do that right now.

In fact, I am getting a little tired. More detailed discussion can be given, showing how this construction allows to construct a graded 2${H}^{*}$-algebra $H$ of discrete path space differential forms, which would give rise to a 2-spectral triple $\left(A,H,d\right)$.

More later.

Posted by: Urs on February 14, 2005 8:17 PM | Permalink | Reply to this

### Re: 2-NCG: an example (discrete strings)

Very nice gentleman! :)

Kea, I am glad to see you and Urs having such fascinating discussions. I am reading through things now and I doubt I will have anything intelligent to say, but wanted to poke in and say, “Hello!” I always felt that the discrete work Urs and I did would find its place in loop space, but distractions kept me from making any progress. It’s all great fun though :)

Keep it up!

Posted by: Eric on February 16, 2005 6:53 PM | Permalink | Reply to this

### Re: HDAII

I really appreciate you explaining this, but as far as I can tell my objections all stand. A quote from the conclusions of HDAII: “While the general study of n-Hilbert spaces will require a deeper understanding of n-category theory…”

Exactly! On page 2 it is pointed out that Hilb is a ring category. These structures were studied in detail in the thesis and related papers of Joyce from 2000 onwards. Unfortunately I don’t even have journal references to give you, except for

“The Racah-Wigner category” W.P. Joyce, P.H. Butler, H.J. Ross; Can. J. Phys. 80 (2002) 613-632

Anyway, quoting HDAII: “so we expect it to play a role in 2Hilb theory similar to $ℂ$ in Hilbert space theory”

This is a simplified idea of categorification, a fact of which I am sure JB is well aware.

The sort of categories being discussed here are used by computer scientists, following the old work of Barr (LNM 752) on *-autonomous categories (symmetric monoidal closed with a duality structure). See for instance the recent seminal work on quantum computation:

“A categorical semantics of quantum protocols” Samson Abramsky, Bob Coecke; http://arxiv.org/abs/quant-ph/0402130

But, I’m digressing. On pages 52+ of HDAII there is a discussion of Categorified Fourier Transforms and Pontrjagin duality. I mentioned this above. Pontrjagin duality is secretly an example of Stone duality in a lattice theoretic context. This has to be more fundamental than the HDAII description because it recognises the importance of decribing the logic.

Posted by: Kea on February 14, 2005 10:02 PM | Permalink | Reply to this

### Re: HDAII

as far as I can tell my objections all stand

We need to do something about it then, because, unfortunately, I still don’t quite see what your objections even are. I am open to being pushed in a more promising direction, but I have to be able to understand what is wrong with the old direction and what is supposed to be better about the new one. And, in fact, what this new direction would be in the first place.

For instance you write:

Exactly! On page 2 it is pointed out that Hilb is a ring category. These structures were studied in detail in the thesis and related papers of Joyce from 2000 onwards.

So what is the objection here? What’s wrong with Hilb being a ring category? (Or with Joyce having studied them? :-) That is precisely one property that we want it to have in order to interpret it as a 2-Hilbert space.

The only objection that I can identify as such is this:

Anyway, quoting HDAII: ‘so we expect it to play a role in 2Hilb theory similar to $ℂ$ in Hilbert space theory’

This is a simplified idea of categorification, a fact of which I am sure JB is well aware.

There is some freedom in what to categroify and how. But identifying $\mathrm{Vect}$ with a categorification of $ℂ$ is certainly reasonable.

One’s choice of categorification is as reasonable as the results that one can extract from it are useful and interesting. In this case I think this makes John Baez’s identification reasonable.

But if you propose another notion of 2-Hilbert space which you find better suited I will be interested in being taught about it!

But, I’m digressing. On pages 52+ of HDAII there is a discussion of Categorified Fourier Transforms and Pontrjagin duality. I mentioned this above. Pontrjagin duality is secretly an example of Stone duality in a lattice theoretic context. This has to be more fundamental than the HDAII description because it recognises the importance of decribing the logic.

This is a little too fast for me. It seems that your objection here is that there is a way to make the constructions in HDA2 more fundamental by generalizing them somhow. Sure, generalizing something is always a good thing. (It doesn’t imply however that the non-generalized thing is not useful.) So how would you go about putting HDA2 on more fundamental footing?

