The n-Category Café

What is a physical theory? We might be able to agree that whatever it is, we demand it to have at least the following ingredients:

a) it provides us with a thing called the space of states of a physical system;

b) it provides us with a thing called the collection of sensible propositions about the states of the physical system;

c) it provides us with a way to evaluate any proposition on any element of the space of states such as to obtain something like a truth value which is a measure for the degree to which the proposition holds for that state.

What is a proposition?

I’ll essentially review my review here, which did receive a bit of positive feedback.

A little (maybe a little more) reflection shows that a good way to characterize the nature of propositions about some collection, $V$, is to realize that propositions $P$ about $V$ should be equivalently described by two properties:

$\bullet$ every proposition $P$ maps every element in $V$ to the truth value in the collection $\Omega$ of truth values;

and

$\bullet$ every proposition corresponds precisely to the sub-collections $$S_P \subset V$$ for which the proposition $P$ “holds”.

So in order to be able to talk about propositions we need to work internally to a context $T$ which has the at least the necessary properties for this to make sense. Such a context is, by definition, called a topos.

What is a topos.

So a topos is a category $T$ with the property that it contains a an object $\Omega$, such that morphisms from any other object $V$ into $\Omega$ corespond precisely to subobjects of $V$:

$$(P : V \to \Omega) \leftrightarrow (S_P \subset \Omega) \,,$$

and has on top of that the right properties for this statement to make good sense in the first place.

Given a topos $T$, we can nicely satisfy our requirements a), b) and c) by picking a state space object in $T$.

What is a state space object?

A state space object is a pointed topos; a topos $T$ together with a fixed chosen object $$V \in Obj(T) \,.$$

Given any state space object $V \in Obj(T)$, we define

$\bullet$ the collection of states to be the elements of $V$, i.e. the morphisms $$\psi : 1 \to V$$ in $T$ (here $1$ denotes the terminal object in $T$);

$\bullet$ the collection of propositions to be the morphisms $$P : V \to \Omega \,.$$

This is nice, because there is a beautifully obvious evaluation of proposition on states now, taking values in truth values, namely the very composition of these two kinds of morphisms $$1 \stackrel{P(\psi)}{\to} \Omega := 1 \stackrel{\psi}{\to} V \stackrel{P}{\to} \Omega \,.$$

Notice that this is not really specific to physical theories. It is rather just the mere minimum of structure to reason about anything at all: to make propositions.

Do we need more than that?

Chris Isham and Andreas Döring propose that the right topos in which to find the space of states of a quantum mechanical system with Hilbert space $H$ is the topos of presheaves on the category of abelian subalgebras of the bounded endomorphism algebra on $H$, with their inclusions.

As also Bas Spitters and Chris Heunen and confirm from a different perspective, this topos contains a rather interesting and useful object usually called $\Sigma$: the presheaf which sends each abelian subalgebra to the collection of algebra morphisms from there to the ground field.

Chris Isham and Andreas Döring suggest that this $\Sigma$ is the right model for the space of states of the given quantum system, in the above sense.

But is it? How do we decide this?

Here we decide this easily: we already know what the collection of states should be, in order for everything to make sense: it should just be the collection of unit length elements in $H$.

So, by the above, we should check if the morphisms $\psi :1 \to \Sigma$ from the terminal object into $\Sigma$ are in bijection with the elements of $H$.

But that fails dramatically, in general: if $\mathrm{dim}(H) \gt 2$ one finds that there is no morphism $1 \to \Sigma$.

this fact, the absence of any states in $\Sigma$, has been realized and emphasized by Chris Isham and J. Butterfield, to be equivalent to what is known as the Kochen-Specker theorem.

So do we need more that that?

At this point there are precisely two options:

a) We remain convinced that $\Sigma$ is the right thing to think of as the space of states of the quantum system with Hilbert space $H$. In that case, we have to conclude that all of the above nice abstraction of the concept of propositions is not a good abstraction of the concept of propositions after all.

b) We conclude that $\Sigma$ is not the object of states of the quantum system with Hilbert space $H$. In that case we have to search another space of states object, hence another pointed topos (which might, or might not, be the same topos but with another singled out object of states).

Chris Isham and Andreas Döring go for a). They have the intuition that this $\Sigma$ should be regarded as an object of states after all, and that hence the formalism needs to be adapted to suit those needs.

