January 9, 2008

The Concept of a Space of States, and the Space of States of the Charged n-Particle

Posted by Urs Schreiber I’ll talk about the topos-theoretic approach to the notion of the space of states of a physical system, recall the proposed answer by toposophers Döring, Isham, Landsman and others, suggest a simplified proposal and discuss it for the generalized $\sigma$-model class of physical systems which I am referring to as the charged quantum $n$-particle.

I’ll start with a detailed introduction that is supposed to make the discussion self-contained. The contribution that I would like to really discuss here with $n$-Café-readers is the last third, which starts with the paragraph title A topos theoretic state object for the charged $n$-particle?

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

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

- 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.

Posted at January 9, 2008 9:58 PM UTC

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Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle Hi Urs

It was good meeting you for the first time yesterday.

I also just had a look at this $n$-café contribution and now understand better your question to me. Within the context of what Andreas and I are doing there is a simple answer to your query: which is to take the pseudo-states that I mentioned briefly yesterday.

It is the pseudo state (which is essentially what we formally called a ‘truth object’) that does what you want, and this is a type of minimal subobject of $\Sigma$. It is the closest one can get to the non-existent global elements of $\Sigma$. As I remarked briefly yesterday, in the classical case this is fine since the statement $s\in K$ is equivalent to $\{s\}\subset K$ where $K$ is any subset of $S$ and hence it is quite ok to think of microstates as being minimal (but non empty) subobjects of $S$!

This is explained in detail in a monster encyclopaedia article that Andreas and I are currently writing but when this will finally hit the archives I am not sure. If you like, I could cut out the pseudo-state chapter and send that to you separately.

Very best regards

Chris

Posted by: Chris Isham on January 10, 2008 2:32 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Dear Chris,

it was also for me very enjoyable being at the workshop and in particular meeting you. I wish I could have joined you for the “pub session” in the evening.

It was good meeting you for the first time yesterday. I also just had a look at this $n$-café contribution and now understand better your question to me. Within the context of what Andreas and I are doing there is a simple answer to your query: which is to take the pseudo-states that I mentioned briefly yesterday.

Okay, thanks. I’d have to learn more about how pseudo-states work in detail. I think my question was trying to ask if we are sure that we need to pass from states to pseudo-states in order to find a good state space object, or whether maybe the ordinary notion of states is sufficient if we do not insist that $\Sigma$ plays the role of the state space object.

I may be very wrong about this. I am just wondering.

This is explained in detail in a monster encyclopaedia article that Andreas and I are currently writing but when this will finally hit the archives I am not sure. If you like, I could cut out the pseudo-state chapter and send that to you separately.

I would certainly be interested! Thanks!

Posted by: Urs Schreiber on January 10, 2008 2:35 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Urs wrote:

Question: Under which conditions on $D$ is $hom(C^{op}, D)$ a topos? I assume it is sufficient that $D$ itself is a topos??

(We’re talking about Grothendieck toposes here, I guess.) Sure, that’s sufficient. I’m almost tempted to say it’s necessary as well, in the somewhat lame sense that if someone claimed to have a category $D$ which was not a topos but where $D^{C^{op}}$ was a topos, my eyebrows would shoot up in surprise. I’m not sure how easy it would be to justify my surprise.

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.

I’m not quite getting this. By ‘monoidalness’, you mean that $\mathbf{phas}$ carries a monoidal structure? Which is getting lost in the translation to presheaves?

If so, would the Day convolution product on the presheaf category, that is induced by the monoidal product on $D$, be at all relevant to this matter?

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

My knee-jerk reaction is, “sure.” But then I worry a little: the exponent is the category of all groupoids, with no bounds on size? That would give me pause – I’d fret a little over cartesian closure (and I must admit, I find set-theoretic issues very annoying to consider; I should bite the bullet and sort out some of these issues once and for all, some day).

Posted by: Todd Trimble on January 10, 2008 2:41 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

I’m not quite getting this. By ‘monoidalness’, you mean that phas carries a monoidal structure? Which is getting lost in the translation to presheaves?

Yes, exactly!