Posted by: Urs on February 15, 2005 1:17 AM | Permalink | Reply to this

### Physical Principles?

It seems to me that you just want to rewrite String theory in its natural language. I don’t see that there’s much physics in String theory. Now, I have a confession to make. I don’t pretend to understand it very well yet, but the reason I am studying tetracategories and lattices and so on is that I am quite convinced that one of the fundamental principles of Quantum Gravity (it makes no sense to talk about unification a la Strings because gauge theories aren’t at all fundamental) is something I call QGC. That is, “quantum general covariance”. General relativity must be respected. Strings don’t do this. QGC asks “what would spacetime-matter duality look like in a background independent quantum informational theory?” It seems there is an answer. There is one big thing on which Strings and Spin Foams agree: quantum gravity can interchange small and large scales. How does one represent this mathematically whilst respecting GR? I’ll let you think about this for a while, and then maybe I’ll continue trying to convince you that this pure category theoretic approach is absolutely essential.

In the end, everything is cohomological in the sense of Street et al descent theory, sometimes known as topos cohomology because, to begin with, one uses omega-cat objects in a topos.

By the way, a topos is really a generalised idea of a (commutative) space (this is the way category theorists view it, not just me). The categorical structure based on quantum logics should therefore describe NCG.

I probably wouldn’t bring it up here if it wasn’t for the fact that recent M-theory papers seem quite excited about the idea of a ‘purely cohomological’ M-theory (Sati? for instance).

So my question to you is: where is this spectral triple stuff heading PHYSICALLY?

Posted by: Kea on February 14, 2005 10:33 PM | Permalink | Reply to this

### Re: Physical Principles?

General relativity must be respected. Strings don’t do this.

I don’t think statements like this are correct. See for instance my explanation here.

sometimes known as topos cohomology

I don’t know anything about topos cohomoloy, but I’ll be glad to be educated. Currently I find it hard enough trying to understand the details of elliptic cohomology.

a topos is really a generalised idea of a (commutative) space

I don’t see in which sense. $\mathrm{Set}$ is a topos. Is there a non-trivial meaningful sense in which the category of all sets is a generalized commutative space?

I probably wouldn’t bring it up here if it wasn’t for the fact that recent M-theory papers seem quite excited about the idea of a ‘purely cohomological’ M-theory (Sati? for instance).

Yes! This is certainly an interesting development. There is need for ‘2-cohomology’ in M-theory.

As far as I understood Sati is arguing that the SUGRA 3-form ${C}_{3}$ should be described by an abelian (Chern-Simons) 2-gerbe.

Using this Aschieri and Jurco argued for the coupling of nonabelian 1-gerbes ($\sim$ nonabelian 2-bundles).

So my question to you is: where is this spectral triple stuff heading PHYSICALLY?

First of all, as I have said before, ordinary spectral triples are essentially the same as (supersymmetric) quantum mechanics.

For the reasons mentioned further above it seems plausible that strings require a couple of concepts to be stringified. For instance ‘nonabelian strings’, whether they arise on stacks of 5-branes or in the context of elliptic cohomology, should be described by 2-gauge theory, which indeed seems to be the case.

The Dirac operator on loop space is related to elliptic cohomology. This operator is just the supercharge on the string worldsheet. It seems what is really needed is a 2-Dirac operator, a functor on a 2-presheaf of sections of some spin 2-bundle.

Up to a couple of days ago I thought I was the only one on earth who ever mentioned the word ‘2-Dirac operator’. Then Nils Baas told me that he is in the process of writing a paper on this guy! :-)

Here is a riddle: ‘What has an algebra, a Hilbert space and a Dirac operator?’

Right, that’s a spectral triple. Now what has an algebra of 2-sections and a 2-Dirac operator? I’d say we want 2-spectral geometry/2-NCG here.

In the toy example of discrete 2-NCG that I have already mentioned it is already possible to construct that 2-Dirac operator and play around with it.

Then, there is the possibility that a spectral 2-action of a spectral 2-triple might be a way to capture superstring spacetime dynamics. If true, that would be a big deal due to the generalizations possible with spectral geometry, I think.

Posted by: Urs on February 15, 2005 8:01 PM | Permalink | Reply to this

### Re: GR

When I say GR, I don’t mean the usual differential geometric formulation. I mean the physical principles. What would Einstein have written down if he had 20th century mathematics? Penrose understood the importance of this question - hence twistors. I have been writing about this on PF. In particular, as Stachel explains, the formulation of Einstein’s equations were delayed until he understood the significance of the fact that spacetime points had no physical existence.

Posted by: Kea on February 15, 2005 10:20 PM | Permalink | Reply to this

### Re: GR

When I say GR, I don’t mean the usual differential geometric formulation. I mean the physical principles.

