December 5, 2007

The Principle of General Tovariance

Posted by Urs Schreiber

Next Friday Klaas Landsman will give a talk here on

The principle of general tovariance: Physical laws should look the same in whatever topos they are defined.

He has made nice slides available online.

In my words, what “tovariance” is supposed to mean is this:

A physical theory should be formulated internally in terms of “arrow theory”: a physical theory should be an abstract diagram with certain properties (usually: some parts of it are required to be certain limits or colimits), such that it may be internalized (interpreted) in a suitable ambient category.

“Suitable” ambient categories might in particular be topoi. Or maybe just Barr exact categories (as in Glenn’s Realization of cohomology classes in arbitrary exact categories) as Igor Baković kindly emphasized to me.

Therefore “the principle of general tovariance” is close to my heart. I have talked about attempts to extract the right arrow theory of quantum field theory of $\sigma$-model type (including all gauge theories) in the series on the charged quantum $n$-particle, which is currently being continued as On BV-Quantization.

In all modesty, I thought of this as an attempt to do for quantum (field) theory what Lawvere has tried to do for classical continuum mechanics: find its right arrow theory. See for instance his Toposes of Laws of Motion which we once discussed a bit here.

Like the principle of general covariance which guided Einstein in the formulation of the classical theory of gravity, the principle of general tovariance becomes rather tautologous after becoming accustomed to it: I think the fact that these ideas once were held as “principles” will in the future just be taken as indication that one had been terribly confused before “discovering” the principle.

Of course the recent ideas by Andreas Döring and Chris Isham on understanding the noncommutativity of the quantum phase space as the result of being internal to a certain topos, which we talked about quite a bit here (I, II, III, IV, V) enters Landsmann’s discussion prominently.

Isham and Döring seem to have seriously sparked new interest in the idea of internalized physics. I wish I could attend the workshop on Categories, Logic and Physics in London next January.

Posted at December 5, 2007 4:18 PM UTC

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Re: The Principle of General Tovariance

There’s a big puzzle surrounding this principle, namely: are we supposed to take our theory of physics as formulated in a specific topos, or a variable topos?

The second alternative seems rather far-out, since we normally fix a topos — or more intuitively, a “mathematical universe” — before diving in and formulating a theory of physics. And of course, 99.9% of us fix this topos to be the category of sets!

But, if we decide to work in a specific topos, what’s the point of ‘general tovariance’? Sure, it can be logically clearer and helpful to our thinking to formulate theories of physics using commutative diagrams, as Urs loves to do. But in general relativity calculations, we really do switch between coordinate systems, so general covariance becomes a very practical thing. In physics, when would we switch between topoi?

(Foundations-of-math people will point out that ‘the category of sets’ is not really a fixed topos, since there are different versions of set theory. But, physicists quite rightly doubt whether these various versions of set theory make any difference in concrete problems. The topoi that are really interesting are the ones that are drastically different from any of these ‘categories of sets’.)

Perhaps Landsmann’s answer to my big puzzle might be this. We need to formulate our theories of physics to work in a variable topos, because the correct topos is observer-dependent.

We get a little sense of that here:

However, the ‘inner’ topos in this picture is not variable and observable-dependent, but fixed by our physical theory, which involves a C*-algebra $A$. So, we can still ask: what does formulating our theory in an arbitrary topos actually buy us here?

Posted by: John Baez on December 5, 2007 5:40 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

There’s a big puzzle surrounding this principle, namely: are we supposed to take our theory of physics as formulated in a specific topos, or a variable topos?

Hm, isn’t that saying: “There is a big puzzle surrounding this principle, namely: are we supposed not to follow it, or to follow it?”

if we decide to work in a specific topos, what’s the point of ‘general tovariance’?

I think there is still a point to it, even if we’d never switch the topos: it guides us to the right nice formulations.

To stay withing the metaphor: even if I do differential geometry just only on $\mathbb{R}^n$ and nothing else, I might find (actually, I do find) it convenient to use the general abstract definition of differential forms, exterior derivative, Lie derivative, etc, instead of (only) the coordinate-ridden way.

In physics, when would we switch between topoi?

I dropped little hints in my entry above indicating that, personally, I feel that the issue is actually not so much about topoi as such, but rather about internalization in general.

I think the main point is, that we want to formulate our concepts such that we can internalize them seamlessly in desired ambient categories.

These ambient categories might well be topoi, and then I might on top of everything else have the pleasure of being able to give lots of vexing philosophical problems nice neat precise mathematical meaning and solutions, but the internalization step is important and useful even apart from that.

So here is a practical simple example of the sort we have been talking about a lot:

for a big chunk of physics, we want at least three things:

a) an object in our ambient category which plays the role of parameter space

b) another object in our ambient category which plays the role of target space

c) the internal hom object between these two objects: the space of fields.