(You did get it! :-)

But I am quite aware that my formulation is not good. I was really getting into a hurry when writing that. Originally I had intended to write a detailed nice exposition of that idea which I wanted to share. I started typing on the Eurostar. When it got out of the tunnel and Brussels was approaching I found myself working on the “review” part at the beginning of my entry. I barely made it to the end of the entry before I had to jump on the next train which took me out of WLAN reach. )

If so, would the Day convolution product on the presheaf category, that is induced by the monoidal product on $D$, be at all relevant to this matter?

Ah, invaluable! That sounds promising. I’ll have to remind myself however, how exactly that works. Hm, I should be able to figure that out…

I find set-theoretic issues very annoying to consider; I should bite the bullet and sort out some of these issues once and for all, some day

John’s general advice was

it doesn’t pay to worry too much about classes.

The general attitude even seems to be:

there is no need to ever worry about classes. Since whenever you would find yourself run into a problem, you simply utter “universe” and everything is as before.

I must admit that I have never been formally educated on the kind of set-theoretic reasoning needed here, and it all remains always pretty mysterious to me.

I could make a couple of ever more shocking confessions here, concerning set theory. At the heart of it seems to be the most shocking of all:

If I really think about it, I realize that I don’t know what a set is.

Posted by: Urs Schreiber on January 10, 2008 3:11 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

John’s general advice was

it doesn’t pay to worry too much about classes.

The general attitude even seems to be:

there is no need to ever worry about classes. Since whenever you would find yourself run into a problem, you simply utter “universe” and everything is as before.

Believe me, I am in complete sympathy both with the advice and with the general attitude. For me the ideal would be to have a totally laid-back attitude toward such matters, but also, whenever some smart-aleck comes along and starts making trouble with set-theoretic quibbles, to be completely prepared to deal with them on the spot and with calm, professional, grown-up dispatch. I don’t feel I’m there yet – I’m still a kind of adolescent who just wants to whine that I don’t want to be bothered by such things (while recognizing that some of the issues here are not quite so spurious after all).

If I really think about it, I realize that I don’t know what a set is.

I’m with you there, too. But I’m in no mood to get into one of these wars over Platonism and such. (Shut up and calculate! :-) )

Posted by: Todd Trimble on January 10, 2008 5:53 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Urs said:

there is no need to ever worry about classes. Since whenever you would find yourself run into a problem, you simply utter “universe” and everything is as before.

And then Todd said:

Believe me, I am in complete sympathy both with the advice and with the general attitude. For me the ideal would be to have a totally laid-back attitude toward such matters, but also, whenever some smart-aleck comes along and starts making trouble with set-theoretic quibbles, to be completely prepared to deal with them on the spot and with calm, professional, grown-up dispatch. I don’t feel I’m there yet – I’m still a kind of adolescent who just wants to whine that I don’t want to be bothered by such things (while recognizing that some of the issues here are not quite so spurious after all).

I’m also very sympathetic, but recently my heart has hardened a bit. But it’s only happened over the last few months, so I can’t say that I’m sure I’m right. My point of view now is that there are situations (probably many of them) where there are genuine set-theoretic issues and you can’t get around them by saying the magic word ‘universe’. (I’m speaking in general here. I wasn’t actually following the discussion that led to this.)

For instance, suppose you have a functor $F:C\to D$ that preserves all limits. Then $F$ has a left adjoint as long as a certain set-theoretic condition is satisfied. But what if it’s not? You might say Damn the torpedoes, we’ll just take a larger universe $V$ and look at the $V$-version of our category $C$. Call it $VC$. Then we’ll have a functor $L:D\to VC$ which is essentially the left adjoint of $F$. Here I’m assuming $C$ is defined in some reasonable way in terms of sets—in other words, sets in our original universe.

But you don’t yet have an adjoint pair, because you’ve changed $C$ to $VC$. Now usually you’ll be able to $V$-ify $F$ by just putting “$V$-” at the right places in the original definition. But then the new functor $VF$ will take values in $VD$, the $V$-ified version of $D$, and then $VF$ won’t have a left adjoint for the same reason as before but $V$-ified. Obviously taking an even bigger universe will create the same problem but another level higher.

You might say OK we can’t have a true pair of adjoint functors but you’ll have two functors $L: D\to VC$ and the original $F$ which are pretty close. For most applications, I probably wouldn’t disagree, even though inside I might be disappointed that you’d settle for anything less than a true adjunction. But now suppose you want to iterate the functor $LF$, maybe because you want to do a bar construction. You can’t because $LF$ goes from $C$ to $VC$. You could probably keep choosing bigger and bigger universes $V_n$, and then make $n$-th “iterates” $C\to V_{n}C$. Then there are questions like How sensitive is what you did to your choice of the universe $V$? In fact, there are pages in SGA 4 devoted to these things.

