February 8, 2007

Isham on Arrow Fields

Posted by Urs Schreiber

In

Chris J . Isham
A New Approach to Quantising Space-Time: I. Quantising on a General Category
gr-qc/0303060

the author considers the concept of an arrow field on a category.

I recall his definition, reformulate it in arrow-theoretic terms, discuss the connection of “arrow fields” to ordinary vector fields and describe how to generalize it to a notion of covariant transport of sections of bundles with connection.

Chris Isham defines (beginning of section 3.1) an arrow field on a category $Q$ as (my paraphrasing) a section of the source map. So it is an assigmnet $v : \mathrm{Obj}(Q) \to \mathrm{Mor}(Q)$ which sends each object $o$ in $Q$ to a morphism $o \stackrel{v(o)}{\to} t(v(o))$ starting at $o$.

Here is comment on that:

we know that something which assigns morphisms to objects is likely to be a natural transformation.

Since we want this assignment to be a section of the source map, it is likely a natural transformation starting at the identity functor $\mathrm{Id}_Q$ on $Q$ $\array{ & \nearrow \searrow^{\mathrm{Id}\;\;} \\ Q & \;\,\Downarrow^v & Q \\ & \searrow \nearrow_{\mathrm{exp}(v)} }$ and ending at some endofunctor of $Q$, which I have given the name $\mathrm{exp}(v)$.

Chris Isham goes on to note that there is a natural monoidal structure on such “arrow fields”: given two arrow fields $v_1$ and $v_2$, we obtain a new one by assigning to each object $o$ of $Q$ the composite arrow $o \stackrel{v_1(o)}{\to} t(v_1(o)) \stackrel{v_2(t(v_1(o)))}{\to} t(v_2(t(v_1(o)))) \,.$

Here is a comment on that:

this monoidal structure is the obvious monoidal structure on natural transformations of the above form $\array{ & \nearrow \searrow^{\mathrm{Id}\;\;} & & \nearrow \searrow^{\mathrm{Id}\;\;} \\ Q & \;\,\Downarrow^{v_1} & Q & \;\,\Downarrow^{v_2} & Q \\ & \searrow \nearrow_{\mathrm{exp}(v_1)} & & \searrow \nearrow_{\mathrm{exp}(v_2)} }$

Finally, in section 3.2, the author defines an action of arrow fields on objects in the obvious way: each arrow field maps an object $o$ to the target of its value at that object: $v : o \mapsto t(v(o)) \,.$

Here is a comment on that:

notice that this is the action of the restriction of the functor $\mathrm{exp}(v) : Q \to Q$ that is the target of the natural transformation $v$.

In the last paragraph of section 3.2, Chris Isham remarks that the action of arrow fields on objects of $Q$ is like the action of the diffeomorphism group $\mathrm{Diff}(\mathrm{Obj}(Q))$.

Using the above re-formulation in terms of natural transformations, we can make this precise as follows:

Let’s consider the identity component of $\mathrm{DIff}(\mathrm{Obj}(Q))$ and look at “smooth families” of arrow fields that contain the trivial arrow field.

So let $Q = P_1(X)$ be the path groupoid of a smooth space $X$, and let $\Sigma(\mathbb{R})$ be the the additive group of real numbers, regarded as a category with a single object.

Write $\mathrm{Flow}(X) \subset \Sigma(\mathrm{End}_\mathrm{Cat}(P_1(X)))$ for the sub-category of endomorphisms of $P_1(X)$ of the form $\array{ & \nearrow \searrow^{\mathrm{Id}\;\;} \\ P_1(X) & \;\,\Downarrow^v & P_1(X) \\ & \searrow \nearrow_{\mathrm{exp}(v)} } \,,$ where everything is smooth.

As we have just seen, each of these can be regarded as defining an “arrow field” on $P_1(X)$.

Notice how these arrow fields $v$ look like in this case: they assign to each point $x$ in $X$ a path $x \stackrel{v(x)}{\to} t(v(x))$ in $X$. Moreover, $\mathrm{exp}(v) : P_1(X) \to P_1(X)$ is the functor which sends any path $x \stackrel{\gamma}{\to} y$ to $\mathrm{exp}(v) : \left( \array{ x \\ \;\;\downarrow\gamma \\ y } \right) \;\;\;\;\; \mapsto \;\;\;\;\; \left( \array{ x &\stackrel{v(x)^{-1}}{\leftarrow}& t(v(x)) \\ {}^\gamma \downarrow\;\; \\ y &\stackrel{v(y)}{\to}& t(v(y)) } \right) \,.$ We might want to think of this as the adjoint action of arrow fields on morphisms.

Now, in order to get a smooth family of such, consider smooth functors $v : \Sigma(\mathbb{R}) \to \mathrm{Flow}(X) \,.$ These send $v : t \mapsto \array{ & \nearrow \searrow^{\mathrm{Id}\;\;} \\ P_1(X) & \;\,\Downarrow^{v(t)} & P_1(X) \\ & \searrow \nearrow_{\mathrm{exp}(v(t))} }$ in a way that respects translation on the real line: $v(t_1 + t_2) : x \mapsto x \stackrel{v(t_1)(x)}{\to} t(v(t_1)) \stackrel{v(t_2)(t(v(t_1)(x)))}{\to} t(v(t_2)(t(v(t_1)(x)))) \,.$ And smoothly so.

I think such smooth functors $v : \Sigma(\mathbb{R}) \to \mathrm{Flow}(X)$ are precisely in bijection with ordinary vector fields on $X$.

To me, this is an arrow-theoretic notion of vector field alternative to that used in synthetic differential geometry.

If “arrow fields” on $Q$ are to be thought of as vector fields, they should act on functions on $\mathrm{Obj}(Q)$ by some kind of translation.

At the beginning of section 4.2, Chris Isham considers the obvious action of an arrow field on a function: $v : C(\mathrm{Obj}(X)) \ni \psi \mapsto (o \mapsto \psi(t(v(o)))) \,.$

My final comment here shall be that this action of arrow fields on functions also has a nice arrow-theoretic formulation in terms of the above natural transformations. This formulation allows in particular to readily see how arrow fields act by covariant transport on sections of fiber bundles over $\mathrm{Obj}(Q)$, in the case where a bundle with connection over $Q$ is present. Such a covariant translation along arrow fields is conidered around equation (67) in Chris Isham’s text.

The relevant diagram, however, is a little hard to draw here. I am discussing it in section 1.2,

of the document which accompanies the discussion here.

Here is a snapshot of the end of that section:

Posted at February 8, 2007 5:38 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1154

Re: Isham on Arrow Fields

I’ll incorporate the following comments tomorrow, when I am less tired:

The entire discussion in section 4.3 of Chris Isham’s paper amounts to saying that with the morphisms of $Q$ regarded as paths between its objects (which is the relevant interpretation for Isham’s purposes), a functor $\mathrm{tra} : Q \to \mathrm{Hilb}$ is a hermitean vector bundle with connection on $\mathrm{Obj}(Q)$.

However, this is not the way Chris Isham puts it. The connection part he calls a “multiplier” and instead of thinking of a parallel transport functor he thinks of this functor as a presheaf.

I don’t really follow the reasoning behind this. Thinking of the above functor as a presheaf would imply thinking of $Q$ as a site. But this is not the role played by $Q$ in the rest of the discussion:

$\mathrm{Obj}(Q)$ rather plays the role of the configuration space of some physical system (and not of a site of open sets on that configuration space).

The discussion around equation (83) then amounts to noticing that a transformation $1 \to \mathrm{tra}$ is not just a section of this bundle, but a flat section.

This is a general issue, which I also do discuss for instance in that Rosetta stone section 1.2:

when we want to form the space of states in our arrow-theoretic formulation of quantum mechanics, we need to pass from the category of “configurations and paths between configurations”, which in my terminology is $\mathrm{cob} = [\mathrm{par},\mathrm{tar}] \,,$ the category of functors from parameter space into target space (for the present case the parameter space is simply the point $\mathrm{par} = \bullet$ so that $[\mathrm{par},\mathrm{tar}] \simeq \mathrm{tar} := Q$)

to the subcategory $\mathrm{conf} \subset [\mathrm{par},\mathrm{tar}] \,,$ the true configuration space, which contains only morphism between configurations that we want to regard as physically equivalent.

That’s the difference between all “cobordisms” (processes) between configurations (morphisms in $\mathrm{cob}$) and the morphisms in $\mathrm{conf}$, which are the processes that are “pure gauge”.

So, this is why I keep going on about why specifying a physical system arrow-theoretically involves not just choosing a target space $\mathrm{tar}$ and a bundle with connection $\mathrm{tra} : \mathrm{tar} \to \mathrm{phas}$ on that, but also a sub-category $\mathrm{conf}$ of the category of all processes in target space. That’s why I define # a physical system as a situation of the form $\left( \mathrm{par} \stackrel{\gamma \in \mathrm{conf}}{\to} \mathrm{tar} \stackrel{\mathrm{tra}}{\to} \mathrm{phas} \right)$ and then give a prescription for computing the space of states which involves first pulling back $\mathrm{tra}$ (Isham’s “multiplier”!) back to configuration space and then taking its sections.

All this is, secretly, the issue discussed around equations (83)-(84) in Chris Isham’s paper.

It’s a not a particularly complicated issue, but I think it pays to try to extract a clear picture here.

I am making this comparison with Isham’s ideas here also in the hope that it will help our mutual understanding, in the light of John’s remarks at the end of this comment.

Posted by: urs on February 8, 2007 9:13 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

Urs schreibt:

instead of thinking of a parallel transport functor he thinks of this functor as a presheaf.

I don’t really follow the reasoning behind this. Thinking of the above functor as a presheaf would imply thinking of Q as a site.

One doesn’t need the notion of a site for a presheaf. A presheaf (of categories, say) on a category $C$ is simply a pseudofunctor $C^{op} \to \mathbf{Cat}$. The site machinery is simply to see if compatible things paste over covers. That said, Isham’s POV is still a bit mysterious, since then the “mulitplier” is a presheaf on $Q^{op}$.

Posted by: David Roberts on February 9, 2007 12:49 AM | Permalink | Reply to this

Re: Isham on Arrow Fields

One doesn’t need the notion of a site for a presheaf.

Sure, right. But given any contravariant functor, you would not call it a presheaf unless you are going to think of its domain as a site.

My point is just this:

of course, since a presheaf is nothing but a contravariant functor, Isham is free to call his contravariant functor a presheaf – but it is not a natural point of view on the situation he considers. I think.

On the other hand, if we think of this functor (contravariant or not is not that important, since in standard applications the domain is a groupoid anyway) as the parallel transport of a bundle with connection, then everything falls into place.

Posted by: urs on February 9, 2007 7:38 AM | Permalink | Reply to this

Re: Isham on Arrow Fields

Hi everyone

One of my PhD students just drew my attention to this site. Perhaps I may add a few remarks to the discussion.

Firstly, there seems to be some confustion between a sheaf and a pre-sheaf: it is the former that requires a site; a presheaf on a category C is simply defined to be a functor from that category to the category Set, there is no need for a Grothendieck topology. At least, that is the definition that is used by all texts on topos theory:-)

When writing my paper I was well aware of the similarity to a connection. Indeed that was part of the original motivation. I saw my work as a big generalisation to a categorial context of Mackey’s work on induced representations of semi-direct product groups . There are two ways of seeing what a Mackey representation looks like. One is in terms of a principal bundle and associated vectors bundles: if you like, that is the ‘differential geometry’ route. Various (flat) connections can arise naturally in this approach.

However, there is another approach which Mackey himself favoured, and this is more functional analytic in form(in fact, I am not sure if Mackey even realised there was a fibre bundle version). In particular, even if the underlying principal bundle is non-trivial, there still exist measureable cross-sections (although, of course, no smooth ones). Mackey used such such to ‘de-bundlise’ (:-)) the construction and, when you do that, the ‘multiplier’ appears naturally from the representation of the isotropy group used in the representation of the semidirect product.

I decided to generalise Mackey’s, measure theoretical, approach rather than the differential geometry one because, in my case, the base space is the objects in the category, and this is far from being a manifold.

Actually, the referee of my paper remarked approvingly (I think:-)) that what I was really doing was constructing a vast generalisation of Mackey theory.