Then you have to be careful, I think. If by physical principle you mean ‘general covariance’ it is important to note that this is not at all unique to gravity. 3D Chern-Simons theory is ‘general covariant’ in a natural way, for instance, and if you tried hard enough you can turn any theory in a ‘general covariant form’.

GR is really more than some general philosophy on backgrounds etc. It is the idea that the dynamics of gravity is given by an action which is the integrated curvature scalar $\int R\phantom{\rule{thinmathspace}{0ex}}\mathrm{dvol}$.

For instance if the action where $\int {R}^{2}\phantom{\rule{thinmathspace}{0ex}}\mathrm{dvol}$ or $\int {R}_{\mu \nu }{R}^{\mu \nu }\phantom{\rule{thinmathspace}{0ex}}\mathrm{dvol}$ or something you would still have all the general philosophy of GR in place, like ‘general covariance’, ‘background independence’, equivalence principle and whatnot, but it would be a different theory.

What would Einstein have written down if he had 20th century mathematics?

Or he might have noticed that the $\beta$-functional equations of conformal 2-dimensional $\sigma$-models involve the Ricci curvature…

But, ok, I find it really hard to speculate what ‘Einstein would have done…’. I have enough trouble figuring out what I would do! ;-)

(BTW, I should note that I currently have to work on machines which cannot display MathML and which do not allow me to install the required software. So I cannot check if the equations I type parse correctly. If you find anything unreadable please let me know.)

Posted by: Urs on February 16, 2005 12:11 AM | Permalink | Reply to this

### Topos as a space

$\mathrm{Set}$ is the same as sheaves over a point. A typical topos is a sheaf category. It therefore characterises a space. Then the elementary toposes of Lawvere-Tierney generalise this.

Posted by: Kea on February 15, 2005 10:27 PM | Permalink | Reply to this

### Re: Topos as a space

$\mathrm{Set}$ is the same as sheaves over a point. A typical topos is a sheaf category

Ah, ok. I guess you told me that before, but I did not get it.

Hm, now what would be the topos given by the category of sheaves over, say, Riemannian surfaces, for instance?

Or, for starters, what is the topos given by the category of sheaves on the reals $ℝ$?

Do I assume correctly that $\mathrm{Cat}$ would be a 2-topos? Can I realize $\mathrm{Cat}$ similarly to $\mathrm{Set}$ as a 2-topos given by a 2-category of, hm, 2-sheaves/stacks over something, like over a 1-object category?

See, it helps me when you insert more details concerning your ideas about topoi.

But then, I thought the point of topoi is that they are about logic. In which sense it is useful to think of them as being about spaces?

Posted by: Urs on February 15, 2005 11:54 PM | Permalink | Reply to this

### Reals

If I might repeat: the reals are not basic. In axiomatic topos theory it is not even true that the Dedekind reals are the same as the Cauchy reals. Fortunately starting with $ℕ$ one can get to the rationals. $\mathrm{Set}$ has an axiom of infinity. That gives $ℕ$.

Why do you want reals anyway? The rule is: localise, localise. What’s wrong with ${ℚ}_{p}$ instead?

Some people (eg. John Corbett) study the Dedekind reals in the context of ‘quantum numbers’. That’s quite nice. Quick definition (McLarty, Johnstone): the Dedekind reals are a subobject of ${\Omega }^{ℚ}×{\Omega }^{ℚ}$. They are not ‘decidable’, ie. we can’t say “either R = \neg R or NOT(R = \neg R) for all R of type $ℝ$”.

Posted by: Kea on February 16, 2005 4:03 AM | Permalink | Reply to this

### Hilbert spaces

Urs: But if you propose another notion of 2-Hilbert space which you find better suited I will be interested in being taught about it!

OK. Let’s give it a go. I’m guessing this conversation is going to go on for a while, and I apologise if sometimes I am busy doing other things such as things that I am supposed to be doing right now.

First, I guess I need to convince you that the important thing about Hilbert spaces is their lattice structure. The best reference for this, recently pointed out to me by Frank Valckenborgh, is the book

“Orthomodular Lattices” by G. Kalmbach (Academic 1983)

which begins with an historical introduction into some of von Neumann’s work.