I don’t know about you, but all this seems like the mathematical equivalent of postulating the existence of larger and larger physical universes (Grothendieck had a real knack for naming things), each one of which is inaccessible from the previous one. But we shouldn’t push our problems off on the gods living in the next universe—we should solve them in this one!

In my (single) experience with such things, the solution was to replace the categories in question with slightly smaller versions of themselves, in which all the objects are suitably accessible. (The categories were sheaf categories on large sites.) Then you have to show that $F$ preserves this property and that $F$ restricted to the small categories has an adjoint. This probably requires some work, but that’s good! It’s much better to do an honest day’s work and get a paycheck than to pretend your problems will be solved tomorrow!

Another benefit of doing this is the usual one in ironing every wrinkle out of the foundations: your attention is focused on the truly essential things, rather than things that are merely closely related.

Posted by: James on January 12, 2008 3:36 AM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

If I really think about it, I realize that I don’t know what a set is.

I’ve come to think of sets as given by objects and an equality on these objects. In fact this is how Errett Bishop defines sets in constructive mathematics («Foundations of constructive analysis», 1967) : for him a set is given by three components :

1. a description of how to construct its elements
2. a description of how to prove that two elements are equal
3. a proof that this equality is an equivalence relation.

If there is only the description of the elements, without the equality, it is called a preset (or a type in type theories, used to formalize constructive mathematics; in these type theories, sets are sometimes called setoids). See, for example,

This is of course very different from the concept of “set” in the cumulative hierarchy of sets of ZF or BG; ZF-sets are in fact subsets of a universe, with absolute everywhere-defined equality and membership relations. This seems to me a much too strong assumption. With Bishop’s sets, each set has its own equality, which is defined by the “user”, and there is no way to compare elements of independent sets, and there is no way to define equality of sets by extensionality.

Bishop’s way of doing set theory is astonishingly categorical. Bishop distinguishes operations and functions. Operations (or rules) which associate to each element of a type (or preset) an element of another type. Functions are operations between sets which preserve the equality (like a functor preserves the arrow-structure). He defines subsets in the categorical way: a subset of a set A is an injection into A. And the quotient of a set A by an equivalence relation R is the set with the same elements as A, but where the equality is replaced by R : to identify two elements, one add an “equality” between them. This is much similar to quotient categories, where to identify two objects one adds an isomorphisms between them.

In fact Bishop’s notion of set is categorical, in the sense that it is stable under equivalence : if A is a set viewed as a discrete category, any category equivalent to A is also a set.

In constructive type theories, categories are defined as a type of objects together with, for each pair of objects A, B a set of arrows from A to B, plus composition and identities. See for example “Bishop’s set theory” or

What is important for me (and for Makkai :

is that we do not include in the definition an equality on objects, i.e. in general a category do not have a set of objects (thus there is a difference between general categories and internal categories in Set, which do have a set of object). Then the problem that a category of all sets should have a set of objects, which would be a set of all sets, disappears. I think this way of defining categories could free us of these size issues. The quantitative distinction between small and large categories is replaced by a qualitative distinction between categories with (internal categories) or without (general categories) a set of objects. Nikolai Durov (chapter 9) seems to have a similar point of view.

Unfortunately, type theorists keep making size distinctions between types (which I find difficult to understand, since there is no way to compare the size of two types, since injections or bijections need an equality to be defined), and use small types for the sets and big types for the categories.

Posted by: Mathieu Dupont on January 10, 2008 10:03 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

This sounds very interesting, Mathieu. Let me ask a naive and perhaps stupid question (which maybe I could figure out myself if I thought hard about it).

Something that is conceptually fundamental in category theory is the ability to take presheaves on a general category: the construction $Set^{C^{op}}$ (plus the Yoneda embedding, etc.). Is this sort of construction “permissible” in the general abstract framework you and Makkai advocate?