Of course, there are more ways of ‘skinning a cat’ as the old English proverb goes, and it is always good to look at something in more than one way.

Kind regards to all

Chris Isham

Posted by: Chris Isham on February 15, 2007 6:21 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

Dear Chris Isham,

many thanks indeed for your comment!

I very much like the philosophy of your approach of “quantizing on a category”. Essentially, one should consider a quantum system as living on the category of paths (“trajectories”, “histories”) in its configuration space.

Without really being aware of your work, I have spent lots of time lately thinking about something very similar #.

My motivation was mainly to understand how a functor characterizing some kind of parallel transport of some physical object gives rise to the functor that describes the quantum propagation of that very object.

For that reason I felt particularly sensitive concerning your remarks on that structure you call a “multiplier”. Therefore my corresponding remark.

You write:

Firstly, there seems to be some confusion between a sheaf and a pre-sheaf: […]

David Roberts raised that point already. I don’t really think there is any confusion concerning the math. But maybe there is some difference in the phiosophy concerning terminology. I had replied to David’s (and now your) remark here.

What I think is helpful is to consider the general idea of “quantizing on a category” for simple familiar examples, like the free non-relativistic particle. Understanding that well enough would help me with applying the same formalism to more ambitious tasks, like quantizing general relativity by considering a configuration space category of posets representing spacetimes, as you do in gr-qc/0304077.

For the single nonrelativistic particle propagating on some target space $X$, we would take the “category that we quantize on” to be the category whose objects are the points in target space (the configurations of our particle) and whose morphisms would be some suitable kind of paths (for instance smooth Moore paths) in $X$.

If the particle is “charged”, under an electromagnetic field, say, there would in addition be a parallel transport functor # in the game, which sends each such path to the corresponding phase change of the particle caused by that background field.

Given that, there is, I think, a canonical arrow-theoretic way to send this “classical” data to the corresponding quantum theory:

For instance the space of states of the particle would be obtained by restricting the parallel transport to constant paths in $X$ and then “pushing it to a point”. This produces # the space of sections of the (potentially nontrivial) vector bundle that represents the background field (a line bundle in the case of electromagnetism).

On that space of states, we can naturally act with certain correspondences, coming from restricting a piece of worldline of the particle to its endpoints. This does reproduce the familiar propagator of our charged particle #.

It is this setup, together with similar consistency checks for more interesting cases, that provides a certain confidence in the interpretation of the various abstract ingredients that go into “quantizing on a category”.

For instance, I believe this does shed light on the issue that you are dealing with in particular in gr-qc/0306064, where you notice that if the “category we quantize on” has Hom-sets with more than one element, then the interpretation of these morphisms as inducing momentum operators breaks down.

Judging from the example of the single particle mentioned above, a reason for this apparent problem is suggested: the morphisms correspond in general to arbitary paths that the “system” (here: the particle) may trace out as it evolves. Only the “infinitesimally short” such paths would induce momentum operators of the kind they appear in, for instance, equation (1.2) and (1.3) of gr-qc/0306064.

I believe this can be made precise as follows: as I indicate in the entry above, there is a way to identify the “arrow fields” that you consider with true vector fields in the case that we have smooth functors which I denoted $v : \Sigma(\mathbb{R}) \to \mathrm{Flow}(X)$ above.

One can see that these describe the flow along a vector field $v$ on $X$. These flows act on the states of our system in a natural way. Differentiating this action with respect to the single paramter defining the flow gives the familiar action of the momentum operator associated to the vector field $v$.

It is this differentiation process which evades the problem you mention, that many paths may go between to given points: as we differentiate, all paths starting at a given point in the limit look like tangent vectors emanating radially away from that point.

Of course all this applies directly only when we have a smooth structure on our category, which does allow us to perform this differentiation.

For the example of the free particle we do. For many other examples, too.

But we might imagine applying that formalism even to examples where the configuration space category of our system does not have a sensible smooth structure.

As far as I understand, this is certainly the case for the categories of posets that you consider.

I would still think, though, that we get “momentum operators” even in these cases from looking at group homomorphisms $\Sigma(G) \to \mathrm{Flow}(X)$ (noticing that the way the category $\mathrm{Flow}(X)$ is constructed it exists quite generally).

For instance, we might imagine that the configutration space category of our system is a discrete cell complex. The morphisms of the “category we quantize on” would be generated from edges of some underlying graph, say.

In such a situation, we might take $G = \mathbb{Z}$ the integers, and consider vector fields on configuration space to be group homomorphisms $\Sigma(\mathbb{Z}) \to \mathrm{Flow}(X) \,.$ Noticing that $\Sigma(Z)$ is the groupoid freely generated from the graph containing a single edge, we might demand this to even be a graph map.

If we do so, we do recover a discrete analog of differentiability: the above group homomorphism will send a single parameter step in $\mathbb{Z}$ to an “elementary” morphism in the category we are quantizing on. If the edges of the underlying graph for instance locally form a cubical lattice, there will locally only be unique paths between pairs of points and we again retrieve a well-defined notion of momentum operator.

I am aware that this example of “quantization on a graph category” might not be directly relevant to the categories of poses that you consider. But I did mention it to indicate how the intuition that can be gained from “quantization on a category” of the single point particle suggests how to deal with more general cases.

Well, or so I think. You have certainly thought longer about these issues than I have, so maybe you would rather disagree with what I say here.

Posted by: urs on February 16, 2007 11:23 AM | Permalink | Reply to this

Re: Isham on Arrow Fields

Allow me to join in the fun and make some comments too :-) Some of these comments are just rehashes of stuff I said earlier in response to John’s comments - these latter are certainly relevant for our current discussion!

Firstly, I found the idea of an “arrow field” on a category an interesting idea. It seems so simple, I thought it was strange I personally had never actually encountered this idea while reading about categories and physics. Thats embarassing for me - I’m sure the n-cafe hosts thought about these things when they were toddlers ;-)

Perhaps I hadn’t seen it before because its not a particularly invariant construction : two equivalent categories will have different monoids of arrow fields. For example, the action groupoid $G$ of $\mathbb{Z}_2$ on a two element set is equivalent to a one object groupoid $\mathbb{Z}_2$, while

(1)$Arr(G) = (\mathbb{Z}_2)^8, \quad Arr(\mathbb{Z}_2) = \mathbb{Z}_2.$

(I think!)

Anyway, I still find it a bit weird. Lets work it out in simple examples. Suppose a finite group $G$ acts from the left on a finite set $X$. As usual, one thinks of this via the action groupoid $G_G (X)$, [ Ed : how do you get calligraphic letters? ] whose objects are elements $x \in X$ with morphisms $g \cdot x \stackrel{g}{\leftarrow} x$ depicting the action of $G$ on $X$.

Now let $Arr (G_G (X))$ be Chris Isham’s group of arrow fields . An arrow field is a choice of an arrow emanating from each object $x$, and you compose them in the obvious way.

In our example, it consists of set-wise maps $s : X \rightarrow G$ with the composition law

(2)$s_1 * s_2 (x) = s_1 (s_2(x) \cdot x).$

This seems such a fundamental example I find it a bit emabarassing that I had never thought about this before : that the set of maps from $X$ to $G$ forms a group .

Secondly, suppose we’re given a finite groupoid $G$ and a representation $\rho : G \rightarrow Hilb$, which we think of as a vector-bundle-with-connection over $G$. (Hint to first-timers : everyone at the n-cafe knows that I go on and on about representations of finite groupoids, ever since my supervisor published a paper about these issues).

Given the representation $\rho$, we have the space of sections $\Gamma_G(\rho)$, and the space of flat sections $\Gamma_G^0(\rho)$. The former is what Chris Isham calls the ‘space of states’, at least in his first two papers.

Now the arrow fields $Arr(G)$ act on the space of sections $\Gamma_G (\rho)$. The corresponding thing which acts on the flat sections $\Gamma_G^0 (\rho)$ is the (objects of) the automorphism 2-group $Aut(G)$ of $G$. (The full automorphism 2-group acts on $Rep(G)$).

Thirdly, Chris Isham defines an inner product on the space states $\Gamma_G (\rho)$ by

(3)$(s_1, s_2) = \sum_{x \in Ob(G)} (s_1(x), s_2(x))$

where these latter inner products are taken in the Hilbert spaces $\rho(x)$. He mentions that “it is necessary to develop a proper measure on the set $Ob (G)$”.

Indeed! That is precisely what all of us at the n-cafe were talking about in the discussion of Tom Leinster’s Euler characteristic of categories.

In the case when $G$ is a groupoid, the appropriate measure to use is $\frac{1}{|x \rightarrow |}$. Thus the inner product on the space of sections should be modified to

(4)$(s_1, s_2) = \sum_{x \in Ob(G)} \frac{(s_1(x), s_2(x))}{|x \rightarrow |}.$

When $G$ is a poset, its also well known what measure to use (see Tom Leinster’s paper).

Finally, in Chris’s third paper he suggests that an alternative definition of the space of states of a category $G$ would be as functions on the space of arrows. This is nothing but $\mathbb{C}[G]$, using John’s notation from this post.

In conclusion : Chris Isham’s ideas mesh well with the stuff n-cafe folks have been talking about!

Posted by: Bruce Bartlett on February 16, 2007 6:58 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

For example, the action groupoid $G$ of $\mathbb{Z}_2$ on a two element set is equivalent to a one object groupoid $\mathbb{Z}_2$, while $\mathrm{Arr}(G) = \mathbb{Z}_2^8 \;\;\;\;\;\;\; \mathrm{Arr}(\mathbb{Z}_2) = \mathbb{Z}_2$

Isn’t $G$ the codiscrete category (precisely one morphism for every ordered pair of objects) on two objects? Doesn’t that give four different arrow fields on $G$?

it’s not a particularly invariant construction

I suggested to think of arrow fields as natural transformations starting at the identity endomorphism on our category. That’s manifestly invariant under equivalence. (You would probably enjoy to think of this as an element of the 2-trace of an endomorphism of our category :-)

“it is necessary to develop a proper measure on the set $\mathrm{Ob}(G)$”.

Indeed! That is precisely what all of us at the $n$-cafe were talking about in the discussion of Tom Leinster’s Euler characteristic of categories.

In the case when G is a groupoid, the appropriate measure to use is $\frac{1}{|x\to|}$

Hm, sure? Makes me a little nervous that the measure is supposed to be determined by topology alone. That’s not how it works for ordinary quantum mechanics! I may be wrong, but consider this:

In the case of ordinary quantum mechanics of a point particle, we would want the measure on the space of objects to be that coming from a Riemannian metric on that space.

Remarkably, for ordinary quantum mechanics we also choose a measure on the space of morphisms. Namely the Wiener measure (or rather its complex version) associated to the Riemannian metric on the space of objects.

Posted by: urs on February 16, 2007 10:19 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

Urs wrote:

Isn’t G the codiscrete category…?

Yes - thanks! Sorry, I suppose I meant $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ acting on a 2-element set. You know what I mean - two objects, four morphisms going out from each object.

You suggested thinking of arrow fields as natural transformations starting at the identity endomorphism. I suppose this only works when the category is a groupoid. Nevertheless its a very helpful point of view! Its weird : it says that an arrow field on a category is the same as a single arrow in another category. Heh its obvious I suppose, but one needs to enjoy the small things in life.

Posted by: Bruce Bartlett on February 17, 2007 12:45 AM | Permalink | Reply to this

Re: Isham on Arrow Fields

You suggested thinking of arrow fields as natural transformations starting at the identity endomorphism. I suppose this only works when the category is a groupoid.

It will coincide with the more component-based notion of arrow field as “section of the source map” if the category is a groupoid.

If the category is not a groupoid, transformations starting at the identity endomorphism may still exist, but there may be less of them than there are sections of the source map.

Its weird: it says that an arrow field on a category is the same as a single arrow in another category

Yes, but notice that to some extent this is just what happnes when you put that monoidal structure on arrow fields and then realize than any element in a monoid is equivalently an arrow in a category with a single object.

Posted by: urs on February 18, 2007 12:25 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

You suggested thinking of arrow fields as natural transformations starting at the identity endomorphism. I suppose this only works when the category is a groupoid.

It will coincide with the more component-based notion of arrow field as “section of the source map” if the category is a groupoid.