There are a couple of different ambient categories – some of them happening to be topoi (but all of them playing a different role compared to the topoi that for instance Döring and Isham consider) – that have immediate practical relevance:

a) we might interpret the above in the context of categories internal to finite sets. This is the right choice for instance if we want to describe finite group field theory, as in Simons’s published and Bruce-and-Simon’s unpublished work.

b) we might interpret the above in the world of $\infty$-groupoids internal to either Top or sheaves on manifolds or the like. This would be the right choice for more sophisticated applications than finite group field theory. Most everything ever considered in physics should live here.

c) essentially equivalently but more tractable, we don’t internalize to Lie $\infty$-groupoids but to their Lie $\infty$-algebroids. Most everything ever considered in QFT has actually (more or less secretly) been spelled out here.

The point I consider as important is:

When we really understand QFT, it will mean that we do not have to rethink everything when we switch from considering finite-group gauge theories to the harmonic oscillator to Chern-Simons theory to string field theory. Rather, we take one powerful concept, formulated in terms of abstract diagrams, and intepret it in the relevant context.

The examples I give here are of course rather different from what Landsman seems to have in mind and what Isham-Döring consider.

The topoi they consider I would consider as a “back end” of the internal physics of the kind I am thinking of: namely after we have set that up and turned the crank, it will spit us out certain morphisms somewhere, which go between one object that models a category of cobordisms and another object that models a category of vector spaces.

Then, I might want to interpret that result in the way Isham-Döring do in order to be able to say: “this projection operator $P$ on the vector space I have obtained here can be thought of as an supobject $\delta P$ of the spectral presheaf, which is the “phase space object” in the topos of presheaves on abelian subalgebras of $B(H)$.”

Doing so would enhance the above procedure of internal physics, which is a map from

(collections of: paramater space, target space, background field)

to

(collection of: cobordism object, vector space object, morphism between them)

to something that takes value in

(collections of: phase space object, valuation object, physical observable object).

That’s how I am thinking about it. I am thinking that internalization in physics is practically relevant when, but also already long before we reach the point where we might want to utter the words “measurement problem”.

Posted by: Urs Schreiber on December 5, 2007 6:59 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

John wrote:

There’s a big puzzle surrounding this principle, namely: are we supposed to take our theory of physics as formulated in a specific topos, or a variable topos?

Urs wrote:

Hm, isn’t that saying: “There is a big puzzle surrounding this principle, namely: are we supposed not to follow it, or to follow it?”

That big puzzle applies to every principle! So, it certainly applies here.

But my worry was a bit different. The principle of general tovariance says “Physical laws should look the same in whatever topos they are defined.” And my question is: what does it mean to follow this principle? And what does it do for us if we follow it?

Does it merely mean that “The laws of physics should be written in terms of geometric logic”? Geometric logic the sort of logic that applies in any topos. This logic lets us define ‘a thing of type $X$ in a topos $T_1$’ and then transport it across any geometric morphisms $T_1 \to T_2$ to get ‘a thing of type $X$ in the topos $T_2$’.

We could write down a theory of this sort and then work with it in a specific topos $T$; that’s not far from what physicists already do. But then how does the principle help us do anything? I guess it still means our theory transforms nicely under geometric morphisms $T \to T$, which are a bit like ‘diffeomorphisms’ in general relativity. Is that the point?

Or, does it mean that “No experiment can tell what topos we’re in: all topoi are equally good!” This would be a quite drastic principle, verging on crazy. But, it’s this crazy idea that’s more strictly analogous to the principle of general covariance, which says “no experiment can tell what coordinate system we’re in: all coordinate systems are equally good”. And of course the principle of general covariance seemed quite crazy too at first.

I dropped little hints in my entry above indicating that, personally, I feel that the issue is actually not so much about topoi as such, but rather about internalization in general.

Well, I understand your ideas much better than Landsman’s. The way I think of it, it’s always good to stip any concept down to its bare essentials, so it can apply to the widest collection of contexts. Doing this systematically forces us to understand the relation between concept and context.

It’s easy to believe that a context is a category, or category with extra structure, or $n$-category with extra structure. The more surprising fact, due ultimately to Lawvere, is that a concept can also be seen as an $n$-category with extra structure. This allows us to say that an instance of the concept $A$ in the context $B$ is just a map

$f: A \to B$

where $A$ and $B$ are $n$-categories with extra structure, and $f$ is an $n$-functor preserving this extra structure.

This sounds scary at first, but it really just means this ‘$f$ paints a picture of $A$ in the frame $B$’.

Lawvere would say ‘$f$ is a model of the theory $A$ in $B$’.

Anyway, all this becomes a very nice way to pursue all sorts of pure and applied mathematics — as soon as one overcomes the fear of abstraction that plagues so many scientists these days.