Posted by: Todd Trimble on January 10, 2008 10:40 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Without any constraint on what can be done with such sets and categories, yes. There is a set of natural transformations between two presheafs, since we can define an equality on the type (or preset) of natural transformations. So there is a category of presheafs. You could also “prove” that the limit of any functor $F:D\to \mathrm{Ens}$ exists, without restriction on $\D$ : it is the set of natural transformations from the constant functor to 1 and $F$ (i.e. the set of cones). So the category of sets would be (completely) complete. Freyd’s proof that a small-complete small category (or a large-complete large category) is a lattice doesn’t apply to this situation, because it uses a product indexed by a set larger than the set of arrows of the category. But the category of Bishop’s sets has no set of objects, so no set of arrows.

But a big problem remains : if a category $C$ is (completely) complete, then a generalization to complete categories of Knaster-Tarski theorem for complete lattices would implies that every functor $F:C\to C$ has a fixed point $Fx\simeq x$. In particular, if Set is complete, the covariant powerset functor would have a fixed point $\mathcal{P}x\simeq x$, which would contradict Cantor’s theorem.

So we must put somewhere a limitation. Reintroducing some kind of “size” limitation on sets could prevent this fixed point to be in Set, but perhaps something else could also do it.

In fact I’m inclined to consider that the subset functor goes intrinsically to Ord, (the category of orders, defined like sets, but without the symmetry of the relation, i.e. orders are types with an order relation [without antisymmetry, which has no meaning since it requires a preexistent equality]), and not to Set. I would be ready to put some kind of limitation which would prevent to define the powerset functor from Set to Set, and allow it only from Set to Ord. But this is a very radical solution : the category of sets wouldn’t be a topos anymore.

Posted by: Mathieu Dupont on January 11, 2008 2:16 AM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

we do not include in the definition an equality on objects, i.e. in general a category do not have a set of objects

Instead such a general category has a preset of objects, if I understand you correctly?

Over two years ago Toby Bartels told me about his thoughts about using presets to base category theory on and how he was thinking about whether or not that would help circumvent set theoretic issues.

Posted by: Urs Schreiber on January 10, 2008 10:54 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Yes, type = preset. I prefer to say type, because a type is not only a preset, but also a precategory, a pre-2-category, and so on. A priori, a type can bear any such structure.

Posted by: Mathieu Dupont on January 11, 2008 2:20 AM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Over two years ago Toby Bartels told me about his thoughts about using presets to base category theory on and how he was thinking about whether or not that would help circumvent set theoretic issues.

Unfortunately, I’ve decided that they don’t! Like Mathieu, I was thinking about Freyd’s argument that the category of sets can’t be completely complete, realising that it doesn’t go through (at least not naively) if equality of objects in inaccessible. Unfortunately, the Burali-Forti paradox still goes through, and Mathieu points out the Knaster–Tarski paradox (if I may call it that) as well.

I am definitely in favour of founding mathematics (this is the Hilbert-style notion of what foundations are) on presets rather than sets; right now, I’m considering a version of the Calculus of Inductive Constructions without identity types. And although this is associated with constructivism (see the title of Bishop’s book, for example), it works perfectly well in nonconstructive mathematics, although admittedly excluded middle can have odd consequences. (For example, in my CIC, it implies countable choice; how odd is that?)

Posted by: Toby Bartels on February 27, 2008 3:44 AM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Day convolution product on the presheaf category, that is induced by the monoidal product on $D$

By the way, this is something I was wondering about recently a bit, when I kept bothering Tom Leinster with questions about presheaves on $\omega Cat$:

the fact that $C$ is monoidal should stronly affect the point of view I have on presheaves on $C$. Shouldn’t it?

There is nothing like a monoidal functor $C \hookrightarrow Set^{C^{op}} \,,$ is there.

But if $C$ is closed monoidal , I had begun thinking that probably we then want to look at $C^{C^{op}}$ instead of $Set^{C^{op}}$ because the Yoneda embedding for closed monoidal $C$ would naturally take values in $C$.

So what’s the right way, generally, to look at this situation – presheaves on closed monoidal categories?

Posted by: Urs Schreiber on January 10, 2008 3:24 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Urs wrote:

the fact that $C$ is monoidal should strongly affect the point of view I have on presheaves on $C$. Shouldn’t it?

Um, yeah, I guess!

There is nothing like a monoidal functor $C \hookrightarrow Set^{C^{op}},$ is there.