By the way, in this context, a natural transformation starting and ending at the identity endomorphism on our category would define a loop field on the category: a choice of loop over every object.

Posted by: urs on February 20, 2007 9:12 AM | Permalink | Reply to this

Re: Isham on Arrow Fields

I am at the airport on my way to Sweden, where I will visit Jens Fjelstad in Karlstad. On my way to the airport, I had the leisure to think.

Above, Bruce declared that the mysterious measure we have to invoke in order to do the path integral (when we “quantize on a category”) is always the weighting of the kind we have been discussing.

This is certainly true – and well known – for topological field theories, like Dijkgraaf-Witten theory.

Still, I objected to the statement in its generality, pointing out that for simple non-topological systems like the ordinary free particle, the measure instead derives from a measure induced by a (pseudo-)Riemannian metric on some target space.

Now I realize that this is not a contradiction at all! Both statements should be right.

The weighting should be the general “right” measure, because it somehow models the results one would obtain had one replaced $\mathbb{C}$ by $\mathbb{C}\mathrm{Set}$ and computed an ordinary colimit.

If, for instance, we assume that that the automorphism group at some vertex acts freely on these sets, then indeed their colimit has cardinality given by the original cardinality divided by the numer of automorphisms.

(There should be a better way to say this. I will re-read some stuff Tom Leinster recently wrote and then maybe try to say the above again, in more generality. Possibly I am only now beginning to fully appreciate the impact of Tom’s idea to the program of arrow-theoretic quantization.)

So, if we expect a certain measure to appear in a certain example, we should probably try to understand the true category of configurations of that example! For the right choice of category, the canonical “weighting”-measure should then coincide with the one we are expecting to see.

On my way to the airport I played around with various discretizations of the free particle, which were such that the canonical groupoid measure reproduced the Wiener measure on paths in an appropriate limit.

If everything is right, the Wiener measure should in fact be (maybe a suitable continuum limit) of a weighting on a category whose objects are parameterized paths in some Riemannian space and whose morphisms are certain automorphisms of these paths – for instance certain notions of reparameterizations.

I think this can be done, but none of the solutions I found looks particularly natural or elegant.

I imagine chances are higher that we find a nice natural structure for the relativistic free particle.

So, consider the poset $\mathrm{worldvol}_n = 1 \to 2 \to \cdots \to n$

as a model for a worldline (of “parameter length $n$”) of a particle.

Moreover, let $M^d$ be $d$-dimensional Minkowski space, choose some sub-lattice of that and consider it as a poset $\mathrm{tar}_{M^d}$ by demanding that $x \to y$ if and only if $y$ is in the future of $x$. (This is how Sorkin and Isham model spacetimes by posets.)

Now, the category of all histories $\mathrm{hist} = [\mathrm{worldvol}_n,\mathrm{tar}_{M^d}]$ is again a poset. An object is a timelike path of $n$ straight steps in spacetime.

A morphism beween two such paths with fixed endpoints describes how to push the source path closer to the lightcone. $\mathrm{hist}$ is itself a poset.

Question: What is the weighting on the subcategory $\mathrm{hist}_x^y$ of all paths whose endpoints sit at some fixed points $x$ and $y$ in target space? How does it behave asymptotically for large $n$?

(Guess what I am hoping the answer will be…)

Posted by: urs on February 22, 2007 9:02 AM | Permalink | Reply to this

Re: Isham on Arrow Fields

This conversation is way over my head, but something you said made me think you were somehow relating connectivity on a graph (via weighting) and geometry (via Reimannian measure). Did I totally miss the mark? That would resonate with a few thoughts stashed away in the back of my mind somewhere.

Posted by: Eric on February 22, 2007 2:27 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

The connection I’m thinking of is more like:

randomness -> geometry

but the weightings have an interpretation as generating “random motion on a graph”, so the neat link would be

weighting -> randomness -> geometry

Is that vague enough for you? :)

Posted by: Eric on February 22, 2007 2:31 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

Yes, somehow this amounts to something that we have vaguely been thinking of before: the structure of the morphisms induces a measure on the space of objects.

Above I was considering this phenomenon on the space of “paths” of a physical system. But if that is the way it seems to be, one would certainly be tempted to apply the same sort of reasoning to categories modeling spacetime itself.

So if we decide to model pseudo-Riemannian spacetime as a poset (with arrows connecting those points that can be connected by a timelike or lightlike travelling particle), i.e. if we specify a light cone structure on spacetime, then Tom Leinster’s formula (for the case that everything is finite at least) actually provides us, for free, also with a volume measure on that spacetime.

Let’s see, how was that: light-cone structure together with volume is almost the same structure as a (pseudo-)Riemannian metric. Right?

Posted by: urs on February 22, 2007 5:16 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

Urs wrote:

Question: What is the weighting on the subcategory $hist(x,y)$ of all paths whose endpoints sit at some fixed points $x$ and $y$ in target space?

After reading this, I tried my hand at solving it, but I sadly failed . It brings up the following questions about Tom’s paper:

(a) Given finite categories $X$ and $Y$ with weightings (and coweightings), is there a weighting on $[X,Y]$?

(b) Suppose a category $X$ has a weighting. What can we say about the existence of a weighting on some full subcategory $A \subset X$?

I’m sure I’m embarassing myself here … but I didn’t know the answer to these, and failed to find the answer explicitly written in bold letters in Tom’s paper. For (a), I tried the formula

(1)$w_{[X, Y]}^F = \sum_{x \in X} w_Y^{F(x)}$

which tries to calculate the weighting on $[X,Y]$ in terms of that on $Y$. “$F$” stands for a functor $F : X \rightarrow Y$.

I also tried one of the form

(2)$w_{[X, Y]}^F = \sum_{x \in X} \frac{w_Y^{F(x)}}{w_X^x}$

which tries to use both the weightings on $X$ and $Y$. Sadly, I couldn’t make headway.

In reply to (b), note that just restricting the weighting on $X$ to $A$ doesn’t work : in general there will be arrows emanating from $A$ to $X$, even though $A$ is a full subcategory.

I think Urs’s idea of a measure on paths (with fixed endpoints) in posets, using Tom’s weightings, is a nice idea - it also ties in with something John wrote recently about inegrating over $hom(a,b)$. Probably I’m revealing some sad misunderstanding about the Euler characteristic in my questions above, but if so, there are probably others like me in the same boat!

Posted by: Bruce Bartlett on February 22, 2007 5:45 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

Just a few more thoughts regarding the above. Urs has suggested that when trying to calculate the classical path integral

(1)$\langle b | U(t) | a \rangle = \int_{paths \gamma : a \rightarrow b} e^{-S[\gamma]} D \gamma$

that the appropriate measure to use on $Paths(a,b)$ - the space of paths $\gamma$ from $a$ to $b$ - is given by Tom’s weightings on the category $Paths(a,b)$:

(2)$e^{-S[\gamma]} D\gamma = w^\gamma.$

Okay, thats at least my interpretation - perhaps lousy - of what Urs is actually saying.

In particular, the path integral computes as the Euler characteristic of $Paths(a,b)$:

(3)$\int_{paths \gamma : a \rightarrow b} e^{-S[\gamma]} D \gamma = \chi (Paths(a,b)).$

To be precise, we talk about paths in a poset called “tar” for target space, which we think of as functors $\gamma : \worldvol_n \to tar$ where

(4)$worldvol_n = 1 \rightarrow 2 \rightarrow \cdots \rightarrow n.$

We set $Hist := [worldvol_n, tar]$. The question is, what is the weighting on $Paths(a,b)$ - the full subcategory of Hist consisting of paths such that $\gamma(1) = a$ and $\gamma(n) = b$?

It seems there are three answers (see my questions above):

1. Use the weighting restricted from Hist. Its not a weighting on Paths(a,b), but who cares! Sum it up and see what happens.

2. Paths(a,b) is itself a poset, and we can calculate a weighting for it. Use this.

3. Do something else.

If you use option (2), then the path integral always is 1, for any $n$ and for any $a,b \in tar$. That’s because the path $(a b b \cdots b)$ is a terminal object in $Paths(a,b)$. Thus the Euler characteristic of $Paths(a,b)$ is 1.

If you use option (1), my intution is that the path integral always gives zero. That’s because inside Hist, there is always a morphism $(a b b \cdots b) \rightarrow (b b b \cdots b)$.

Posted by: Bruce Bartlett on February 22, 2007 7:13 PM | Permalink | Reply to this

measure on paths

Urs has suggested […]

That’s the nice thing about discussions: ideas arise that none of the individuals would have come up with by themselves. Here we see Bruce attributing an idea to me which I was attributing to him. Apparently it was given to us from elsewhere… :-)

If you use option (2), then the path integral always is 1

Right! So that’s not what we want.

I tend to think that what we should really be looking at is more like your option (1), where the measure is that induced by the measure on free paths.

To figure this out, we should think properly about which procedure we really ought to be looking at: there is a general pull-push process of quantization, which, as you know, I tried to formulate in the arrow-theoretic terms needed here.

When you look at this prescription, you see that it involves a push-forward operating on all of $\mathrm{hist}$ in a certain way.

I should try to work that out in more detail for the case of hand where $\mathrm{worldvol}_n = 1 \to 2 \to \cdots \to$ and $\mathrm{tar}$ is a “causal poset” of Minkowski spacetime.

A formula which might be helpful for doing this is the one Tom Leinster mentions here.

Posted by: urs on February 23, 2007 8:18 AM | Permalink | Reply to this

Re: measure on paths

I wrote:

To figure this out, we should think properly about which procedure we really ought to be looking at

Okay, so when doing this pull-push, what we seem to end up computing (but see below) is, for each point (object) in target space, an integral over all paths ending at that point, namely over the pullback of some state to these paths times a phase associated to the paths: $\psi(y) = \int_{t^{-1}(y)} \mathrm{tra}(x \stackrel{\gamma}{\to} y)(\phi(x))) \; e^{-S_{\mathrm{kin}}(\gamma)}D\gamma \,.$ As before, $e^{-S_{\mathrm{kin}}(\gamma)}D\gamma$ would be the measure on the space of paths, which we would like to see if it can be understood as a weighting on the category of these paths in a suitable way.

(I write $\mathrm{S}_\mathrm{kin}$ to emphasize that it is really just the so-called “kinetic” part of the action which is supposed to be part of the measure on the space of paths. In the above formula, for instance, $\mathrm{tra}(\gamma) := e^{i S_{\mathrm{bg}}(\gamma)}$ is the (exponentiated) action corresponding to the coupling to the background gauge field encoded by the functor $\mathrm{tra}$.)

So it would seem we should be looking for the measure on the full subcategory $\mathrm{hist}^y \subset [\mathrm{worldvol}_n,\mathrm{tar}_{M^d}]$ of all paths ending at some point $y$.

Sadly, this category still has a terminal object, namely the constant path sitting at $y$. This, as Bruce remarked, would imply that we have a weighting which is concentrated entirely on that constant path (all others having weight 0).

But, on the other hand, the above is still not quite what we are supposed to be looking at. Rather, we should be looking at the pull-push of transformations $e$ through $\array{ && \mathrm{hist}\times \mathrm{worldvol} \\ & {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} \\ \mathrm{conf}\times \mathrm{par} && \Leftarrow && \mathrm{conf}\times \mathrm{par} \\ & {}_{\mathrm{ev}} \searrow && \swarrow_{\mathrm{ev}} & \downarrow \\ && \mathrm{tar} && \downarrow \\ && \mathrm{tra}\downarrow\;\; & \stackrel{e}{\Leftarrow}& \downarrow \\ && \mathrm{phas} & \leftarrow } \,,$ for the case that $\mathrm{phas} = {}_{\mathrm{FinSet}}\mathrm{Mod} \,,$ as I described here.

When you write out what this means (best for the simple case where $\mathrm{tra}$ is a trivial line bundle and hence assigns the canonical 1-dimensional $\mathrm{FinSet}$-vector space to each point, i.e. $\mathrm{FinSet}$ itself), you find that this leads to a colimit on a functor on the category of paths which takes values not quite in $\mathrm{FinSet}$ itself, but in a version of $\mathrm{FinSet}$ where morphisms may be “twisted” in a certain way!