So, if Landsmann’s ‘principle of general tovariance’ is just a weird way of saying this sort of thing, I’m all for it, except insofar as it privileges ‘topoi’. Topoi are a wonderful sort of context, but they are not the right context for everything.

Posted by: John Baez on December 5, 2007 8:43 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

the relation between concept and context.

[…]

this sort of thing, I’m all for it,

Yes, it’s this kind of principle which I mean, too. And, clearly, I learned it from you! :-)

But Landsman’s slides made me feel it worthwhile emphasizing this again. In my unbounded bluntness, I was suggesting that this is what he might really mean.

And the way you put it (much better than I did) it sounds very tautologous. Which is good. A good principle of xyz-variance should really be a dumb tautology once we really live up to it.

(Like it becomes hard, nowadays, after a modern introduction to general relativity, to appreciate what it took to propose the principal of general covariance: it is just the obvious statement that when talking about smooth spaces you use the language of differential geometry.)

Here is one piece of evidence that the “principle of tovariance” should really be nothing but the principle you mention, the – er, let me give it a name - principle of concept and context:

what’s the main point of Heunen & Spitter’s A topos for algebraic quantum theory, which Landsman is reviewing from slide 12 on?

No, I’d say. The main point is this statement:

Noncommutative geometry is commutative geometry internal to a suitable context.

Where it so happens that the context in question is a topos.

Posted by: Urs Schreiber on December 5, 2007 9:53 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

By the way, there is a nice companion to Heunen & Spitter’s result that noncommutative $C^*$-algebras are commutative $C^*$-algebras suitably internalized:

In

P. Bouwknegt, K. Hannabuss, V. Mathai, C*-algebras in tensor categories

the authors discuss how non-associative $C^*$ algebras are associative $C^*$-algebras internal to a suitable topological monoidal category.

(Incidentally, these non-associative $C^*$-algebras arose in the study of topological T-duality: it turned out that there are some string backgrounds with a gerbe on them whose T-dual is neither a commutative nor a non-commutative geometry, but a non-associative geometry.)

Posted by: Urs Schreiber on December 5, 2007 10:06 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

I like this idea because — if I remember correctly — the octonions become a commutative associative C*-algebra in a suitable symmetric monoidal category.

Posted by: John Baez on December 6, 2007 3:10 AM | Permalink | Reply to this

Re: The Principle of General Tovariance

Mathai is an old pal of mine

I didn’t know that. Or maybe I did and forgot it again.

He was visiting Hamburg when they were thinking about finding an ambient category in which their non-associative $C^*$-algebra became associtive. He, Konrad Waldorf and myself spent a while in front of the blackboard in my office trying to figure it out.

Alas, in those minutes we didn’t quite see how to do it exactly. That would have been fun…!

Anyway, it seems to me that this result nicely helps to put Heunen&Spitters and Isham & Döring’s work in perspective.

Is the Bouwknegt-Hannabus-Mathai associativizing context a topos? At least it doesn’t seem to matter much, at least as far as their result is concerned.

Most probably it would matter if and when we’d start to address something like the “measurement problem” or the “observer problem” in the context of nonassociative geometry.

So we have the principle of context and concept at work from the start (things should be formalized internally as much as possible) and for special applications we may need to demand that the context has the properties of a topos.

So here is my suggestion:

Principle: When formulating geometry and physics, follow the principle of context and concept (formalize internally, as much as possible). When you reach the point where you want to address the measurement problem of quantum mechanics, make sure that the context has the properties of a topos, in order to be able to extract the logic of observations predicted by the model.

Posted by: Urs Schreiber on December 6, 2007 10:33 AM | Permalink | Reply to this

Re: The Principle of General Tovariance

Noncommutative geometry is commutative geometry internal to a suitable context.

Is it possible to apply this statement to Connes’ program?

Posted by: David Corfield on December 6, 2007 9:38 AM | Permalink | Reply to this

Re: The Principle of General Tovariance

I paraphrased Heunen and Spitter’s result that every noncommutative $C^*$-algebra is a commutative $C^*$-algebra internal to a suitable category as:

Noncommutative geometry is commutative geometry internal to a suitable context.

Is it possible to apply this statement to Connes’ program?

I’d expect that. I don’t know for sure. But that’s what I expect.

In fact, in my infinite naivety, I was expecting this already – let me see… – from the 12th of January 2005 on. I misunderstood something Marni Sheppeard (who is often calling herself “Kea” on the web, like in the comment she dropped below, but who is also blogging under her real name here ) said, which sounded to me like

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.

Back then Kea disagreeed with this, replying:

No. Ordinary toposes are not good enough to do NCG.

But I think that Heunen-Spitter’s actually show that topoi are in fact “good enough”. At least for noncommutative topological spaces.