Sure there is! That’s the beauty of Day convolution. An incredibly useful technique.

Let me start with a very simple example, to explain how this works. Let’s consider graded sets, i.e., functors $F: \mathbb{N} \to Set$, where $\mathbb{N}$ is the usual discrete category with monoidal product given by addition.

The convolution product in this case is given by the simple formula

$(F * G)(n) = \sum_{i + j = n} F(i) \times G(j).$

A slighly fancier way of putting it is

$(F * G)(n) = \sum_{i, j} F(i) \times G(j) \times hom_{\mathbb{N}}(n, i + j),$

and now Day convolution extends this idea, with a general monoidal category $C$ replacing $\mathbb{N}$.

Bluntly stated, the coend formula for the Day convolution product of two presheaves $F, G: C^{op} \to Set$ reads as follows:

$(F * G)(e) = \int^{c, d \in Ob(C)} F(c) \times G(d) \times hom_{C}(e, c \otimes d)$

where the tensor product appearing on the right is the monoidal product of $C$. The Day convolution $F * G$ has the nice property that it preserves colimits in the separate arguments $F$ and $G$, i.e., we have canonical isomorphisms

$colim_j (F_j * G) \stackrel{\sim}{\to} (colim_j F_j) * G,$

$colim_j (F * G_j) \stackrel{\sim}{\to} F * (colim_j G_j),$

where the colimits are taken over some category $J$. An abstract way of putting this is that the Day convolution is a certain tensoring or weighted colimit:

$(F * G)(e) = (F \times G) \otimes_{C \times C} hom(e, \mu(-))$

where $\mu = \otimes_C: C \times C \to C$ is the monoidal product. Tensoring with a module preserves colimits, as do the functors $- \times G$ and $F \times -$; therefore Day convolution preserves colimits in its separate arguments.

Since $F$ and $G$ are themselves (weighted) colimits of representables, we may as see how Day convolution behaves when applied to representables. This is directly relevant to your question about the Yoneda embedding being monoidal.

Let me break this down a little. The precise way in which a presheaf $F: C^{op} \to Set$ is a canonical colimit of representables is given by the formula

$F(-) = \int^c F(c) \times hom(-, c)$

which is one of a million manifestations of the Yoneda lemma, which (it may be no exaggeration to say) could very well be the deepest ‘triviality’ known to humanity. For those who like this sort of thing, we can express this as the statement that $hom_C$ is an identity 1-cell in the bicategory of categories and bimodules (aka profunctors) between them:

$F \cong F \otimes_C hom.$

Anyway, this coend expression of Yoneda is fantastically useful in calculations. It says that whenever you see homs in coend formulas, you can “Yoneda-reduce” according to the scheme

$\int^c F(c) \otimes hom(d, c) \rightsquigarrow F(d).$

[Aside: “Yoneda reduction” rhymes with “eta reduction”; I think of it that way.]

Okay, let’s put Yoneda reduction to use. Take $F = hom(-, c)$ and $G = hom(-, d)$ to be representable functors, and Day-convolve them:

$(hom(-, c) * hom(-, d))(e) = \int^{c^\prime} \int^{d^\prime} hom(c^\prime, c) \times hom(d^\prime, d) \times hom(e, c^\prime \otimes d^\prime).$

Apply Yoneda reduction once to the inner coend, to get rid of the dummy variable $d^\prime$:

$\int^{c^\prime} hom(c^\prime, c) \times hom(e, c^\prime \otimes d).$

Now apply Yoneda reduction again, to get rid of the other dummy variable $c^\prime$; we get

$hom(e, c \otimes d).$

So, we have

$hom(-, c) * hom(-, d) \cong hom(-, c \otimes d)$

which says precisely that the Yoneda embedding is a monoidal functor. If you layer on extra structure, e.g., if $C$ is symmetric or braided monoidal, then Day convolution inherits symmetric or braided monoidal structure, and the Yoneda embedding becomes a symmetric or braided monoidal functor.

All this stuff, including the coend calculations, should be in Brian Day’s thesis. (Well, not the braided stuff of course, but you get the idea.)