Hm…

Posted by: urs on February 23, 2007 10:34 AM | Permalink | Reply to this

Re: measure on paths

I have thought a little more about this issue of finding a lift of a measure set to a category such that a weighting on the category exists and reproduces the original measure on the space of objects.

I would like to see if this can be applied to posets modelling pseudo-Riemannian spacetime (inspired by these ideas).

In the above above we did run into some first problems.

Pausing for a moment and reflecting on what might have gone wrong in these attempts, I noticed that already my model for the 1-dimensional space was severely broken. Fixing this might be a key to making progress.

Namely, I said I wanted to consider the poset $\mathrm{worldvol}_n = \{1 \to 2 \to 3 \to \cdots \to n\}$ as a model for the interval $[0,n]\subset \mathbb{R}$.

But that badly fails in modelling the standard measure on $\mathbb{R}$: since $\mathrm{worldvol}_n$ has a terminal object, $n$, it has a weighting which is 1 at that object and 0 elsewhere. So from the point of view of measure the space $\mathrm{worldvol}_n$ is no different than a single point.

In the context of the exercise we are doing here, this should mean that we have to throw in additional morphisms unti we do get the weighting we desire.

On the other hand, we don’t want to ruin the poset-nature of $\mathrm{worldvol}_n$, in order to retain the nice physical interpretation that we are after.

I think there is one nice and natural way out: we pass to a mild blend between a poset and a groupoid!

So let’s consider categories which have at most one morphsim going between different objects, but where each object may have nontrivial automorphisms.

This implies that composing every automorphism with a non-automorphism just reproduces that non-automorphisms. So we do can consistently put nontrivial automorphism group on the objects of a given poset and still have a category.

So consider a new category $\mathrm{worldvol}'_n$ which is generated from the poset $1 \to 2 \to \cdots \n$ and a vertex group of order $i$ on the $i$th object, subject to the relation that composing any automorphism with a morphism coming from the poset just reproduces the latter.

Then it is easy to see (calculating backwards from the object $n$) that a weighting on this category is given by the constant function which sends every object to $1/n$.

This is indeed the weighting that resembles the standard measure on the interval $[0,1]$.

One can proceed completely analously for causal posets obtained from finite truncations of lattices inside $d$-dimensional Minkowski space, and probably for any causal subset of any pseudo-Riemannian manifold. Adding suitable automorphisms groups to the objects allows one to pretty much arbitrarily adjust the weighting at that object.

So let $\mathrm{tar}'_{M^d}$ be a category of this kind. Then we should redo the exercise we discussed above and try do understand the induced weightings on the functor category $\mathrm{hist} = [\mathrm{worldvol}'_n,\mathrm{tar}'_{M^d}] \,.$

I haven’t managaed to make any progress on that yet, but at least the full subcategories $\mathrm{hist}^y \subset \mathrm{hist}$ of paths that end at some point $y$ in target space no longer contain a terminal object and hence have a chance of carrying a non-pathological weighting.

In closing, I’ll mention the following observation:

By throwing in all those extra automorphisms into our poset, we are effectivly enlarging the automorphims group of the identity endofunctor on our poset.

In a slightly different but rather similar context, such automorphisms, have been addressed as supersymmetries.

Not sure what this means, but maybe something to keep in mind.

Posted by: urs on February 24, 2007 12:43 PM | Permalink | Reply to this

Re: measure on paths

Hi Urs,

Would you mind writing one or two sentences describing what you are trying to do in terms even I can understand? You guys are way over my head at the moment, but occasional I read something that resonates with things I’ve thought about. For example, it almost seems like you are trying to build a discrete model for R. I’m probably just confused…

Eric

Posted by: Eric on February 24, 2007 4:33 PM | Permalink | Reply to this

Re: measure on paths

Hi Eric,

Would you mind writing one or two sentences describing what you are trying to do in terms even I can understand?

I’ll try to do that!

Essentially everything in the game here is quite elementary (except possibly for the proof of Tom’s theorem, but the statement of the theorem is certainly elementary), so if you feel that

You guys are way over my head at the moment

then this is certainly a defect on “our” side (my side, more likely).

So, this is the idea:

As you know better than I do, some people are fond of the idea that pseudo-Riemannian spaces can equivalently be thought of as posets equipped with a measure.

As for instance described in

Rafael Sorkin, Causal Sets: Discrete Gravity .

Now, along comes Tom Leinster with a theorem which says that for a given category there is, under some conditions at least, a natural notion of measure induced by the set of morphisms on the set of objects.

What is so very nice about this measure it that it translates colimits of functors over that category into integrals of functions over the objects of that category.

Maybe you’re not familiar with the notion of colimit. But never mind for the moment and just accept that this is an interesting property. The upshot is that the morphisms of a category (under suitable conditions) do induce in a god-given way a measure on the space of objects.

Now, people around Sorkin have tried to induce a measure on their posets by considering their elements to be embedded into some space and measuring their density there.

But I am now saying: given Tom Leinster’s result, there is a much more intrinsic and more natural way to think of our poset as being equipped with a measure (see this comment by Tom Leinster for what is so good about this, and maybe my remark on why this might be interesting for physics).

Or almost. To get measures of the kind we expect to see using Tom’s result, it turns out that we need to modify the space of morphisms of the poset a little.

Above, I was talking about how one can take a poset and add to it lots of automorphisms (invertible arrows that start and end at a given point) such that the measure induced on these points by Tom’s theorem is more like the kind of measure we expect to see (less pathological than when we don’t do this).

Ah, by the way, maybe the term “measure” is confusing here: it is really just a fancy word, meant to be suggestive but being total overkill: we are really talking about finite sets. A measure on that is just a function that assigns a weight to each point – a number that we want to think of as the “volume” associated with this point.

So, all you need to know in order to participate in this game is what Tom’s theorem says how such an assignment of numbers is induced by a collection of morphisms on our set.

The definition is indirect: we say that a function $w : x \mapsto w(x)$ on our set of objects is a measure induced from the morphisms on these objects, if for every object $x$ the following condition holds: $\sum_{y} |\mathrm{Hom}(x,y)| w(y) = 1 \,.$ Here the sum is over all objects and the term $|\mathrm{Hom}(x,y)|$ just denotes the number of morphisms from $x$ to $y$.

For simple examples of categories, it is easy to solve this equation recursively by working backwards from objects that have very few outgoing morphisms.

If you need some simple examples, try Tim Silverman’s discussion here (but beware that he is talking about coweightings, where we replace $\mathrm{Hom}(x,y) \leftrightarrow \mathrm{Hom}(y,x)$).

So, for instance, for the category with two objects and a single nontrivial morphism going between them $0 \to 1$ the weighting is $w(1) = 1$ $w(0) = 0$ Notice that there is an identity-morphism on every object which we do take into account. So there is one morphism emanating from $1$ and two from $0$.

Now, this weighting is not what we expect to be a discrete version of the standard measure of the interval $[0,1]$.

But now throw in one additional morphism $1 \stackrel{f}{\to} 1$ and require that $1 \stackrel{f}{\to} 1\stackrel{f}{\to} 1 = 1 \stackrel{\mathrm{Id}}{\to} 1$ and that $0 \to 1\stackrel{f}{\to} 1 = 0 \to 1 \,.$

This way, we now have two morphisms emanating from 1, both having the object 1 as target. This implies that now $w(1) = 1/2 \,.$ Also, there are still only two different morphisms emanating from 0, one of them being now weighted by $1/2$. It follows that also $w(0) = 1/2 \,.$

This way, we do indeed recover the obvious discretization by two elements of the standard measure on the interval $[0,1]$.

Posted by: urs on February 24, 2007 6:41 PM | Permalink | Reply to this

Re: measure on paths

And so on.

So what I called $\mathrm{worldvol}_3 = \{1 \to 2 \to 3\}$ is the category with three objects $1,2,3$ and nontrivial morphisms $1 \to 2$ $2 \to 3$ and $1 \to 3$ such that $1 \to 2 \to 3 = 1 \to 3 \,.$

I want to think of this again as a discrete model for the interval $[0,1]$.

As you can easily see, the Leinster-measure on the set $\{1,2,3\}$ induced by the morphisms of the above kind is again “pathological”:

$w(1) = 0$ and $w(2) = 0$ and $w(3) = 1 \,.$

To “correct” that, I throw in extra autmorphisms, namely $2\stackrel{f}{\to} 2$ and $3\stackrel{g_1}{\to} 3$ and $3\stackrel{g_2}{\to} 3$ such that $2\stackrel{f}{\to} 2\stackrel{f}{\to} 2 = 2 \stackrel{\mathrm{Id}}{\to} 2$ and $3\stackrel{g_1}{\to} 3 \stackrel{g_1}{\to} 3 = 3 \stackrel{g_2}{\to} 3$ and $3\stackrel{g_1}{\to} 3 \stackrel{g_2}{\to} 3 = 3 \stackrel{\mathrm{Id}}{\to} 3$ and $2 \to 3 \stackrel{g_i}{\to} 3 = 2 \to 3$ and $2 \stackrel{f}{\to} 2 \to 3 = 2 \to 3$ and $1 {\to} 2 \stackrel{f}{\to} 2 = 1 \to 2 \,.$ Now the object 3 has three morphisms emanating from it, all of them with target being 3 itself. This implies, by Tom’s formula, $w(3) = 1/3 \,.$ Working our way backwards, we find that this now implies also that $w(2) = 1/3$ and that $w(1) = 1/3 \,.$

This is the kind of measure we expect to see when we discretize the interval $[0,1]$ by three equidistant points.

Posted by: urs on February 24, 2007 7:05 PM | Permalink | Reply to this

Re: measure on paths

Thanks for the explanation! :)

I think I understand a little better now the goal. However, I am not so sure that Tom’s weightings should be used as the measure. The weightings determine a random process. I am going to keep repeating that until you acknowledge it because I think it is a key to understanding it :)

This random process deduced from the weighting is closely connecting with the measure and I think there is likely some way to deduce a measure from Tom’s weighting, but I am not so sure it is as simple as what you outlined. To get things to work out, you have to pull a few too many tricks and God doesn’t play like that :)

The simplest random process I can think of is a random walk on a binary tree, i.e. a 2-diamond. The fundamental building block for a 2-diamond is a single branch.

Beginning with three nodes: 1,2,3 and only two arrows 1->2 and 1->3, what happens if we turn Tom’s magic loose on it? I suspect we would get w(2) = w(3) = 1/2.

I’ll try to work it out myself, but I am Mr Mom again and time is short (baby is sleeping now :)).

Thanks again!

Posted by: Eric on February 25, 2007 12:38 AM | Permalink | Reply to this

Re: measure on paths

Ok. Really quick…

For the single branch I get

$k^1 = 0$, $k^2 + k^3 = 1$

and

$k_1 = 1$, $k_2 = k_3 = 0$.

I interpret the first set of equation as saying that the probability of having a particle enter node 1 is 0. The probability of a particle entering node 2 is 1 minus the probability of a particle entering node 3. Makes perfect sense to me.

I interpret the second set as saying that the probability of a particle leaving node 1 is 1 and the probability of a particle leaving nodes 2 or 3 is zero. Also makes perfect sense.

Now if you turn the branch sideways and view it along the time direction, it looks like a little spatial segment with 3 nodes and two edges.

$\chi(branch) = \chi(segment) = 3 - 2 = 1$

$k^2 + k^3 = k_1 = \chi(branch)$.

To relate this stuff to what is already familiar, I think you need to consider that the act of applying a morphism (or traversing an arrow) requires one step in time. So when we think of the graph/category as some manifold, we should consider it as a manifold extruded in time. Like I said before, to study a line segment $[0,n]\subset R$ I think you should think of it as a 2-diamond.

I think what you were trying to cook up might be related to the “projection of a diamond along the time direction”.

Baby woke up!

Posted by: Eric on February 25, 2007 1:47 AM | Permalink | Reply to this

Re: measure on paths

I am not so sure that Tom’s weightings should be used as the measure.

That was my first reaction to Bruce’s claim, too.

Then I was converted.

By these two observations:

1) There is a discrete toy version of Chern-Simons theory, called Dijkgraaf-Witten theory. It is useful as a key for understanding the structure of other quantum theories in that it is so simple and yet so rich.

Its path integral is just a finite sum over finitely many configurations.

These configurations happen to be connected by invertible arrows, which describe gauge transformations between them.