But I find it hard to believe that in a context where we can internalize a commutative $C^*$-algebra one shouldn’t be able to internalize a commutative spectral triple, too.

Posted by: Urs Schreiber on December 6, 2007 10:17 AM | Permalink | Reply to this

Re: The Principle of General Tovariance

I haven’t thought much about toposes from the NCG point of view, but I did go to an interesting seminar last year by Paolo Bertozzini from Thailand, who was advocating ‘modified toposes’ from a thorough analysis of the concept of spectral triple. Unfortunately, I cannot track down a paper on this.

Posted by: Kea on December 6, 2007 6:50 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

Wait! I found some slides by Bertozzini.

Posted by: Kea on December 6, 2007 7:08 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

Kea wrote:

I found some slides by Bertozzini.

Thanks!

I spoke with one of his collaborators, Roberto Conti, in Oberworlfach a bit, and some of the stuff on these slides reminds me of these conversations.

Roberto Conti had an idea how to define “spaceoids”, which I liked a lot, but which I was asked to keep top secret.

Now it seems to me that these spaceoids are what Bertozzini is announcing on slide 25: the thing that complets the horizontal categorification (= many object version) of the Gelfand duality theorem.

(I don’t need to remind anybody here about the vertical categorification of the Gelfand theorem, do I ?)

Posted by: Urs Schreiber on December 6, 2007 7:29 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

I wrote:

But I find it hard to believe that in a context where we can internalize a commutative $C^*$-algebra one shouldn’t be able to internalize a commutative spectral triple, too.

Well, Chris Heunen and Bas Spitters of course essentially apply the the abstract nonsense theorem which appears as their proposition 1, cited from Johnstones Sketches of an Elephant:

Fact. Let $T$ be a geometric theory, and $\mathbf{C}$ any category. Denote by $\mathbf{Mod}(T,\mathbf{T})$ the category of $T$-models in the topos $\mathbf{T}$ with $T$-homomorphisms. There is an isomorphism $\mathbf{Mod}(T,\mathcal{S}^{\mathbf{C}}) \simeq \mathbf{Mod}(T, \mathcal{S})^{\mathbf{C}}$.

In words: for suitable theories (“geometric theories”), presheaves of their models are the same as models of them in presheaves.

So I guess somebody should check:

Question: Is a spectral triple a geometric theory?

Does anyone know?

(I guess I should be able to figure this out by doing some reading. But if anyone familiar with the concept of “geometric theories” can quickly check this, I’d be grateful.)

Because if it is, then it’s clear how to copy Heunen-Spitters’ approach to realize noncommutative spectral triples as commutative spectral triples internal to a suitable topos:

simply let the topos be that of presheaves on the category of commutative sub-spectral triples of the given spectral triple, under inclusion.

Well, one would need to decide on the right notion of category of spectral triples first. As Paolo Bertozzine discusses from slide 16 of this collection on, there are some choices to be made here.

But anyway. Make your choice, and I’ll make your noncommutative spectral triple a commutative one internal to some topos.

If the concept “spectral triple” is a geometric theory, that is…

Posted by: Urs Schreiber on December 6, 2007 7:44 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

Urs almost wrote:

Principle: When formulating geometry and physics, follow the principle of concept and context (formalize internally, as much as possible). When you reach the point where you want to address the measurement problem of quantum mechanics, make sure that the context has the properties of a topos, in order to be able to extract the logic of observations predicted by the model.

Perhaps unsurprisingly, I like this version of the principle!

I wonder what some of the toposophers (Chris Isham, Andreas Döring, Klaas Landsmann) would think of it. In his post, Andreas emphasized the role of formal languages. As far as I can tell — but I’m extrapolating a bit here — he wants to describe a physical system using an axiom system $A$ written in the language of geometric logic; he then has the freedom to look for models of this axiom system inside various topoi $T$. I just want to say a word about how that relates to the ‘principle of concept and context’.

Clearly $A$ is the ‘concept’ here — a bunch of statements about the system we’re trying to describe — while $T$ is the ‘context’. But, we can put this idea into the framework I was describing if we form the ‘classifying topos’ $Th(A)$ of the axiom system $A$. Then a model of $A$ in the topos $T$ is the same as a geometric morphism

$f: Th(A) \to T$

A geometric morphism is just a specially nice sort of functor between topoi, which preserves a bunch of the structure that topoi all have. So, as I was advocating in my post, we can see a model of our axiom system $A$ in the topos $T$ as a structure-preserving functor. This is just what Lawvere called ‘algebraic semantics’.

I would be interested to hear if Andreas agrees with this, since he didn’t actually speak of an ‘axiom system written in geometric logic’ — he just spoke of a ‘language’. But, I think we need more than a ‘language’ to describe a physical system; we need a bunch of statements in this language: axioms. And, while there are various ways to do logic in a topos, the most flexible and interesting way seems to involve geometric logic.