A lattice is a poset with 0 and 1 and binary operations $\wedge$ and $\vee$. Traditionally, quantum logic is characterised by lattices that are also equipped with an orthocomplement which satisfies

(i) $U$ in $V$ implies $¬V$ in $¬U$

(ii) $¬¬U=U$

(iii) $U\vee ¬U=1$

(iv) $U\wedge ¬U=0$

These rules force Booleanness in topos logic. Orthomodular logic on the other hand replaces distributivity by the law

(1)$U\mathrm{in}V\to U\vee \left(¬U\wedge V\right)=V$

An orthomodular lattice is characterised by the rule “if $U$ in $V$ and $V\wedge ¬U=0$ then $U=V$”.

Orthomodular lattices characterise Hilbert spaces. Inclusion of subspaces acts as an order relation, which we think of as a directed arrow. An orthogonal subspace $¬U$ of a subspace $U$ of a Hilbert space $H$ satisfies the complement rules. Intersection represents $\wedge$, and $U\vee W$ is the smallest subspace of $H$ containing the union of $U$ and $W$. These operations are associative and commutative. The distinction between $\vee$ and set theoretic union breaks distributivity (consider three orthogonal subspaces).

It turns out that a lattice is orthomodular but not distributive iff it does contain sublattices that look like diamonds and pentagons, but does not contain a hexagonal sublattice made of distinct $\left\{0,1,U,V,¬U,¬V\right\}$.

Sorry, must go.

Posted by: Kea on February 15, 2005 3:30 AM | Permalink | Reply to this

### Re: Hilbert spaces

So in summary Hilbert spaces are described by ‘nondistributive lattices’?

Fine, so what do I have to deduce from that?

What is your definition of categorified Hilbert space? If you can hand me one I’ll see if I can construct a spectral 2-triple using your version of categorified Hilbert spaces.

Posted by: Urs on February 15, 2005 8:05 PM | Permalink | Reply to this

### definitions?

A two minute answer? OK. But I find your terminology difficult. When you say ‘categorify’ you’re talking about something that I don’t accept is essential. Anyway, thinking about the physics first:

Heisenberg said that in some sense every particle contains every other particle - hence particles are not fundamental. What is fundamental in quantum physics? Let us say it is a calculus of propositions.

A proposition $\varphi$ is a statement that may be interpreted with respect to given variables to have truth values, which in the topos $\mathrm{Set}$ are the values true and false. Given a set $S$, for every proposition $\varphi$ with restricted quantifiers (see McLarty’s book, or Mac Lane and Moerdijk) there exists a set $X$ with $x\in X$ if and only if $\varphi \left(x\right)$ is true and $x\in S$. This is the comprehension scheme of restricted ZF set theory.

A long time ago Gray (see LNM 391 and references therein) considered the categorical analogue of this problem and was led to the development of 2-categories, in particular to the tensor product of 2-categories which has the novel feature of being dimension raising. Until one has a need for, say, tricategorical structures this dimensional instability may be completely ignored, and usually is in applications of higher categories to quantum gravity. But here the Gray tensor product is seen to have physical significance. The tensor product of more and more, let us call them, ‘fundamental systems’ requires higher and higher categorical dimension. This is why particle number should not be conserved in QFT.

You were wondering about $\mathrm{Cat}$. Which one? Normally people work in a series of Grothendieck universes. This is a way of getting around Russell type paradoxes and doing everything in set theoretic mathematics. Fix a set $U$. A ‘small’ category is one where the Hom sets are in the universe $U$. One can talk about the category of small categories. But this doesn’t include Top, for instance. So, one can talk about the category of large categories.

I’m an advocate of axiomatising things, because both physics and topos theory says to do this. That means we can do away with Grothendieck universes. Set is easy enough to axiomatise. Categories of categories are harder. McLarty has done some work on this, inspired by Lawvere and Benabou and others. No doubt there is a lot more out there than I am aware of.

This context suggests the following. Think of ${\mathrm{Vect}}_{ℂ}$ as a kind of ‘linear topos’ for now. The number field replaces the terminal object in $\mathrm{Set}$. Axiomatise this like a topos. Scalars, for instance, become arrows. Recall that monic arrows represent subobjects. The ‘lattices’ are the sort we want.

If you really want the 2-categorical analogue…..computer scientists have studied bilinear logic (ie. non-commutative linear logic). According to the above paragraph, this should correspond to something like braided monoidal categories with biproducts. This connection has been studied by Blute (see “Hopf Algebras and Linear Logic” and refs therein) under the heading Biautonomous Categories. It is shown that $\mathrm{Mod}\left(H\right)$ for a non-cocommutative Hopf algebra $H$ is biautonomous. More interesting is a topologised version of this (we need topologies to do cohomology) and a suitable restriction of the topologised reps gives Yetter’s (J. Sym. Logic 55,1 (1990) 41-64) bi*autonomous categories if the Hopf algebra has an involutive antipode.