It turns out that the right weighting to use for this finite sum is precisely the Leinster-measure on this category of configurations and gauge transformations!

2)

In general, doing the path integral amounts to pulling something up a correspondence and then pushing it down the other way. For functions just stating this requires making additional specifications. For (set-valued) functors, though, the push-forward is a colimit. So in this case we may choose to not intervene and just compute this colimit as it comes to us. Then Tom’s theorem kicks in and tells us that the colimit (at least in case a couple of assumptions are satisfied) is a sum weighted by that Leinster-measure.

After I realized that, I had the strong feeling that this is telling us that the bare and unintervened push-forward, but of functors instead of functions, is the right way to do the path integral.

This would mean that we never choose a measure by hand, but that it is determined by the morphisms that connect the possible paths of our system.

Put the other way: this would mean that we have found the right category of paths of our system if its Leinster measure reproduces the measure we expect to see anyway.

So therefore the exercise we are talking about here: how does a category look like such that it qualifies as the correct configuration space for the free relativistic particle by producing a Leinster measure which does (approximate at least) reproduce the measure on paths of a relativistic particle?

The weightings determine a random process. I am going to keep repeating that until you acknowledge it because I think it is a key to understanding it :)

Okay, I do understand what you mean. The condition on the weightings to sum up to one at every point is like a condition saying that the probabilities for a particle sitting at that point to move one step in all possible ways add up to one.

[…] but I am not so sure it is as simple as what you outlined. To get things to work out, you have to pull a few too many tricks and God doesn’t play like that :)

Right, I began pulling tricks. But I felt good about this, because these were tricks not invented for their own sake but for the sake of finding the answer to a god-given question: what is the category of maps from “worldlines” into “target spaces” such that its Leinster-measure reproduces the measure appearing in the path integral of the free particle?

As I tried to indicate with my two observation above, this question seems to be god-given.

To solve god-given questions I am allowed to play whatever dirty trick I can think of!

Beginning with three nodes: 1,2,3 and only two arrows $1\to 2$ and $1\to 3$, what happens if we turn Tom’s magic loose on it? I suspect we would get $w(2) = w(3) = 1/2$.

No, unfortunately not. What makes this counting go wrong is the fact that Tom’s machine works for categories, not for graphs. As a result, the composites of your morphisms need to be defined, too, and identity morphisms need to be present. Then in your example you get the poset with one morphism (the identity) emanating from 3, two emanating from 2 and three emanating from 1. The Leinster weight on that is concentrated in the last object, 3, because this is “terminal”, meaning that every other object has a unique morphism into it. Whenever any category has a terminal object, its Leinster weight is concentrated on that object.

Posted by: urs on February 25, 2007 9:32 AM | Permalink | Reply to this

Re: measure on paths

Urs, I don’t think you understood Eric’s example. It has 2 nontrivial arrows: one from 1 to 2 and one from 1 to 3. There’s nothing from 2 to 3.

If that is what Eric meant, then there’s an initial object, so coweighting (1, 0, 0). And the weighting is (-1, 1, 1).

Posted by: David Corfield on February 25, 2007 10:41 AM | Permalink | Reply to this

Re: measure on paths

Urs, I don’t think you understood Eric’s example.

Oh, thanks! Right, I misread Eric’s example. Sorry. I should not have. It is so fundamental. Eric will again rightly complain that I keep forgetting this (even ignoring it when it is in front of my eyes)…

Posted by: urs on February 25, 2007 10:52 AM | Permalink | Reply to this

Re: measure on paths

So now I maybe understand that Eric is looking for the Leinster weight in categories that come from graphs which are binary trees.

This can easily be computed recursively in the height of the tree:

all objects at top height have weight +1.

all objects at top height minus 1 have weight -1.

all objects at top height minus 2 have weight -1

all objects at top height minus 3 have weight -1

all objects at top height minus 4 have weight -1

That seems to continue this way: the weight has constant value -1 expect at the topmost vertices (the leafs of the binary tree) that have no further descendants, where it is 1.

Hm, that’s interesting: an essentially homogeneous measure! Fun. This is what I was looking for!

For my interpretation, it is maybe troublesome that a binary tree yields a poset for a rather strange spacetime, but we’ll see.

Posted by: urs on February 25, 2007 11:16 AM | Permalink | Reply to this

Re: measure on paths

the weight [on categories coming from binary trees] has constant value -1 expect at the topmost vertices

Unless I have not had enough coffee, this seems to generalize:

the Leinster weight on categories coming from $n$-ary trees has constant value $(1-n)$ except for the leafs, where is is 1.

The same remains true if instead of categories coming from graphs that are $n$-ary trees we use categories coming from $n$-diamond graphs.

Well, that’s nice. So maybe I should not be looking at posets after all as a model for pseudo-Riemannian spacetime, but at categories generated from just light-cone graphs (instead of posets generated from the future-like graphs).

(Yes Eric, I know what you are thinking now…)

That would still allow me to invoke that theorem which relates pseudo-Riemannian metrics to light-cone structure plus measure. Even easier so, since the light cone structure would be manifest (given by the indecomposable morphisms) while for posets it has to be extracted first.

Hm. Hm.

Hm.

So it is just the (1+0)-dimensional case (the one which I kept going on about above) which makes problems.

Well… if worldlines are problematic, switch to worldsheets!

So then, here is the improved version of the exercise we should look into:

Exercise Let $\mathrm{worldsheet}$ be the category generated from a 2-dimensional diamond graph (like a binary tree, but with every second neighboring vertex identified) and let $\mathrm{tar}_d$ similarly be the category coming from a $d$-dimensional diamond graph.

Question: What is the Leinster-measure on $\mathrm{hist} = [\mathrm{worldsheet},\mathrm{tar}_d]$ ?

Well, possibly we will need to think of $\mathrm{worldsheet}$ as a 2-category in the obvious way, and similarly for for $\mathrm{tar}_d$.

Oh, wait, that’s nice: while we have just seen that we do not want to consider posets generated from graphs, but the full graph categories generated from these graphs, we now see that maybe we want to be looking at the 2-poset generated from the graph! Instead of identifying all parallel 1-morphisms, we identify all parallel 2-morphisms.

This is real fun. It feels like we are making progress.

Posted by: urs on February 25, 2007 11:59 AM | Permalink | Reply to this

Re: measure on paths

I’d come to the same conclusion about weightings on $n$-ary trees. But now

Let worldsheet be the category generated from a 2-dimensional diamond graph (like a binary tree, but with every second neighboring vertex identified)

I’m not sure I understand you definition clearly enough, and certainly not enough to get your $d$-dimensional diamond graph. Can these trees be of any height?

Posted by: David Corfield on February 25, 2007 12:23 PM | Permalink | Reply to this

Re: measure on paths

I’d come to the same conclusion about weightings on $n$-ary trees.

Good, thanks for the confirmation.

I’m not sure I understand your definition [of diamond graph] clearly enough

It’s just a cubical oriented graph, with all edges oriented such that their projection on the main diagonal has the same orientation everywhere.

So, for instance, a 2-dimensional diamond graph has objects being a subset of the obvious lattice $\mathbb{Z}\times \mathbb{Z} \subset \mathbb{R}^2$ with all edges pointing either in the positive $x$ direction or in the positive $y$ direction.

For our interpretation, this would mean that we identify the main diagonal (the line at angle $\pi/4$ from the $x$-axis) as the canonical time-axis).

An illustrative diagram follows…

Posted by: urs on February 25, 2007 12:34 PM | Permalink | Reply to this

Re: measure on paths

A 2-dimensional diamond graph:

$\array{ (n,m) &\to& (n+1,m) &\to& (n+2,m) &\to& (n+3,m) &\to& (n+4,m) \\ \downarrow && \downarrow && \downarrow && \downarrow \\ (n,m+1) &\to& (n+1,m+1) &\to& (n+2,m+1) &\to& (n+3,m+1) \\ \downarrow && \downarrow && \downarrow \\ (n,m+2) &\to& (n+1,m+2) &\to& (n+2,m+2) \\ \downarrow && \downarrow \\ (n,m+3) &\to& (n+1,m+3) \\ \downarrow \\ (n,m+4) }$

This example is special in that it is rather symmetric. This makes it easier to say what the Leinster measure on the categories freely generated from these graphs is, but has otherwise no real significance. Often we will want to think of infinite graphs of this sort.

Posted by: urs on February 25, 2007 12:46 PM | Permalink | Reply to this

Re: measure on paths

Just in order to have the statement explicitly in front of us:

the Leinster measure on the category freely generated from the graph $\array{ (n,m) &\to& (n+1,m) &\to& (n+2,m) &\to& (n+3,m) &\to& (n+4,m) \\ \downarrow && \downarrow && \downarrow && \downarrow \\ (n,m+1) &\to& (n+1,m+1) &\to& (n+2,m+1) &\to& (n+3,m+1) \\ \downarrow && \downarrow && \downarrow \\ (n,m+2) &\to& (n+1,m+2) &\to& (n+2,m+2) \\ \downarrow && \downarrow \\ (n,m+3) &\to& (n+1,m+3) \\ \downarrow \\ (n,m+4) }$ is $\array{ -1 &\to& -1 &\to& -1 &\to& -1 &\to& 1 \\ \downarrow && \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& -1 &\to& 1 \\ \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& 1 \\ \downarrow && \downarrow \\ -1 &\to& 1 \\ \downarrow \\ 1 }$

Posted by: urs on February 25, 2007 2:48 PM | Permalink | Reply to this

Re: measure on paths

So the Euler characteristic is $\frac{n}{2}(3 - n)$, where $n$ is the number of objects along the top row.

Posted by: David Corfield on February 25, 2007 7:32 PM | Permalink | Reply to this

Re: measure on paths

So the Euler characteristic is $\frac{n}{2}(3-n)$, where $n$ is the number of objects along the top row.

Yes, where I would now want to think of this quantity as the “proper volume” of the filled light-cone of diameter $n/2$.

Maybe more interesting than the graph of the above shape, for the applications that we (that is: myself and whoever wants to join in) have in mind, is a Lorentzian cylinder, i.e. piece of a cylindrical 2-dimensional Minkowski spacetime:

$\array{ && && (n,m) &\to& \\ && && \downarrow && \downarrow \\ && (n+1,m-1) &\to& (n+2,m-1) &\to& (n+1,m+1) &\to& \\ && \downarrow && \downarrow && \downarrow \\ (n,m) &\to& (n+1,m) &\to& (n+2,m) &\to& (n+3,m) &\to& (n+2,m+2) \\ \downarrow && \downarrow && \downarrow && \downarrow \\ &\to& (n+1,m+1) &\to& (n+2,m+1) &\to& (n+3,m+1) \\ && \downarrow && \downarrow \\ && && (n+2,m+2) }$ Notice the periodic identification. The Leinster-measure of (the category generated by) this is still of the now familiar form, -1 everywhere except for the future boundary: $\array{ && && -1 &\to& \\ && && \downarrow && \downarrow \\ && -1 &\to& -1 &\to& -1 &\to& \\ && \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& -1 &\to& -1 &\to& 1 \\ \downarrow && \downarrow && \downarrow && \downarrow \\ &\to& -1 &\to& -1 &\to& 1 \\ && \downarrow && \downarrow \\ && && 1 } \,.$

So this Minkowski 2-cylinder has proper volume -6.

If we were in the mood that prevails Sorkin’s text we would maybe say that the cylinder has a proper volume of 6 Planck units squared.

But we are not in that mood currently, are we?

Posted by: urs on February 25, 2007 8:05 PM | Permalink | Reply to this

Re: measure on paths

So this Minkowski 2-cylinder has proper volume -6.

Hm, let’s see. The classifying space of these categories we are talking about are precisely the graphs they come from. Right?

Tom Leinster’s Euler characteristic is supposed to coincide with the ordinary one of the classifing space, if it exists.

The above graph has $V = 10$ vertices and $E = 16$ edges. So $V-E = -6$. Looks good.

Okay, that was just to see if I understand what’s going on…

So, if we continue building our 2-dimensional “causal lattices” by gluing on little squares $\array{ \bullet &\to& \bullet \\ \downarrow && \downarrow \\ \bullet &\to& \bullet }$ to a larger diamond graph, then in most of the cases we will be adding one vertex and two edges and hence increase the “proper volume” of the total thing by -1.