The propose a more flexible generalization of the above notion of propositions, states and their pairing. I will not recall what that generalization is in detail. I will just note that given this generalized notion of state objects, they prove that there is an injection of the elements of $H$ into the generalized elements of their state object.

I think it is clear, and that Chris Isham and Andreas Döring agree, that this inclusions is not, in general, onto.

I summarize this situation then as follows: while with the former definition of state objects, the one described above, $\Sigma$ had fewer elements than $H$ (namely none, in general), with the Isham-Döring flexibilized generalizatoin it has more elements that $H$.

Hence it is still not the space of states we want to see, is it?

I talked about that with both Chris Isham and Andreas Döring, and they agree that this issue requires more attention.

I will now make a suggestion, which seems to me to be natural, given the state of affairs. My suggestion is:

The original notion of properties on spaces of states, the one described abobe, is nice and crisp and elegant and powerful – hence good. Instead of modifying that we should rather question the intuition that $\Sigma$ deserves to be taken as an object of states. $\Sigma$ clearly plays some important role, but maybe not as an object of states. Rather, we should look for a pointed topos such that its object of states does have elements in bijection with the collection of states that we want to see. Only if we fail to find any such pointed topos at all can we be sure that the neat, crisp definition of state objects needs to be modified.

In the remainder I’ll follow that suggestion.

What is the space of states of the $n$-particle?

It strikes me that since a while, long before I appreciated the concept of a collection of states in a topos as a collection of morphisms from the terminal object, I kept emphasizing that the space of states of a quantum system arises as a collection of generalized elements of this form. Now I will argue that this is not a coincidence.

The general idea I gave a talk about at the Fields institute: Quantum 2-States and sections of 2-vector bundles and further elaborated on it in a couple of posts concerned with the setup that I am referring to as the charged $n$-particle, where the notion of state is intimately related to the notion of section of a $n$-bundle with connection using the general logic described in tangent categories and Sections, states, twists and holography.

The “charged $n$-particle” is supposed to be the abstract concept of the class of physical systems which describe

- an $n$-dimensional particle (familiar to many as an “$(n-1)$-brane”) $\mathbf{par}$

- propagating on a target space $\mathrm{tar}$

- where it couples to a background field $\mathbf{tra}$ (for “transport”) exists, which is a morphisms $$\mathrm{tar} : \mathbf{tar} \to \mathrm{phas} \,,$$ where $\mathbf{phas}$ is some object of phases.

The charged $n$-particle is supposed to subsume all physical systems known as $\sigma$-models (including their “non-geometric phases”) but also, crucially, all gauge theories. An $n$-dimensional gauge theory can be regarded as a charged $n$-particle propagating on some kind of classifying space $\mathbf{B} G$. In the sigma-model case the “background field” is just that: what people call a background gauge field that the $n$-particle is charged under, in the gauge theory case the “background field” is what is often called the “twist” of the gauge theory.

Be sure to follow the change of perspective here which that might mean for you: to amplify, when the gauge theory in question is for instance gravity, many people like to call it “background free”, but that’s a different use of the notion of background, namely one on parameter space $\mathbf{par}$, whereas I am talking on a “background” structure on target space $\mathbf{tar}$.

But notice that it is not an accident that there is danger of confusion here, but it is a feature: there is a notion of second quantization of the charged $n$-particle, and it sends a charged $n$-particle with parameter space $\mathbf{par}$ target space $\mathbf{tar}$ to a charged $n'$-particle with parameter space $\mathrm{tar}$.

Anyway, that’s not the topic right now.

The states of the charged $n$-particle.

Given a charged $n$-particle setup as above, internalized into some suitable context, we obtain the following secondary notions:

the configuration space of the $n$-particle is $$\mathbf{conf} := hom(\mathbf{par},\mathbf{tar}) \,,$$ the transgression of the background field to configuration space is the morphism $$tg(\mathbf{tra}) := hom(\mathbf{par},\mathbf{tra}) : conf \to hom(\mathbf{par},\mathbf{phas}) \,.$$

In order to focus on where the phenomena of interest right now happen, let’s maybe assume for simplicity that parameter is just the point, so that $\mathbf{conf}$ coincides with $\mathbf{tar}$ and such that the transgressed background field is precisely the original background field.