In summary,

Definition: a 2-Hilbert space is an object in a cyclic bi*autonomous category.

This isn’t the only possible way of looking at it of course, but I hope it gives you an idea of what happens if one tries to respect the logic.

What do we see in fig 4.10 in Yetter’s paper? Looks a bit like a template. Something else I’d like to talk about…but not right now!

Posted by: Kea on February 16, 2005 3:35 AM | Permalink | Reply to this

### Re: definitions?

This context suggests the following. Think of Vect ℂ as a kind of ‘linear topos’ for now. The number field replaces the terminal object in Set. Axiomatise this like a topos. Scalars, for instance, become arrows. Recall that monic arrows represent subobjects. The ‘lattices’ are the sort we want.

I guess in order to understand your definition of 2-Hilbert space I first have to follow this part. Currently I don’t. How do I axiomatise $\mathbf{Vect}$ like a topos? Maybe if you could spell that out in detail for me it would be helpful.

You wrote:

Definition: a 2-Hilbert space is an object in a cyclic bi*autonomous category

You lost me here. I don’t know what a autonomous category is, not what a biautonomous category is and not what a cyclic biautonomous category is. I won’t be able to read all the books that you cite, much less all the references in these books that you point me to! :-)

Worse, I still don’t see why I would want to read all these books to get your definition of 2-Hilbert spaces because I still don’t see why you think your 2-Hilbert spaces are the right thing to call a 2-Hilbert space.

It would help me to maybe proceed in more detail and in a more focused way to really make progress. If we are just throwing fancy language around we will hardly make much progress with this discussion.

So right now I have no idea what your 2-Hilbert spaces are really like.

What is ‘2-addition’ in these ‘spaces’, what would be the ‘2-inner product’? What would be their set of objects, of morphisms? What would be a morphism between two such 2-Hilbert spaces?

Posted by: Urs Schreiber on February 16, 2005 4:00 AM | Permalink | Reply to this

### Re: definitions?

Goodness. Don’t take this definition too seriously. I wasn’t expecting you to go and read all the references, but I thought perhaps someone would appreciate them. And yes, we should focus on one thing at a time, but it must be something on which we have some common ground. Any suggestions?

Posted by: Kea on February 17, 2005 12:28 AM | Permalink | Reply to this

### Re: definitions?

Yes, it would be nice to talk about something that I can take seriously.

Today John Baez and Toby Bartels explained some aspects of topos theory to me, like how the notion of topos was first invented to describe sheaf-categories and later evolved into something more general and now mostly associated with different kinds of ‘logic’.

It is kind of weird at first how sheaves seem to encode logic, but the two of them gave me some intuition for how to think about it. It’s kind of like objects of the base Grothendiek category (open sets in the ordinary case) describe contexts in which certain things are true and inclusion of $a$ in $b$ implies that what is true in $b$ must be true in $a$, roughly.

(If that does not sound right it is my fault.)

And yes, we should focus on one thing at a time, but it must be something on which we have some common ground. Any suggestions?

I can list some category rekated questions that I would like to better understand and you can see if you can help me with that.

So in my little 2-NCG project I want to deal with ‘2-algebras’ (monoidal categories, essentially, with some extra strucure as seems appropriate) of functors from some category $S$ (to be thought of as a base ‘path’ space) to some target category $T$ (to be thought of as the ‘probability amplitude’ for a path to have a given configuration, if you wish.

This is supposed to be the categorification of ${ℂ}^{*}$-algebras of functions from an ordinary space $S$ into the complex numbers $𝕔$.

Now, there are some good reasons why one might want to choose $T=\mathrm{Hilb}$, as explained in HDA2. But that might not be the only reasonable choice and maybe not the best one depending on the applications that one has in mind. I am wondering if I should think about other target categories.

What is nice about choosing $T=\mathrm{Hilb}$ is that then one can prove the categorified Gelfand-Naimark theorem, which looks like a good thing if one wants to categorify something based on the ordinary GN theorem. But I suspect that much of the theorem will go through for other choices, too.

So one question is what other reasonable choices there are for $T$ and how that would affect the nature of the 2-algebra of functors $S\to T$.

That’s a rather vague and conceptual question, of course. You had indicated some dissatisfaction with $2-{H}^{*}$-algebras, so I was wondering if you had an idea for a different choice here.