The really high level way to say all of this, which is given for example in the Baez-Dolan HDA paper on opetopes and $n$-categories, to the effect that for $C$ a small category, $Set^{C^{op}}$ is the free cocompletion of $C$, and if $C$ is a (symmetric, braided) monoidal category, then $Set^{C^{op}}$ equipped with Day convolution is the free (symmetric, braided) monoidal cocompletion of the (symmetric, braided) monoidal category $C$. This comment is getting pretty long, so I won’t try to explain all this precisely now.

But wait, there’s more:

So what’s the right way, generally, to look at this situation – presheaves on closed monoidal categories?

Well, in that case, the Yoneda embedding isn’t just a monoidal functor, it’s a closed monoidal functor! Let me break this down a little:

Even if $C$ is just monoidal, $Set^{C^{op}}$ with the Day convolution product is closed monoidal – actually biclosed monoidal. This is more or less because the functors $F \star -$ and $- \star G$, being colimit-preserving functors acting on presheaf catgories, have right adjoints; therefore we have

$Set^{C^{op}}(F * G, H) \cong Set^{C^{op}}(F, H/G) \cong Set^{C^{op}}(G, F \backslash H)$

where $H/G$ and $F \backslash H$ are notations for the left and right internal homs. There are weighted limit/end formulas for these internal homs.

Secondly, if $C$ is closed monoidal (say closed on the left side), then one calculate with the help of Yoneda reductions that the Yoneda embedding preserves the left closed structure.

So the picture is about as nice as it can possibly be. Dude: Day convolution rules!

Posted by: Todd Trimble on January 10, 2008 5:10 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

That’s very nice.

If one could obtain plugins for a brain as one can for a browser, I’d get me the Australian Enriched Category Plugin™.

The precise way in which a presheaf $F : C^{op} \to Set$ is a canonical colimit of representables is given by the formula $F(-) = \int^c F(c) \times hom(-,c)$

Ah, you mentioned recently elsewhere that every presheaf is a colimit of representables, and I had intended to ask you about that, but forgot.

Even though I realize that I need to refresh my mind once again about coends, this formula does make some sense to me.

Hm, okay, so presheaves over monoidal categories are monoidal and, better yet, are monoidal completions of the original ones. Very nice. That’s precisely what I’d need for the kind of argument I was using above.

But then I next ran into a problem with terminal objects versus tensor units. Maybe that also goes away by just talking to you a bit :-).

Let’s see. I guess the point is that with $C$ monoidal, there are two different natural monoidal structures on $hom(C^{op},Set)$:

- the one you just described to me, which is inherited from the monoidal structure of $C$

- the canonical cartesian monoidal structure on presheaves, inherited from the cartesian monoidal structure on sets.

Hm, what’s my question, really. Let me see. Suppose $C = Vect$ and consider $hom(Vect^{op},Set)$.

What are the two monoidal structures? Hm. The one coming from the tensor product of vector spaces using Day convolution must still correspond to the tensor product of vector spaces, by what you just explained.

But then there is the standard presheaf tensor product:

$X \times Y : U \mapsto X(U) \times_{Set} Y(U) \,.$ Evaluated on representables that just produces something like the cartesian product of the two sets underlying the representing vector spaces. Hm.

I need to think a bit more (and, first of all: sleep a bit more) before I can get back to you with more sensible comments.

But maybe one quick question which I couldn’t find an answer to at the London workshop on topos-physics:

Is $Cat$, regarded as a 1-category, a topos? If not, is $Grpd$?

Is $\omega Cat$ / $Str \infty Cat$ a topos?

I gather that somebody is working on $\infty$-topoi and am guessing that $\omega Cat$ must be an $\infty$-topos, if the concept is at all to be useul. But is it an ordinary topos?

Posted by: Urs Schreiber on January 10, 2008 8:43 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Some quick flash responses:

Is $Cat$, regarded as a 1-category, a topos? If not, is $Grpd$?

No. No.

(Although you shouldn’t don’t feel bad for asking; I’ve heard a rumor that Grothendieck also once asked whether $Cat$ were a topos.)

Is $\omega$Cat / Str$\infty$Cat a topos?

Nope – not an ordinary topos.

I don’t know what an $\infty$-topos is supposed to be, actually. (Is this a concept due to Jacob Lurie?)

If there’s a notion of $n$-topos, is that supposed to be a generalization of Mark Weber’s notion of 2-topos?