Then, what is a state of the system? This is the issue I kept going on about in Talks at “Higher categories and their application” and in Sections, states, twists and holography and elsewhere: when $\mathbf{phas}$ is monoidal, then so is $hom(\mathbf{tra},\mathbf{phas})$. A state is precisely a generalized element of the background field $\mathbf{tra}$ with respect to this monoidal context, i.e. a morphism $$\array{ & \mathbf{conf} \\ \multiscripts{^1}{\swarrow}{} && \searrow^{\mathbf{tra}} \\ & \stackrel{\psi}{\Rightarrow} \\ \searrow && \swarrow \\ & \mathbf{phas}. } \,,$$ where here $1$ denotes the tensor unit in that hom-category.

(There is another issue here which I have discussed elsewhere at length, but shall gloss over here for not to obscure the point of interest: states are not, in general, morphisms into $\mathrm{tra}$ itself, but into its curvature. But this is irrelevant to the point in question here.)

A topos theoretic state object for the charged $n$-particle?

So this gives a collection of states of the charged $n$-particle. And, crucially, we know that this is the right collection of states. (See for instance The story of quantizing by pushing to a points, Chan-Paton bundles, etc.).

Does it also yield a state object in the sense of pointed topoi as described above.

Not in general, I think. Here is a question to the topos-theory experts:

Question Under which conditions on $D$ is $$hom(C^{op},D)$$ a topos?

I assume it is sufficient that $D$ itself is a topos??

If that’s true, and if $D$ isn’t a topos in the first place, we can always use the Yoneda embedding $$D \hookrightarrow Set^{D^{op}}$$ and embed $D$ into the topos of presheaves on it.

But for $D = \mathbf{phas}$ as above, that may spoil its monoidalness, which was used in the above definition of a space of states.

But we can handle that. Recall, for instance from the discussion at $n$-Curvature that if our background field $\mathbf{tra}$ is a principal $n$-bundle with connection, it is determined by

- a cover $\pi : P_n(Y) \to \mathbf{tar}$

of target space

- a $G_{(n)}$-valued cocycle

$g : Y \times_X Y \to \mathbf{B}G_{(n)}$

for $G_{(n)}$ an $n$-group, determining the bundle structure,

- and the connection itself, which is an $(n+1)$-functor

$$\mathrm{curv} : P_{n+1}(Y \times_X Y) \to \mathbf{B} \mathrm{INN}(G_{(n)})$$

with values the inner automorphism $(n+1)$-group

- and finally a morphism

$\mathbf{tar} \to \mathbf{B}\mathbf{B} G_{(n)}$ whose precise nature is easily understood only when either $G_{n()}$ is sufficiently abelian such that the double supsendion exists in the ordinary sense, or else after we pass from Lie $n$-groups to their corresponding Lie $n$-algebras, where $\mathbf{B}\mathbf{B} G_{(n)}$ is modeled by the dg-algebra $inv(g)$ of invariant polynomials on the Lie $n$-algebra $g$ of $G_{(n)}$ as described at Lie $\infty$-connections and their application to String- and Chern-Simons transport.

These three pices of data have to fit into a diagram $$\array{ Y \times_X Y &\stackrel{g}{\to}& \mathbf{B} G_{(n)} \\ \downarrow && \downarrow \\ P_n(Y \times Y) &\stackrel{curv}{\to}& \mathbf{B}\mathrm{INN}(G_{(n)}) \\ \downarrow && \downarrow \\ P_n(X) &\stackrel{}{\to}& \mathbf{B}\mathbf{B} G_{(n)} }$$ whose meaning is discussed at length in Lie $\infty$-connections and their application to String- and Chern-Simons transport as well as in the slides On String- and Chern-Simons $n$-Transport.

Anyway, the point is that a section of this thing is a trivialization of the middle arrow $\mathrm{curv}$, which always exists since $\mathbf{B}INN(G_{(n)})$ is contractible. (The rest of the diagram above ensures that there is still interesting information on $curv$: we can regard the entire diagram as a pullback of the $n$-groupoid version of the universal $G_{(n)}$-bundle and the trivializablility of $curv$ then corresponds to the contractibility of the total space of that universal bundle, see the end of inner automorphism $(n+1)$-group.)