On another note — Urs wrote:

Noncommutative geometry is commutative geometry internal to a suitable context.

Is it possible to apply this statement to Connes’ program?

I think the answer is very much yes. In fact, this idea is quite old. One place noncommutative geometry shows up is in the study of ‘bad quotient spaces’. Say we have a group $G$ acting on a topological space $X$, and we’re trying to understand the topology of this situation. An obvious thing to look at is the quotient space $X/G$. But, if the orbits of $G$ aren’t closed, this space may not (will not?) be Hausdorff — it’s ‘too folded up on itself’. This causes problems: we’ve got a ‘bad quotient space’.

Connes introduces a substitute for $X/G$, a noncommutative algebra where you take the commutative algebra of functions on $X$ and throw in extra operators coming from the action on $G$.

Another approach, which is somehow secretly equivalent, is to consider $X$ as a space not in the topos of sets, but in the topos of $G$-sets.

Yet another approach, also somehow equivalent, is to form the ‘weak quotient’ $X//G$, where instead of identifying points $x$ and $g x$ we stick in an isomorphism between them. This $X//G$ is a topological groupoid, which in certain cases people often treat as an ‘orbifold’ or ‘stack’.

I’ve never seen anyone come out and clearly say in one place how all three pictures are related — perhaps because I haven’t read enough of the literature, or perhaps because people are a bit shy. But, one can sense the secret relation between noncommutative geometry, geometry internal to a topos, and the geometry of groupoids by looking at the list of papers here:

or Ieke Moerdijk’s papers, for example Foliation groupoids and their cyclic homology and Orbifolds as groupoids: an introduction.

Posted by: John Baez on December 6, 2007 8:17 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

Another clue to the secret relation between noncommutative geometry, geometry internal to a topos, and the geometry of groupoids comes from theorems relating arbitrary topoi to topoi coming from groupoids. Here’s one:

They say that ‘every topoi with enough points is equivalent to the classifying topos of a topological groupoid’. And, they provide references to earlier results along these lines, including a paper by Joyal and Tierney.

I don’t understand the big picture here…

Posted by: John Baez on December 6, 2007 8:29 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

John wrote:

They say that ‘every topoi with enough points is equivalent to the classifying topos of a topological groupoid’.

Incidentally, just before I saw this comment I had a look with Igor Baković (Igor is my temporarily local topos guru, as you may have noticed) at Ieke Moerdijk’s book on classifying topoi. If I understand correctly, the classifying topos of any category $C$ is nothing but presheaves on $C^{\mathrm{op}}$.

For instance the classifying topos of a group $G$ is the category of $G$-sets. But of course that’s nothing but $G\mathrm{Set} \simeq Funct(G^{\mathrm{op}},Set) := Presheaves(\{\bullet \stackrel{g}{\to} \bullet | g \in G\}^{\mathrm{op}}) \,,$ where the op doesn’t really matter

Posted by: Urs Schreiber on December 6, 2007 9:30 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

I know little of this stuff, but Butz and Moerdijk define the classifying space of a topological groupoid $G$ with object space $X$ to be the topos of sheaves over $X$ equipped with a $G$-action (in the hopefully obvious sense).

Posted by: John Baez on December 6, 2007 10:07 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

Butz and Moerdijk define the classifying space of a topological groupoid $G$ with object space $X$ to be the topos of sheaves over $X$ equipped with a $G$-action (in the hopefully obvious sense).

Okay, sounds better. I was probably hallucinating.

As far as I understand the classifying topos $\mathbf{B}G$ should be defined by the fact that for $X$ a space equivalence classes of morphisms $\mathrm{Hom}_{Topos}(Sheaves(X),\mathbf{B}G)$ correspnd to principal $G$-bundles (principal groupoid torsors) on $X$.

I guess from there one should derive what $\mathbf{B}G$ is.

For instance for $G$ a one-object groupoid coming from a topological group, the classifying topos is the category of $G$-sets.

Given a $G$-bunde $P \to X$, we send $G$-sets $S$ to sheaves on $X$ by $S \mapsto \Gamma(P) \times_G S \,,$ where $\Gamma(P)$ is the sheaf of sections of $P$.

Or that’s how I understood it. Unless I am hallucinating again.

Posted by: Urs Schreiber on December 6, 2007 10:44 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

Urs write:

For instance for $G$ a one-object groupoid coming from a topological group, the classifying topos is the category of $G$-sets.

Note that $G$-sets are just what we get from following Butz and Moerdijk’s prescription. They say to use sheaves on the space of objects of the groupoid, equipped with an action of the space of morphisms. But your groupoid has just one object, and a sheaf on a one-point space is just a set, so all we get are $G$-sets.