Therefore the constant Leinster-measure of value -1 in the interior of our categories.

Posted by: urs on February 25, 2007 8:29 PM | Permalink | Reply to this

Re: measure on paths

for the applications that we (that is: myself and whoever wants to join in) have in mind

Maybe I should say again what I am trying to understand here.

Take the category generated from the cylinder above and call it $\mathrm{worldsheet}_\mathrm{cyl} \,.$

Similarly, consider a category generated from a finite convex (say) subgraph of a $d$-dimensional diamond lattice and call that category $\mathrm{tar}_d \,.$ Then consider the category of functors $\mathrm{hist} = [ \mathrm{worldsheet}_\mathrm{cyl}, \mathrm{tar}_d]$ or rather a full sub category $\mathrm{hist}^\gamma \subset \mathrm{hist}$ of all those functors which map the future boundary of our cylinder (those vertices carring Leinster-measure +1) to a fixed collection $\gamma$ of vertices in $\mathrm{tar}_d$.

I would like to understand the Leinster-measure on $\mathrm{hist}^\gamma$.

It would be interesting if that measure would asymptotically (for large size of our cylinder and of the target category) would look like $X \mapsto \exp(- \frac{T}{2}\int_\Sigma X^*ds^2) = \,,$ where $X : \mathrm{worldsheet}_\Sigma \to \mathrm{tar}_d$ is the given object in $\mathrm{hist}^\gamma$, where the integral is supposed to denote the volume of our cylinder as measured by the Leinster measure on target space pulled back along the map $X$, and where $T$ is some constant factor.

In other words, if the measure were proportional to the exponentiated proper volume of the image of the cylinder in target space.

A function of the proper volume of our cylinder is the only natural quantity one could imagine assigning to a functor $X : \mathrm{worldsheet}_\Sigma \to \mathrm{tar}_d$. So it’s not out of the question that something like this could be true. But I have not managed to check this so far.

Posted by: urs on February 25, 2007 8:54 PM | Permalink | Reply to this

Re: measure on paths

Urs wrote in very small part:

diamond graph

Of course, we’ve all read Greg Egan’s book Schild’s Ladder, right? That’s the one that begins with the premise that we develop a loop-like theory of everything in which the vacuum is called the ‘diamond graph’.

Posted by: Toby Bartels on February 28, 2007 7:23 PM | Permalink | Reply to this

Re: measure on paths

diamond graph

I haven’t!

While I am vaguely aware of Greg Egan’s quantum-gravity-inspired literature #, I did not know that he uses the term “diamond graph”.

He has a nice picture of one on his website:

By the way, this book seems to be built on a premise which one sees stated a lot, comparatively, not only in popular accounts, but which I don’t really see the basis for: namely that LQG predicts (explains?) that, somehow, “spacetime is made from graphs”, or the like.

I don’t follow that. The only aspect where graphs come in is in a particular choice of basis. If I choose any other basis, I don’t see any graphs anywhere.

On the other hand, the diamond graphs as they appear in Sorkin et al., as discussed above, do genuinely and explicitly appear as discrete models of spacetime. In these approaches one really has, by construction/postulate/axiom

the quantum graphs that underlie all the constituents of matter and the geometric structure of spacetime #

as Greg Egan imagines it in his book.

Posted by: urs on March 1, 2007 1:07 AM | Permalink | Reply to this

Re: measure on paths

I haven’t either! :)

Neat. I bet if you take a 4-diamond and project it onto a “time slice”, you would get something that looks like this figure. That would be cool if true.

Posted by: Eric on March 1, 2007 1:16 AM | Permalink | Reply to this

Re: measure on paths

Projecting a 3-diamond onto a time slide gives an equilateral triangular graph, which is encouraging. My brain bonks out after three dimensions though :)

Posted by: Eric on March 1, 2007 1:21 AM | Permalink | Reply to this

Re: measure on paths

I bet if you take a 4-diamond and project it onto a “time slice”, you would get something that looks like this figure.

Right, sorry, I said this is a diamond graph, while it is really the dual to a tetrahedral complex.

Hm, I cannot visualize those diamond 4-graphs, either…

By the way, what do you say to that Leinster-measure on 2-diamond cylinders?

Posted by: urs on March 1, 2007 1:30 AM | Permalink | Reply to this

Re: measure on paths

I can’t visualize it, but I compute the angle between any two edges in the resulting graph obtained from projecting a 4-diamond onto a time slice. I get that the angle between each projected edge should satisfy

(1)$\cos\theta = -\frac{1}{3}$

I’m pretty sure that the figure (if that is what Greg calls a diamond graph) corresponds to a projection of what we call a “4-diamond”.

Making a rigorous connection between what we did and what Sorkin et al are doing would make a nice paper (hint hint) :)

The Leinster measure stuff on a 2-diamond cylinder looks very cool. I got stuck because I wanted to relate morphisms to edges of a diamond, but there are no analogs to “automorphisms”, i.e. an edge whose nodes coincide, and got stumped.

Posted by: Eric on March 1, 2007 1:40 AM | Permalink | Reply to this

Re: measure on paths

Another thing I like about what you did with the 2-diamond cylinder is that it is an example of a 2-diamond that is not just a “cubic discretization of $R^2$”. I know that several people who have even attempted to read our paper get turned off early because it seems to only apply to discretizations of $R^n$. This is unfortunate because I think diamonds can represent any manifold of the form $R\times M$, where $M$ is any smooth manifold. Of course, that viewpoint is only necessary if you care about the continuum :)

Posted by: Eric on March 1, 2007 1:50 AM | Permalink | Reply to this

Re: measure on paths

Well, there is the nontrivial global topology. This can certainly be accomodated for.

Then there is the issue of inhomogeneities in the graph.

The undisturbed $n$-diamond graph (or rather the category generated by it) has, as we have found out #, homogeneously associated to it a constant Leinster-volume density of $(1-n)$ per interior vertex.

This means, in order to get target spaces $\mathrm{tar}$ of this sort with more interesting pseudo-Riemannian metrics on them, we’d have to disturb the crystalline nature of the diamond graph.

For instance, try to pick any one edge of the graph and accompany it with another, parallel edge.

That will affect the Leinster-meausure on the source vertex of that edge, and that disturbance in the conformal volume factor will “propagate” along the entire past light cone interior of this vertex through the graph, modifying the metric in that past of the vertex we tampered with.

(Hm, at the vertex where we insert another parallel outgoing edge the measure changes from $(1-n)$ to $(1-(n+1))$, but I cannot right now see the general formula that describes how this effect propagates to all the points in the past…)

The reason why we had been hesitant to consider such disturbances was, that we had a notion of dimension of the graph which was such that at an impurity of the diamond graph it jumped.

In a somewhat different mindset, we might have been excited by that. But anyway. Excitement or not, in the context that I am now imagining, we and our excitement would become secondary anyway, as the theory would just follow the flow of the Tao: we would take any graph as our category $\mathrm{tar}$ and do the pull-push quantization with that. The Leinster-measure would automatically equip that with a metric, and we find some sort of quantum theory. It’s not at us choosing the interpretation of it. It’s given to us, we’d just have to figure it out.

And that’s where I am currently stuck. I have not yet succeeded in determining the Leinster-measure on the configuration space of an $n$-particle propagating on a diamond graph (much less on a perturbed diamond graph).

Posted by: urs on March 1, 2007 11:28 AM | Permalink | Reply to this

Re: measure on paths

[…] but I cannot right now see the general formula that describes how this effect propagates to all the points in the past

Oh, it does not seem to propagate at all:

here is our 2-diamond cylinder from before #, but now at one spot I have added in just one more edge (the two edges at this point now indicated by a double arrow $\Downarrow$). Here is the resulting graph with its Leinster measure $\array{ && && -1 &\to& \\ && && \downarrow && \downarrow \\ && -1 &\to& -1 &\to& -1 &\to& \\ && \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& -1 &\to& -2 &\to& 1 \\ \downarrow && \downarrow && \downarrow && \Downarrow \\ &\to& -1 &\to& -1 &\to& 1 \\ && \downarrow && \downarrow \\ && && 1 } \,.$

Posted by: urs on March 1, 2007 11:55 AM | Permalink | Reply to this

Re: measure on paths

This remains true for arbitrary insertions of an additional edge in the interior: it only affects the Leinster meausure at its source vertex, nowhere else: $\array{ && && -1 &\to& \\ && && \downarrow && \downarrow \\ && -1 &\to& -1 &\to& -1 &\to& \\ && \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& -2 &\Rightarrow& -1 &\to& 1 \\ \downarrow && \downarrow && \downarrow && \downarrow \\ &\to& -1 &\to& -1 &\to& 1 \\ && \downarrow && \downarrow \\ && && 1 } \,.$

Posted by: urs on March 1, 2007 12:00 PM | Permalink | Reply to this

Re: measure on paths

And now, a gravitational Leinster-Sorkin wave propagating on a 2-dimensional cylindrical spacetime:

(Update: in my original comment the following graph was incorrectly weighted. Now it is hopefully right. Compare also the more general statement below.)

$\array{ && && -2 &\to& \\ && && \Downarrow && \downarrow \\ && -1 &\to& -2 &\to& -1 &\to& \\ && \downarrow && \Downarrow && \downarrow \\ -2 &\Rightarrow & -2 &\Rightarrow& -3 &\Rightarrow& -2 &\Rightarrow& 1 \\ \Downarrow && \downarrow && \Downarrow && \downarrow \\ &\to& -1 &\to& -2 &\to& 1 \\ && \downarrow && \Downarrow \\ && && 1 } \,.$ It originates in a pointlike perturbation at the incoming boundary, that produces two wave crests which propagate at the speed of light in both spatial directions. Both run around the spacelike circle once, pass right through each other once and then hit the outgoing boundary.

Posted by: urs on March 1, 2007 12:20 PM | Permalink | Reply to this

Re: measure on paths

And now, a gravitational Leinster-Sorkin wave […]

That Leinster-measure formalism is smarter than I am. I first had the above “gravitational wave” metric wrong, forgetting that at the point where the two crests meet they superimpose and lead to a total proper volume of $-3 = -1_{\mathrm{flat}\;\mathrm{background}} -1_{\mathrm{first}\; crest} - 1_{\mathrm{second}\; crest}$

But the weighting knows about that, Tom’s formula does produce the $-3$ that I got wrong!

Posted by: urs on March 1, 2007 3:56 PM | Permalink | Reply to this

Re: measure on paths

Wild :)

Is it just me, or did you just find a way to quantize Sorkin’s stuff via our stuff?

Posted by: Eric on March 1, 2007 3:56 PM | Permalink | Reply to this

Re: measure on paths

did you just find a way to quantize Sorkin’s stuff

Alas, not yet!

The game we are currently playing here is about background structures. You might consider some big diamond graph as those above, and consider that as a model for the pseudo-Riemannian target space of some $n$-particle. All we are currently doing would serve to define an action for that $n$-particle propagating on that background.

Quantization would then be the next step.

By the way, I first got the measure for the “gravitational wave” background above # wrong, now it should be better.

Posted by: urs on March 1, 2007 4:24 PM | Permalink | Reply to this

Re: measure on paths

By the way, last night I think I figured out a way to handle arbitrary 3-diamonds representing a smooth manifold $R\times M$, where $M$ is any 2-dimensional smooth manifold.

The idea is pretty enough that I can’t imagine it not generalizing to higher dimensions.

1.) Begin with a 2-d manifold $M$ 2.) Construct a Delaunay trianguation and its dual Voronoi 3.) Connect the two in a clever way

The result is an honest-to-goodness 3-diamond :)

As time evolves, the graph alternates between the triangulation and its dual.

If it holds up, I think I proved the claim I’ve been making for the last 2 years.

Any smooth manifold $R\times M$ can be “diamonated”.

Posted by: Eric on March 1, 2007 3:53 PM | Permalink | Reply to this

Re: measure on paths

1.) Begin with a 2-d manifold $M$

2.) Construct a Delaunay trianguation and its dual Voronoi

3.) Connect the two in a clever way

Hey, that’s interesting!

Maybe you can write up (draw) the details and post them here?

