The n-Category Café

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.

Andread Döring wrote:

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.

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.

Urs described your language by:
–
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

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

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 $$Sub(P) \to \Gamma \Omega \,.$$