To see why we need sheaves instead of presheaves, we’d need a fancier example! Perhaps for starters a topological groupoid with just identity morphisms: this is just a topological space.

Posted by: John Baez on December 11, 2007 3:30 AM | Permalink | Reply to this

Re: The Principle of General Tovariance

On the question how and why different topoi show up: in Chris Isham’s and my work, a topos is assigned to each physical system. (Of course I cannot speak for Klaas, since he has not published a paper on this yet, but since his ideas seem very closely related to ours, it might be useful to briefly describe what we do in our work.)

More precisely, each system S has attached to it a certain first-order formal language L(S) that then is represented in suitable topoi. If one wants to treat S as classical system, L(S) is represented in the topos Set. If one wants to treat S as a quantum system, L(S) is represented in a certain topos of presheaves. The base category of these presheaves is given by the poset of unital, abelian von Neumann subalgebras of the von Neumann algebra of observables of the system S.

Let us concentrate on the quantum case for a second. In order to speak about the relations between different systems, we need morphisms between the associated presheaf topoi - geometric morphisms. In our fourth paper (arXiv:quant-ph/0703066v1), we suggest to consider an abstract category of systems, which is a symmetric monoidal category. The task is then to find translations between the languages associated to different systems and between the representations of these languages in the topoi of presheaves belonging to the systems. The construction of such translations involves geometric morphisms. It will be interesting to see if there are closer relations to the work on symmetric monoidal categories done by Samson Abramski, Bob Coecke, John Baez et. al. Another interesting open question is which structure the collection of presheaf topoi has. One may expect that there is one largest system, the ‘universe’, which contains all other systems as subsystems. How this is encoded precisely on the level of the presheaf topoi remains to be worked out.

Another question (and I know that Klaas is interested in this) is the relation between the quantum and the classical description. I strongly suppose that Klaas’ principle of tovariance is closely related to (or maybe can be made precise by) the use of formal languages to describe physical systems. Each language L(S) contains a symbol that in representations becomes a ‘state-space’ object (which need not be a space, of course), a symbol that in representations becomes an object of ‘values’, and function symbols between them, which in a representation become arrows between the objects. This leads to a structural similarity between the classical and quantum description that usually is not given. I guess that Klaas wants to promote this structural similarity to a principle. One may consider geometric morphisms or more general arrows between the topoi for the quantum and for the classical description and hopefully gain some new insight into the quantisation/dequantisation problem.

The work of Bas Spitters and Chris Heunen (arXiv:0709.4364v1) to which Klaas refers is concerned with C*-algebras rather than von Neumann algebras, since there are more constructive results on C*-algebras, in particular Chris Mulvey’s and Berhard Banaschewski’s constructive Gel’fand duality. On the other hand, Bas and Chris have to use AW*-algebras for the measure-theoretic part of their work, since one needs enough projections there. This is quite close to von Neumann algebras.

Bas and Chris do not consider formal languages at all. Their work is mathematical and shows very nicely how our constructions can be understood in a constructive context. The basic idea, namely to consider the abelian parts of a larger, non-abelian structure and to construct presheaves or functors thereon, goes back to Chris Isham’s work with Jeremy Butterfield on the Kochen-Specker theorem in ‘98-‘02. Physically, this kind of description is useful for a single, quantum-mechanical physical system. If one wants to consider more than one system and/or one system in different descriptions (like classical and quantum), one needs several topoi. Klaas wants to formulate a principle for how physical descriptions of systems have to look like. Chris and I have suggested formal languages as a tool to do that. Of course, a lot of work remains to be done.

Posted by: Andreas Doering on December 6, 2007 3:50 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

Each language L(S) contains a symbol that in representations becomes a state-space’ object (which need not be a space, of course), a symbol that in representations becomes an object of values’, and function symbols between them, which in a representation become arrows between the objects. This leads to a structural similarity between the classical and quantum description that usually is not given. I guess that Klaas wants to promote this structural similarity to a principle.

We might want to promote this to a statement of the following kind. We would like to say something like:

a) a physical system is a topos $T$;

b) a phase space object is an object $\Sigma$ in $T$ such that this and that holds;

c) a number object is an object $R$ in $T$ such that this and that holds;

c) the space of observables is a subobject of $\mathrm{hom}_T(\mathrm{Sub}_\Sigma, \Omega)$ such that this and that holds.

Preferably, we’d want to say

* “Mechanics in the system $T$ is a diagram in $T$ such that this and that part of it is a limit or colimit.

Given such a diagram, we call this object of it the phase space object, that object of it a valuation object, and so on.”

It seems that the question is to figure out what the “this and that”-conditions should be.