Yes, you’re quite right that one needs to be careful to distinguish various monoidal products on $Set^{C^{op}}$, if $C$ is monoidal. But it must be late over there; get some rest! :-)

Posted by: Todd Trimble on January 10, 2008 9:17 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Hi Urs,

Great to meet you at last, however briefly!

Is Cat, regarded as a 1-category, a topos?

If we’d known that, maybe we wouldn’t have had so much trouble working out what a sub-2-group was back in the Klein 2-geometry days.

We were told that Cat forms a 2-topos.

What are $\infty$-toposes (I though that had won over ‘topoi’) like?

Posted by: David Corfield on January 10, 2008 9:25 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

What are $\infty$-toposes (I though that had won over ‘topoi’) like?

I don’t know. All I know is that I had heard that some people, Jacob Lurie for instance, I guess, is talking about such things.

I guess an $n$-topos should be an $n$-category that “behaves like $(n-1)Cat$”.

Hm. Is the 2-category of all topoi a 2-topos?

Okay, we are not getting anywhere here until we find somebody who actually knows something about $n$-topoi. :-)

Unless we (re)invent it ourselves, of course.

Posted by: Urs Schreiber on January 10, 2008 10:58 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

I thought Lurie’s work was on $(\infty, 1)$-toposes. We’ll have Tom to answer to if we call them $\infty$-toposes.

Regarding 2-toposes, if Cat forms one of those, what is the equivalent of the subobjects of an object? Instead of a subobject classifier there is a discrete opfibration classifier. So what are the discrete opfibrations over a category, and in what way does such a thing resemble a subobject?

Fibres as sets rather than as truth values?

Posted by: David Corfield on January 11, 2008 9:02 AM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

I thought Lurie’s work was on $(\infty ,1)$-toposes. We’ll have Tom to answer to if we call them ∞-toposes.

You certainly will. Lurie calls them $\infty$-toposes: tut tut.

As I understand it, $\mathbf{CAT}$ is a 2-topos in the sense of Weber. (I don’t know about the sense of Lurie.) The analogue of the subobject classifier is $\mathbf{Set}$. For any set $A$, there is a canonical map $A \to 2^A$ sending $a$ to the characteristic function of $\{a\}$; the analogue one dimension up is the Yoneda embedding.

So if $2$ is replaced by $\mathbf{Set}$, truth values (elements of $2$) are replaced by sets. You can talk about variable truth values, e.g. the answer to “is it raining?” varies in space and time. Similarly, as you know, you can talk about variable sets. And a truth-value is a $(-1)$-category, whereas a set is a $0$-category. So it’s a very natural progression.

Posted by: Tom Leinster on January 11, 2008 5:08 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Can you tell me how exactly the pullback condition $\array{ S_0 &\hookrightarrow& S \\ \downarrow && \downarrow^{\chi_{S_0}} \\ 1 &\stackrel{true}{\to}& \Omega }$ for the subobject classifier is weakened as we go to 2-toposes?

I recall we once had a sdiscussion here somewhere about 2-subobjects, where David Roberts provided a bunch of helpful information, but I cannot find it right now.

Hm, I am not even sure. What is the set “true”? Must be the 1-element set, I hope.

Is there any good intuiitive way to think of set-valued truth values? Is that what you meant by “variable truth values”? I must say I do not understand what you mean by that. But I haven’t really thought hard about it either.

Last question: whatever a 2-topos is, I am hoping that all categories of prestacks with values in Cat qualify. Right?

Posted by: Urs Schreiber on January 12, 2008 11:18 AM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

We’re having a related chat over here.

Posted by: David Corfield on January 12, 2008 3:40 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Urs wrote:

Is there any good intuiitive way to think of set-valued truth values? Is that what you meant by variable truth values? I must say I do not understand

I’ll reply over on the 2-topos thread.

Posted by: Tom Leinster on January 12, 2008 5:14 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Dear Urs,

I am sorry to have missed what must have been a great event. (I was too busy becoming a father of two healthy sons!)

Regarding your question about the state space. Let me recall that in our framework states correspond to integrals on the internal spectrum. In the Hilbert space formalism these states need not be elements of the Hilbert space (pure states’) but correspond to trace class operators (states’). Although Sigma does not have have any points there are plenty of integrals on it. Perhaps this is relevant to the question you ask.

Bas

Posted by: Bas Spitters on January 11, 2008 4:03 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Hi Bas,

thanks a lot for your message.

in our framework states correspond to integrals on the internal spectrum.