Posted by: urs on March 1, 2007 4:54 PM | Permalink | Reply to this

Re: measure on paths

I’ve sent a little writeup with drawings to your googlemail. Feel free to post them if you like.

Posted by: Eric on March 1, 2007 6:40 PM | Permalink | Reply to this

Re: measure on paths

Okay, I think when I have the time, I’ll collect the stuff we have here into a new entry.

Until then, here is Eric’s note on how apparently every 3-manifold $\Sigma \times \mathbb{R}$ may be diamonated.

Posted by: urs on March 1, 2007 6:54 PM | Permalink | Reply to this

Re: measure on paths

I can’t count :)

On the first page, I said “five sides”, when it is obviously “six sides”. Doh!

Posted by: Eric on March 1, 2007 7:13 PM | Permalink | Reply to this

Re: measure on paths

Hmmm…

Well that was neat and encouraging, but it is not obvious that every such constructed cell is necessarily going to be a 3-diamond. It’ll be something amenable to the calculus, but maybe not a 3-diamond.

Hmmm…

Not sure.

Posted by: Eric on March 1, 2007 7:23 PM | Permalink | Reply to this

Re: measure on paths

it is not obvious that every such constructed cell is necessarily going to be a 3-diamond.

Do you, or don’t you, get a graph with the property that every vertex has exactly three incoming and three outgoing edges? It seems you do, right?

That might be all we want, really, even if that’s not strictly a diamond graph.

Posted by: urs on March 1, 2007 8:05 PM | Permalink | Reply to this

Re: measure on paths

Yeah. I would be happy if it turned out that there were exactly three incoming and outgoing edges, but even that is not obvious to me. I used up too many brain cells thinking about that first case.

I THINK that every node of that one 3-diamond should be the source, i.e. furthest in the past, node for a new cell. That new cell may or may not have the desired properties. It’s not obvious to me. I worked 19 hours straight yesterday and only slept 4 hours before coming in this morning, so I’m not hitting all cylinders :)

Posted by: Eric on March 1, 2007 8:37 PM | Permalink | Reply to this

Re: measure on paths

Let’s see, I am thinking the following more general statement about Leinster-measures on graphs is true, but it would not hurt if somebody cross-checked this:

Say a graph is a diamond graph with multiplicities (or come up with a better terminology yourself :-) if it is like a diamond graph, but such that between each to vertices $a$, $b$ that may be connected at all (following the diamond structure), we now allow several edges $a \stackrel{f_i}{\to } b$.

So such a graph comes with a function $f : V \to \mathbb{N}$ which tells us how many edges are emanating from each vertex.

Then:

Proposition: The Leinster measure $d\mu : V \to \mathbb{R}$ on such a diamond graph with multiplicities is given by $d\mu(x) = 1 - f(x) \,.$

Posted by: urs on March 1, 2007 4:30 PM | Permalink | Reply to this

Re: measure on paths

So, this more general statement

$d\mu : x \mapsto 1 - f(x)$ nicely incorporates some of the things that we ran into before in a more unsystematic fashion:

1) For every point $x$ on the outgoing boundary we have $f(x) = 0$ hence $d\mu(x) = 1$.

2) One of the issues that I was fighting with above was that the obvious 1-dimensional graph $\array{ x_1 \to x_2 \to x_3 \to x_4 \to x_5 }$ (which I wanted to use as a model for the worldline of some particle) had a “pathological” measure $\array{ 0 \to 0 \to 0 \to 0 \to 1 } \,.$

Now, we see (what, of course, could have been just as easily seen directly, had I not been so dense) that we can simply remedy this by considering multiplicities.

So let $a \Rightarrow b$ denote the situation where two edges go from $a$ to $b$ and consider the graph $\array{ x_1 \Rightarrow x_2 \Rightarrow x_3 \Rightarrow x_4 \Rightarrow x_5 } \,.$

Its Leinster-measure is $\array{ -1 \Rightarrow -1 \Rightarrow -1 \Rightarrow -1 \Rightarrow 1 } \,.$

Okay, great. This allows me to simplify my excercise a little:

Exercise: Let $\mathrm{worldline}$ be the above graph category and let $\mathrm{tar}$ be our 2-diamond cylinder.

What is the Leinster measure on $\mathrm{hist} = [\mathrm{worldline},\mathrm{tar}]$ ?

Posted by: urs on March 1, 2007 4:43 PM | Permalink | Reply to this

Re: measure on paths

I’m probably wrong, but I don’t see a problem with the initial weight you assigned to your model of $[0,n]\subset R$. If you interpret what you did as a 1-diamond, then that is the only weighting that makes sense (to me). The weighting (I think) should be thought of as defining a random motion on a graph. If you have just a 1-diamond, there is no choice but to move forward. That is why the weight is 1 on the forward node and 0 and the back node.

If you want to consider a weighting that would correspond to random motion on $[0,n]\subset R$, you need to go to a (subset of a) 2-diamond.

I’m very very likely just confused…

Eric

Posted by: Eric on February 24, 2007 5:01 PM | Permalink | Reply to this

Re: measure on paths

I don’t see a problem with the initial weight […]

This depends on what we are after.

The game which I am currently playing is this:

Reverse-engineer the Leinster-assignment of measures to categories:

In other words:

given a space $X$ (I can only handle finite sets $X$ for the moment, so assume that) with a measure $d \mu$ (i.e. for finite sets a function $d\mu : X \to \mathbb{R}$) find a category $C$ with $\mathrm{Obj}(C) = X$ such that it has a unique Leinster-measure $w$ such that $d\mu = w \,.$

In particular, do this for categories coming from causets obtained from (subsets of) pseudo-Riemannian spaces, obtained by including addtional morphisms.

For these cases, Leinster-measures which are concentrated entirely on a single point are certainly “problematic”.

But you may have in mind an entirely different way to use the Leinster-measure to model something. That’s certainly fine.

What I find so nice about the interpretation that I am talking about here, is that it gives a way to do “integration of functions on the space of objects of a category” in a god-given way:

So assume $X$ is the space of objects of some well-behaved category $C$ which does have a Leinster-measure. Consider any function $f : X \to \mathbb{N} \,.$ Then think of each integer as a placeholder for an isomorphism class of finite sets. So consider instead a functor $\tilde f : C \to \mathrm{FinSet}$ which assigns to the object $x \in X$ any finite set $\tilde f(x) \in \mathrm{FinSet}$ of cardinality $|\tilde f(x)| = f(x) \,.$ If the (or a) Leinster-measure $d\mu = w$ of $C$ exists, then the corresponding integral of $f$ over $X$ would be $\int_X f \; d\mu = \sum_{x \in X} w(x) f(x) \,.$

Assume that this assignment of sets to objects of $C$ can be extended to a “good” functor $\tilde f : C \to \mathrm{Set}$ which assigns maps of sets to morphisms of $C$.

Here “good” means that this functor has the properties that go into the assumptions of Tom Leinster’s proof of the following formula #:

$|\mathrm{colim}_C \tilde f| = \int_X f\; d\mu = \sum_{x \in X} w(x) f(x) \,.$

(This is one gap in my reasoning: I have no idea under what conditions this extension of $f$ to a good functor really exsist. But let’s keep our finger’s crossed and assume that this works under mild conditions – or else that we can slightly modify the setup such that it does work, for instance by passing to $\mathbb{C}\mathrm{Set}s$ or something like that.)

Posted by: urs on February 24, 2007 7:46 PM | Permalink | Reply to this

Re: measure on paths

In the Bombelli presentation you linked to above, there is a reference to a recent paper I hadn’t seen:

The ideas of spacetime discreteness and causality are important in several of the popular approaches to quantum gravity. But if discreteness is accepted as an initial assumption, conflict with Lorentz invariance can be a consequence. The causal set is a discrete structure which avoids this problem and provides a possible history space on which to build a “path integral” type quantum gravity theory. Motivation, results and open problems are discussed and some comparisons to other approaches are made. Some recent progress on recovering locality in causal sets is recounted.

The sad thing is there is no reference to our paper! :)

In a way, I think what we did can be thought of as a “calculus on a causal set”. In fact, that is why I called it “Discrete Differential Geometry on Causal Graphs” :)

The product of an edge a->b and c->d is nonzero only if b = c, which would imply a and d are causally connected and the result is the causal 2-diamond containing a->b and b->d.

Here’s a quote from the bottom of page 56 in our paper:

Causal sets. The above shows that the graph operator singles out a causal structure on our discrete space. We could more generally consider scalar multiples of the preferred metric operator (4.45):

(1)$\hat{g}_G(V ) := V(x) \hat{g}_G.$

These describe geometries where each discrete lightcone is identified by $\hat{g}_G$ and carries a spacetime volume given by $V(x)$. This is the data used in causal set theory [24] to describe spacetime geometry. We here see that the formalism used here, with the preferred role that the graph operator $G$ plays in it, naturally makes contact with concepts known from causal set theory.

It seems like you are trying to relate Tom’s weighting to our $V(x)$. Like I said here a few posts back, I think it is possible (and would be great!), but maybe not quite the way you’ve attempted so far.

Posted by: Eric on February 25, 2007 6:45 AM | Permalink | Reply to this

Re: measure on paths

It seems like you are trying to relate Tom’s weighting to our $V(x)$.

Exactly!

And with the help of your comments, I think I now see how it works:

Tom’s weighting (also known as “the Leinster-measure” :-) on graph categories of $d$-dimensional dimanond graphs takes the constant value $(1-n)$ everywhere except at the leafs of the graph.

An overall factor (and sign) is, for our purposes, completely an issue of convention. Hence this says that we get a homogenoeus measure on our spacetime model, constant at each point.

Very nice!

I think it is possible (and would be great!), but maybe not quite the way you’ve attempted so far.

Yes, you are completely right. What I tried applied to the 0+1-dimensional case, which seems to be ill-behaved in the present context.

For $n$-particles with $n \geq 2$ all abstract nonsense seems to be on our side, though.

Posted by: urs on February 25, 2007 12:24 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

But given any contravariant functor, you would not call it a presheaf unless you are going to think of its domain as a site.

But what else would you call it? Especially when the codomain of the functor is implicit in a given discussion, ‘presheaf’ is a very convenient short term to use, so people do even when they have little interest in topoi and the like. Even ‘presheaf of groups’ (say) often works more nicely than ‘contravariant functor to Grp’.

However, in case it makes you feel better, any category may be made into a site in a trivial way, by declaring that each covering family consists of a single isomorphism. Then every presheaf is a sheaf.

Posted by: Toby Bartels on February 16, 2007 11:12 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

[…] ‘presheaf’ is a very convenient short term to use, so people do even when they have little interest in topoi and the like. Even ‘presheaf of groups’ (say) often works more nicely than ‘contravariant functor to Grp’.

However, in case it makes you feel better, any category may be made into a site in a trivial way, by declaring that each covering family consists of a single isomorphism. Then every presheaf is a sheaf.

Uh, that makes me feel worse! :-)

By the same argument, every functor is more conveniently called a pre-co-sheaf, which may always be thought of a co-sheaf in a trivial way.

Of course it is true that we often say “presheaf” for contravariant functor. But since both terms are exactly equivalent as far as the technical definition goes, there must be some other motivation for using one or the other terminology.

In general, I don’t want to quibble about this terminology business. As long as we agree on what these terms mean, that’s fine with me.

But in the present context, I do think it is worthwhile thinking about the terminology.

The point is that in the present context we are not so much into proving theorems (yet), but rather into mathematical model building: there is a physically motivated concept we want to capture (quantization) and we are trying to identitfy the best mathematical language to model that.

For this endeavour, it is helpful to find that terminology which most suggestively indicates how the mathematical structures chosen are to represent the physics that they are supposed to model.

For instance: is it really important that the functor that Chris Isham considers is in fact contravariant? Is that something forced upon us by our goal to model a physical concept, or is it maybe rather a result of some previous choice of conventions?

Posted by: urs on February 18, 2007 12:46 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

Of course it is true that we often say “presheaf” for contravariant functor. But since both terms are exactly equivalent as far as the technical definition goes, there must be some other motivation for using one or the other terminology.