For instance, I gather that you have shown in your work with Chris Isham that, in the case that $T = PreSh(V(H))$ (the topos of presheaves on the poset of abelian subalgebras of bounded operators $B(H)$ on a Hilbert space $H$ wrt inclusion) the following is true:

every quantum state, namely every vector $\psi \in H$, gives rise to a morphism from subobjects of the phase space objects to global sections of the suboject classifier $T^\psi : Sub(\Sigma) \to \Gamma \Omega \,.$

It seems to me that what we’d eventually want is the converse of this statement: we’d want to characterize states as certain elements of, say, the object $\mathrm{Hom}(Sub(\Sigma),\Gamma \Omega)$, which is defined itself as an object of $T$ by some abstract properties.

Then we could say:

see, if I internalize this diagram here in the topos $T$ of presheaves over $V(H)$, then I find, by turning the crank, that a state is this and that (should be a vector in $H$), an observable is this and that (a self-adjoint element of $H$).

But if I internalize instead in the topos $T'$, then I find, by turning the crank, that, instead, a state is something else (a point in the phase space of the classical harmonic oscillator, say) and an observable is now no longer a self adjoint operator, but instead something else.

I had the vague impression that this kind of internal characterization of aspects of physical systems is what Klaas Landsman has in mind when he speaks about “tovariance”.

Actually, it seems to me that the relevant question is open even for classical systems, so it might be worthwhile thinking about these first:

as you mentioned in a talk, every classical state $\psi$ on classical phase space $P$ gives rise to a morphism $\Psi_p : Sub(P) \to \Omega$ in the topos of sets (namely the map which sends each subset of phase space which contains the point $p$ to $1 \in \Omega = {0,1}$, and all other subsets to 0.

But not every such morphism comes from a classical state this way.

Hence what we’d eventually want is a converse of the above statement: how do we characterize classical states by using just language internal to a topos.

It would, for instance, be good if we could say:

for the classical harmonic oscillator, there is a topos $T$ such that

- phase space $P$ is an object of $T$ characterized by this and that property,

- classical states are precisely the elements of the internal hom from this to that object in the topos, both of which are characterized by having these and those properties.

If we have achieved that, I’d think we could say that we have found a “tovariant” understanding of the concepts system, phase space, observable.

Posted by: Urs Schreiber on December 6, 2007 6:29 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

Yes, Urs, you are right: a real internal description would surely be the nicest thing to achieve. We are thinking about this quite a lot, in particular since we want to regard our scheme as an axiomatic one. I somehow doubt that the axioms will be purely mathematical, we will probably need `physical input’.

One additional difficulty is that we want a scheme based on formal languages that are the same regardless whether we think of the system as a classical system, a quantum system or even something else. The task is to see what to incorporate on the level of the formal language and what will be part of the representation.

If we reach a good understanding of this, then we will have a very general way of constructing physical theories which allows for generalisations that usually would not be seen (probably).

P.S.: When we started, we did not aim at an axiomatisation of physical theories in topoi. We were surprised several times how strong the formalism is, and we keep discovering things, so give us some time (and help, if you like) ;-)

Posted by: Andreas Doering on December 6, 2007 6:55 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

Dear Andreas,

Thanks for your comments. It is nice to see all this attention for the topos ideas. [I should mention that we have some further results and we are working on a revised version of the paper.]

It is indeed nice to compare our two approaches. The application of internal logic to the results in your papers was our starting point (Thanks again for that!).
Some remarks on your description of our paper.

You wrote:

The work of Bas Spitters and Chris
Heunen (arXiv:0709.4364v1) to which
Klaas refers is concerned with
C*-algebras rather than von Neumann
algebras, since there are more
constructive results on C*-algebras

This is not our main reason for focusing n C*-algebras. The first is that the extra generality comes for free (every N-algebra is a C*-algebra). The second is that we believe that C*-algebras are the natural setting for physical theories. [We may disagree on this last point.]

On the other hand, Bas and Chris have to
use AW*-algebras for the
measure-theoretic part of their work,
since one needs enough projections
there.

More precisely, we use AW*-algebras to connect to the measures on projections (as in Gleason’s theorem). The construction also works for C*-algebras, where there may not be enough projections. Such measures do not seem to have been studied before, perhaps because the internal spectrum was missing. In short, the restriction to AW*-algebras is only used to connect to previous work.

Bas and Chris do not consider formal
languages at all.

I would say that we use the Mitchell-Benabou language.

a) a physical system is a topos T;
b) a phase space object is an object Σ in T such that this and that holds;
c) a number object is an object R in T such that this and that holds;

In a sense, we are more conservative then you and Chris. You propose a new language, we stick to the established framework of C*-algebra and derive a topos structure in which your language can be naturally interpreted. Every C*-algebra A induces:
a) a topos T (in which A becomes commutative.)
b) a phase space object, Σ, the internal spectrum of A which is a compact
regular locale;
c) a number object R, the interval domain, i.e. the collection of partially
defined real numbers. This is present in any topos.