As described in Arrow-theoretic differential theory, such a trivialization is a morphism into $curv$, regarded as taking values in $\mathrm{AUT}(INN(G))$ as usual, from the functor that sends everything to the point $\{\bullet\}$.

It’s here that we see the connection to the topos-theoretic state more clearly: instead of mapping from a tensor unit into out background field, we really map from the terminal object now. $$\array{ & \mathbf{conf} \\ \multiscripts{^1}{\swarrow}{} && \searrow^{\mathbf{curv}} \\ & \stackrel{\psi}{\Rightarrow} \\ \searrow && \swarrow \\ & Set^{\mathbf{Grpd}^{op}}. } \,.$$

Then, by the general nonsense on presheaf categories, we know that a “subobject classified transport functor” $$\array{ \mathbf{tar} \\ \downarrow^{\Omega} \\ Set^{Grpd^{op}} }$$ exists.

(If I am right that $hom(\mathbf{tar}, Set^{Grpd^{op}})$ is a topos, that is. Am I??)

So this way the entire setup of the “charged $n$-particle” becomes internal to the topos theoretic framework that Chris Isham and Andreas Döring are emphasizing is useful for interpreting quantum mechanics.

We now know that a proposition about the states of the charged $n$-particle is a morphism $$\array{ & \mathbf{conf} \\ \multiscripts{^{\mathbf{tra}}}{\swarrow}{} && \searrow^{\Omega} \\ & \stackrel{P}{\Rightarrow} \\ \searrow && \swarrow \\ & Set^{Grpd^{op}}. }$$ from the background field to the subobject classifier transport, and each such proposition can be applied to a state of the $n$-particle to yield a generalized truth value in the generalized elements of that subobject classifieer transport.

And the state object $$\array{ \mathbf{conf} \\ \downarrow^{\mathbf{tra}} \\ Set^{Vect^{op}} }$$ internal to our topos would be neither to large nor to small, but just right, and we#d be just using the simple elegant standard pairing of states and propositions by mere composition.

But, on the other hand, the above notion of states is now picking out states regarded as section of the principal bundle underlying the background field, or rather of its version where the fibers have been replaced by the corresponding action groupoids.

There are two possibilities to move that setup back to the vector-like setup which we expect to see in quantum physics:

- the straightforward but less ambitious one is: we pick any linear representation $$\rho : \mathbf{B}G \to Vect$$ which induced a corresponding 2-representation $$\hat \rho : \mathbf{B}INN(G) \to Baez-Crans 2Vect$$ and then form the associated transport everywhere by hitting everything in sight with $$Hom_{INN(G)-Act}(--, \hat(\rho)(\bullet))$$ (I am indebted to Bruce Bartlett for emphasizing to me that this is the, by far, most elegant way for talking about associated $n$-bundles!).

This moves everything to the linear kind of sections that we actually expect. The downside is that under this operation the terminal object in our topos is no longer terminal, but instead again the tensor unit in some category.

- The more ambitious approach which I hope will eventually work out is the one suggested by the Baez-Dolan-Trimble groupoidification program applied to geometric representation theory, which more or less tells us that down on the fundamental level, we should replace vector space by action groupoids anyway. From that point of view the fibers $$INN(G) = G//G$$ of $curv$ would already be regarded as vector spaces with a $G$-representation on them, in some sense.

Summary

Let’s see, what did I say. I said:

- the internalization of state spaces in terms of pointed topoi is pretty cool;

- before generalizing that to something less concise, let’s make sure we have tried to find a pointed topos which does describe quantum state spaces on the nose;

- notice that the conception of quantum state spaces in the context of the “charged $n$-particle” is already formally very close to the topos theoretic one;

- except that there is an issue whether or not we regard states as morphisms from an object which is

a) the terminal object

or

b) the tensor unit.

- In the first case we can identify the space of sections of our $n$-bundle nicely with a topos-theoretic state space – at the cost of not having a vector space of states;

- in the second case we have the familiar vector space of states, but did move away from the standard topos theoretic definition of state by replacing the terminal object with the tensor unit;

- but, finally, the groupoidification program applied to geometric representation theory might actually turn the apparent problem with the first case into a feature. I am deeply in love with this idea, but not sure yet if it will work out.