Then, of course, if I could make you buy into my idea of categorifying NCG there is a host of things that one might want to categorify, construct, conjecture, prove, etc.

Right now I am trying to warm up by thinking about that simple ‘discretized path space’. In the end, of course, I am interested in more sophisticated stuff. Involving true path spaces and algebras of 2-sections of some 2-bundles over them.

It would be nice if one could come up with a reconstruction theorem analogous to the ordinary case. So given some 2-algebra of 2-functions and a 2-Dirac operator represented on some 2-Hilbert space, can we reconstruct the Riemannian 2-geometry of some 2-space, like of some loop space?

These are the kinds of questions that would ineterest me in this context.

Posted by: Urs on February 17, 2005 4:41 AM | Permalink | Reply to this

### more stuff

Hi Urs,

I might be busy this week, moving house.

Other target categories:

Within the framework that you are working in, I cannot see how to improve on Hilb and 2Hilb. Considering the question again anyway…

What ‘categorifies’ $ℂ$?

Let’s say that what you want, first of all, is something that ‘categorifies’ complex-valued functions on compact Hausdorff spaces. There is a book that I really should have referenced above, which is P.T. Johnstone’s “Stone spaces” (Cambridge 1982). Fact: the category of compact Hausdorff spaces is equivalent to a category of locales.

If I might revert to type… Did you know that the paper “A globalization of the Gelfand duality theorem”, B. Banaschewski and C.J. Mulvey 21pp shows that Gelfand duality (using locales) extends to any Grothendieck topos?

Following this paper…

Summary of Definition: a commutative ${C}^{ast}$ algebra in a topos $E$ is a commutative Banach *-algebra $A$ such that for a norm $N$, which is a map from the positive rationals in $E$ into ${\Omega }^{A}$, the following condition holds: for each positive rational $q$ and $a\in A$, $a\in N\left(q\right)$ is the same as $a{a}^{ast}\in N\left({q}^{2}\right)$. As an example, one has the algebra of complex numbers ‘in the topos $E$’, given by $N\left(q\right)$ = $z\in ℂ$ s.t. norm $z$ less than $q$.

The great thing about this is that it puts the complex numbers where they belong, as an object in a topos.

Spaces that are really spectra for finitely generated commutative algebras appear in a really cool commutative triangle, which I don’t believe I can draw here (can we use xypic here?), so in words: there is a functor ‘taking the spectrum’ from ${\mathrm{Alg}}^{\mathrm{op}}$ into a topos called the Big Zariski topos (I don’t know who called it that - maybe Lawvere). The third vertex is the classifying topos ${\mathrm{Set}}^{\mathrm{Alg}}$. There is a functor from the Zariski topos into the classifying topos has an adjoint (ref. Kan’s theorem). The third functor is the Yoneda embedding into the classifying topos. In other words, the spaces you are using are really arrows in $\mathrm{CAT}$.

We already agree that this is the way to go - but what you mean by categorified NCG and what I mean by it are two different things. To me ‘categorification’ means doing everything in purely category theoretic terms right from the start, including worrying about where the real numbers might come from in the reduction of the full theory to classical GR.

Posted by: Kea on February 19, 2005 3:36 AM | Permalink | Reply to this

### Re: more stuff

but what you mean by categorified NCG and what I mean by it are two different things. To me ‘categorification’ means doing everything in purely category theoretic terms right from the start, including worrying about where the real numbers might come from in the reduction of the full theory to classical GR.

So what you mean is really generalization of concepts by fomulating them in a category-theoretic context, while what I was referring to is the dimensional lifting of concepts.

Posted by: Urs on February 22, 2005 5:14 PM | Permalink | Reply to this

### Re: more stuff

I think we both agree that higher dimensional categories are highly relevant, but yes, to me ‘NCG’ isn’t about ‘lifting’ some more basic form of NCG, because basic NCG doesn’t respect quantum logic.

If you want to convince me that 2-Dirac operators as you describe them are important, as opposed to using 2-topologies in a more category theoretic cohomological sense, then you have to convince me that you can describe the logic of observation with them. Or else you need to provide some new physics that I haven’t thought of at all. Do you disagree?

Posted by: Kea on February 23, 2005 10:26 PM | Permalink | Reply to this
Read the post The Principle of General Tovariance
Weblog: The n-Category Café
Excerpt: Landsmann proposes that physical laws should be formulated such that they may be internalized into any topos.
Tracked: December 6, 2007 6:37 PM

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