Bas

Posted by: Bas Spitters on December 13, 2007 8:54 AM | Permalink | Reply to this

Re: The Principle of General Tovariance

Nice set of slides, but why do people keep avoiding the issue of distributivity (and the need for, eg., higher dimensional toposes)?

Posted by: Kea on December 5, 2007 6:55 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

Please restore the slides as soon as it is allowed! I skimmed them last night, and I was looking foward to reading them in detail over the holidays. They are such a nice concise description of a host of mathematical concepts that I have heretofore avoided out of fear! I was really hoping to learn some of this stuff.

Posted by: Scott Carter on December 6, 2007 3:59 PM | Permalink | Reply to this
Read the post A Topos for Algebraic Quantum Theory
Weblog: The n-Category Café
Excerpt: A summary of some key points of Chris Heunen's and Bas Spitter's article.
Tracked: December 6, 2007 11:42 PM

Re: The Principle of General Tovariance

Klaas Landsman gave a really beautiful talk. He kindly sent me the pdf file with the slides he used.

Posted by: Urs Schreiber on December 9, 2007 4:48 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

I have a second to say a word or two about Klaas Landsman’s talk.

One thing he emphasized very nicely was the beauty of the formulation of propositions and their truth values on certain “states” in the toposophical framework.

Let me try to paraphrase a bit loosely, from memory, the way he described that. (Nothing of which is new, or was supposed to be new, but it was nicely put in exposition.)

Of all the axioms of a topos, of course there is one which all the others are built around: a topos is a category $T$ with a subobject classifier $\Omega$.

Any subobject $V \subset A$ in the topos is the pullback along a unique classifying map $A \to \Omega$ of the subobject “truth” inside “all truth values” $1 \stackrel{true}{\hookrightarrow}\Omega \,.$

Now, if $P \in Obj(T)$ is some “space of things” about which we may want to reason (like the phase space of a physical system, for instance), we have the following two nicely compatible notions:

- a proposition about the “elements” of $P$ is a morphism $F : P \to \Omega$ “sending each element of $P$ to the truth value which it has under the proposition”.

- a “state” instead is just an element of $P$ $\psi : 1 \to P \,.$

So, then, the evaluation of a proposition on a state $evaluation : propositions \times states \to truth values$ is just the pairing of these two morphisms, which yields the truth value $F(\psi) : 1 \stackrel{\psi}{\to} P \stackrel{F}{\to} \Omega \,.$

There is more to say. But I am on a shaky public wlan connection and leave it at that for the moment.

Posted by: Urs Schreiber on December 9, 2007 7:48 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

Not so different from what we were finding out about in Progic.

Posted by: David Corfield on December 9, 2007 8:47 PM | Permalink | Reply to this

Re: The Principle of General Tovariance

David Corfield wrote, concerning the elementary sketch of topos logic here:

Not so different from what we were finding out about in Progic.

Right, not so different at all. In fact, what John describes at that link is a generalization. What I mentioned follows by looking at the trivial “probability monad”.

I should have mentioned your series. I must say I got lost somewhere in the Progic project, though. Maybe it’s time for me to look back at the old installments.

Posted by: Urs Schreiber on December 10, 2007 9:42 AM | Permalink | Reply to this

Re: The Principle of General Tovariance

I guess the discussion of pure states by Isham and Döring which I succeeded in reproducing only half-way here fits into this, nicely:

they say a pure state yields a morphism from subobjects of the phase space object $P$ to global section of the subobject classifier $Sub(P) \to \Gamma \Omega \,.$

But subobjects of $P$ are (that’s the whole point of the topos setup), nothing but morphisms $P \to \Omega \,.$ While a global section of $\Omega$ is here meant to be nothing but a morphism $1 \to \Omega$ (1 being the terminal object).

Hence if we had any morphisms $\psi : 1 \to P$ (as we do in $Set$), each of them would give a pure state by precomposition $T^\psi : (P \to \Omega) \mapsto (1 \stackrel{\psi}{\to} P \to \Omega).$

But in other contexts there is not a single $1 \to P$. Hence they resort to looking at more general morphisms $Sub(P) \to \Gamma \Omega \,.$

Posted by: Urs Schreiber on December 10, 2007 9:58 AM | Permalink | Reply to this
Read the post The Concept of a Space of States, and the Space of States of the Charged n-Particle
Weblog: The n-Category Café
Excerpt: On the notion of topos-theoretic quantum state objects, the proposed definition by Isham and Doering and a proposal for a simplified modification for the class of theories given by charged n-particle sigma-models.
Tracked: January 9, 2008 10:26 PM