That’s a good very point, addressing precisely the issue I was wondering about. I should have thought of that, since I did read your paper.

I have no time at all right this moment to look at this carefully, but let me quickly ask this question:

Can you find the “space of integrals on the internal spectrum” internally to the topos? I.e. can you find an object $S$ in the topos such that $Hom(1,S)$ is in bijection with integrals on the internal spectrum?

Posted by: Urs Schreiber on January 11, 2008 4:53 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Urs wrote:
Can you find the “space of integrals on the internal spectrum” internally to the topos? I.e. can you find an object S in the topos such that Hom(1,S) is in bijection with integrals on the internal spectrum?

Yes. This is precisely what I am saying. In my forthcoming paper with Thierry Coquand we do the following constructively:
Given a Riesz space (vector lattice) we define three compact regular locales:
* The spectrum
* The locale of valuations on the spectrum
* The locale of integrals on the Riesz space

The last two locales are homeomorphic. This is a strengthening of the Riesz representation theorem connecting integrals to measures.

In the paper you referred to Chris and I applied this result to the internal self-adjoint part of a commutative C*-algebra, which is a Riesz space. The spectrum we obtain is precisely the Sigma you referred to. The construction of the locale of integrals is completely internal.

Best,

Bas

Posted by: Bas Spitters on January 11, 2008 8:35 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

we define three compact regular locales:

* The spectrum

* The locale of valuations on the spectrum

* The locale of integrals on the Riesz space

[…]

The construction of the locale of integrals is completely internal.

Great. That sounds good. That’s what I am looking for.

You completely internalize commutative functional analysis and obtain non-commutative functional analysis, item by item. That’s great.

Just to be entirely sure:

you have a setup where there is, for any Hilbert space $H$, a topos $T$ with an object $S$ such that

- what’s usually considered states in the quantum mechnical system with Hilbert space $H$ corresponds bijectively to morphisms

$1 \to S$

??

If so, did you talk with Chris Isham and Andreas Döring about this particular point? Can you relate your object $S$, if indeed you have (as I gather you are saying you do) to the “pseudo-states” that Chris Isham mentions?

Posted by: Urs Schreiber on January 12, 2008 12:24 AM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

Dear Urs,

==
you have a setup where there is, for any
Hilbert space H, a topos T with an object S such that

- what’s usually considered states in the quantum mechnical system with Hilbert
space H corresponds bijectively to morphisms
1→S ??
==

That’s correct!

==
Can you relate your object S, if indeed
you have to the “pseudo-states” that Chris Isham mentions?
==

I have not worked out the precise connection with their work, (although it was an important source of inspiration!). If I am not mistaken it works as follows:
Given a state (trace class operator) we obtain an integral on Sigma internally. By our internal Riesz representation theorem we obtain a valuation on Sigma. We may thus consider the sublocales which have measure 1. I believe this is more or less what Chris Isham calls a “pseudo-state”
(One may even take the meet of all those sublocales. In a slightly different context, Alex Simpson calls this meet the locale of random sequences.)

To see why all of this works, we can consider a point of Sigma as Dirac measures on Sigma (apparently these do not exist), but general measures, “smoother” versions of these Dirac measures, do exist internally.

Bas

BTW: What is the right way to quote a message?

Posted by: Bas Spitters on January 12, 2008 8:43 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

What is the right way to quote a message?

Like this:

<blockquote>

… quoted matter …

</blockquote>

Posted by: Tim Silverman on January 13, 2008 1:47 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

It may also depend on which filter one uses. For ‘Convert Line Breaks’, it’s as Tim says. But for the MathML filter I normally use, one needs to put

<blockquote>

<p>

…quotation…

</p>

</blockquote>

For more information, check out the thread ‘TeXnical Issues’ (it’s not just about TeX), permanently in the sidebar under Random Past Entries.

Posted by: Todd Trimble on January 13, 2008 3:02 PM | Permalink | Reply to this

Re: The Concept of a Space of States, and the Space of States of the Charged n-Particle

I was too busy becoming a father of two healthy sons!

Congratulations!

Posted by: Tom Leinster on January 11, 2008 5:08 PM | Permalink | Reply to this
Read the post 2-toposes
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Excerpt: 2-toposes
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