I think that it’s the grammar! If you don’t want to mention the codomain of the functor (only the domain), then you’d rather say ‘presheaf on C’ than ‘functor out of C’ (especially since this ‘out of’ is not so common, and some people would say ‘from’ instead, even with no ‘to’ following). And if you want to mention the codomain only implicitly, then you’d rather say ‘presheaf of groups’ (say) than ‘functor to Grp’ (especially if you haven’t introduced the symbol ‘Grp’ and don’t want to). On the other hand, I agree that ‘functor from C to D’ is superior (poetically) to ‘D-valued presheaf on C’ (which some people would say).

[…] it is helpful to find that terminology which most suggestively indicates how the mathematical structures chosen are to represent the physics […]

I only want to defend ‘presheaf’ (in some contexts) in contrast to ‘functor’. Suggestive terminology like ‘parallel transport’ is much better, once somebody thinks of it. So if that is your point, then I agree entirely —but you can’t blame people for saying ‘presheaf’ until after they read your papers! ^_^

Posted by: Toby Bartels on February 20, 2007 1:19 AM | Permalink | Reply to this

Re: Isham on Arrow Fields

A clarifying PS to my last posting. One reason why one cannot literally use fibre bundle theory for a category is that each object in the category may have a different ‘internal structure’. This means that in the quantisation procedure (a la Mackey) the fibres over different points in the base space would *not* be isomorphic to each other. Whereas, of course, in a normal associated vector bundle, the fibres are isomorphic.

That is why I chose to go the presheaf way. But it is true that, apart from the non-isomorphic fibres/stalks, the definition of a presheaf is very much like a bundle with a connection/parallel transport if one wants to think of it in that way.

Best regards

Chris Isham

Posted by: Chris Isham on February 16, 2007 8:14 AM | Permalink | Reply to this

Re: Isham on Arrow Fields

One reason why one cannot literally use fibre bundle theory for a category is that each object in the category may have a different ‘internal structure’.

I see similar arguments involving nontrivial endomorphisms of the domain category in your papers. I must admit that I do not quite follow the reasoning here. I’ll try to spell out the reasons for that below.

This means that in the quantisation procedure (a la Mackey) the fibres over different points in the base space would not be isomorphic to each other.

Let me see: I believe we are talking about a setup as follows:

there is some category $C$, and there is some functor $F : C \to \mathrm{Hilb}$ (or some other codomain, that’s not really important, but in your papers you consider the category of Hilbert spaces).

Possibly you want to consider this functor as being contravariant instead $F : C^\mathrm{op} \to \mathrm{Hilb} \,.$ In that case, just replace $C \leftrightarrow C^\mathrm{op}$ in the following.

Now, it seems that the question is under which conditions the morphisms in the image of $F$ are isomorphisms. Right?

I got the impression that you are saying that this depends on the nature of the endormorphisms in $\mathrm{Mor}(C)$. But this I don’t see.

On the other hand, if for instance $C$ is a groupoid, then all the morphisms in the image of $F$ are certainly forced to be isomorphisms - even if the groupoid contains nontrivial endomorphisms, as it usually will.

So, for instance, we could consider $C$ to be the “core” of $\mathrm{FinSet}_n$: the category of all $n$-element sets with morphisms all isomorphisms between these.

In that case, we could reasonably say that the elements of the groupoid have “inner structure” (in that they are sets). Still, any functor $C \to \mathrm{Hilb}$ would take values in invertible morphisms.

Of course if we instead take $C$ to be, say, the “core” of $\mathrm{FinSet}$, i.e. the category of all finite sets (no restriction on the number of elements) and all isomorphisms between them, then the image of any functor $F : C \to \mathrm{Hilb}$ would in general associate non-isomorphic Hilbert spaces to sets of different cardinality.

But this phenomenon is due to the fact that in this case $C$ is not connected, i.e. that $\pi_0(C)$ is nontrivial. I wouldn’t address this as a consequence of the “inner structure” of the elements in $\mathrm{FinSet}$.

Maybe to emphasize that point: consider a vector bundle on a space $X$ which is the disjoint union $X = X_1 \cup X_2$ of two spaces. Then the category $P_1(X)$ of (thin homotopy classes of) paths in $X$ would be disconnected, too: $P_1(X) = P_1(X_1) \cup P_1(X_2) \,.$

As a result, a parallel transport functor (describing a hermitean vector bundle with connection on $X$) $\mathrm{tra} : P_1(X) \to \mathrm{Hilb}$ would not have to associate isomorphic Hilbert spaces to objects in $X_1$ and $X_2$: a vector bundle on $X$ might have different rank on $X_1$ and on $X_2$.

Well, anyway. What I am just saying is that the images under some functor of two objects in the domain are isomorphic if these objects are connected by an isomorphism. If not, they may be isomorphic or not.

Posted by: urs on February 16, 2007 11:50 AM | Permalink | Reply to this

Re: Isham on Arrow Fields

Dear urs

You say:

“Well, anyway. What I am just saying is that the images under some functor of two objects in the domain are isomorphic if these objects are connected by an isomorphism. If not, they may be isomorphic or not”

Yes, I agree with that. But for types of system I was thinking of the objects will not be connected by an isomorphism. For example if you take a category $C$ whose objects are posets (causal sets to the physicists). The point is that the hom sets Hom(A,A) for A an object in C can be very $A$ dependent. And, one of my basic requirements is that the Hilbert space over each $A$ carries an irreducible representation of \$Hom(A,A).

I agree what you say about C being a groupoid, but that is not the case for the causal set examples

Also, you say

“But this phenomenon is due to the fact that in this case C is not connected, i.e. that ð 0 (C)ð 0 (C) is nontrivial. I wouldn’t address this as a consequence of the “inner structure” of the elements in FinSet.

I think this is the difference between a mathematician and a physicist :-) You have given a mathematical explanation while I was using a physicist’s description. Again, the causal set example is a good one to consider.

My apologies to all if I am a bit slow to respond to comments some times, but it is hard for me to always find the time to write things (at the moment it is 3:50 in the morning:-))

Best regards

Chris

Posted by: Chris Isham on February 17, 2007 3:55 AM | Permalink | Reply to this

Re: Isham on Arrow Fields

Yes, I agree with that.

Okay, good. Thanks for your response!

Probably this is just another matter of terminology. All I was really tending to take issue with is to attribute the non-isomorphy of images of objects to the “inner structure” of their pre-images.

For instance, if we consider the example of the ordinary free particle (instead of poset quantum gravity) for a moment, we see that a natural choice of “category to quantize over” (namely of category of configurations and paths between configurations) is the category of Moore paths in target space.

This has no nontrivial isomorphisms. And still, each object is interpreted as a “point in a space” which we would not want to think of as having “inner structure” (even though each object does have nontrivial endomorphisms, namely all Moore loops based on that object).

Posted by: urs on February 18, 2007 12:57 PM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: T-Duality
Weblog: The n-Category Café
Excerpt: Topological T-duality as a pull-push transformation of sections of the 2-particle.
Tracked: February 16, 2007 2:15 PM
Read the post QFT of Charged n-Particle: Algebra of Observables
Weblog: The n-Category Café
Excerpt: The algebra of observables as certain endomorphisms of the n-category of sections.
Tracked: February 28, 2007 2:32 AM
Read the post QFT of Charged n-Particle: Gauge Theory Kinematics
Weblog: The n-Category Café
Excerpt: A review of Fleischhack's discussion of a Weyl algebra on spaces of generalized connections.
Tracked: March 1, 2007 2:56 PM

Re: Isham on Arrow Fields

As far as measures on diamond spaces and Leinster weights are concerned:

I feel rather uneasy about this approach for various reasons—which probably just shows I don’t understand what you’re trying to do. But I’ll mindlessly articulate two of my concerns anyway and see where it leads …

First: if the weighting or coweighting is supposed to be some sort of ‘curvature’ which one integrates over the space to get the Euler characteristic as a total curvature, one would expect the weighting or coweighting of an $n$-diamond space to be $0$ everywhere (except perhaps at boundaries), seeing as these are surely supposed to be flat spaces.

Second: in an $n$-diamond space, $n\gt1,$ why would we include only the underlying graph of 0-cells and 1-cells, and leave out the faces, $3$-cells, etc?

Starting by very naively applying this philosophy to $2$-diamond spaces, we say we have a vertex, contributing $1$ to its local Euler characteristic (or ‘curvature’), $2$ incoming (or outgoing) edges contributing $-1$ each, and one incoming face giving a contribution of $1.$ Totalling $1-1-1+1$ we get $0$ as expected.

Less naively, lets try and do this properly.

Consider a $2$-category with $2$ objects $a$ and $b,$ with two morphisms $f, g:a\rightarrow b,$ and one $2$-morphism, $A:f\rightarrow g.$

We’ll consider this to be a $2$-graph, and try to generalise the algorithm we developed for $1$-graphs. Hopefully this should instantiate something sensibly algebraic on the $2$-graph’s $2$-category, though I haven’t tried to work this out yet.

So we try a 2-step procedure. First, assign weights (or co-weights) to the edges, according to the rule “weight = $1$-number of incoming (or outgoing) $2$-edges”. Then we assign weights (or co-weights) to the vertices, taking the weights on the edges into account.

I’ll go with co-weights (based on incoming $n$-edges) since that’s what I’m used to. Step 1 assigns $f$ a weight of 1 (no incoming $2$-edges), and assigns $g$ a weight of $0$ (1 incoming $2$-edge).

Then step 2 assigns vertex $a$ the value 1 (no incoming edges), and assigns $b$ a weight of … well, let’s see … $1 - weight of f - weight of g = 1 - 1 - 0 = 0.$ So the vertex total is $1+0=1$ which is correct, since our $2$-graph is topologically a disc.

So lets add a second $2$-edge parallel to the first, so we get a sphere. Then $f$ is still assigned a weight of $1$ and $g$, with $2$ incoming $2$-edges, gets assigned a weight of $-1.$ Now, checking the vertices, $a$ still has a weight of $1,$ and $b$ has a weight of $1 - weight of f - weight of g = 1 - 1 - (-1) = 1.$ So totalling the weight of $a$ and $b,$ we get $2,$ which, for a sphere, is right.

I wanted to show some pictures of more complicated $2$-graphs but, gads, the graphics capabilities of this interface are worse than ASCII-art, so I just drew some pictures on paper. You may imagine me looking at them now …

A couple of issues arise in the more general situation. What exactly are the $2$-edges? The reasonable answer, I think, is that we make them polygons whose boundaries are made up of the ($1$-)edges. In fact, just to simplify things, I only considered the case where every $1$-edge is part of the boundary of at least one $2$-edge, though I sincerely hope this isn’t essential for the process to make sense. Since the $1$-edges and vertices generate a poset, each polygonal boundary has a single ‘source’ and a single ‘target’ vertex, with respectively two edges of the polygon leaving and two edges entering; all other vertices will have one leaving and one entering. So there is no doubt which morphisms (that is, which catenations of edges) act as source and target of the polygonal $2$-morphism; the only decision to make is which is source and which is target: we need an orientation. In keeping with the situation with edges and vertices, we want the $2$-morphisms to organise the edges as a poset; we want to avoid cycles.

However, the way we are proceeding, we are trying to assign weights to edges, not to morphisms. Hopefully this will fall in some nice inclusion-exclusion way out of the algebra, but at the moment I don’t have an algebra, just a candidate algorithm. Since the $2$-morphism acts between two $1$-morphisms which may well be catenations of edges, we need to decide how the weight gets shared out. A bit of thinking and playing around makes me reasonably confident that we assign -1 to an edge for each incoming (or, dually, outgoing) $2$-morphism whose target morphism has the same target vertex as the edge itself; in other words, to the last edge in the catenation making up the morphism. The other edges don’t get any contribution to their weight due to that $2$-morphism (i.e. polygon).

At this point, I would draw a cube, with its vertices, edges and faces, and start assigning orientations and weights, but I can’t face trying to draw one here, so I will leave this as a slightly fiddly but interesting exercise for the reader, and simply report what I found. The end result is that the edge weights are somewhat opaque in their significance (at least to me) but the vertex weights are of crystalline clarity. I have just one vertex with only outgoing edges, and one vertex with only incoming edges, and each of these gets assigned a weight of 1; the remaining vertices all get assigned a weight of 0. This gives an Euler characteristic of 2, which is right, since the cube is topologically a sphere.

This is now starting to look increasingly Morse-like, so I tried one more configuration equivalent to a sphere: the ‘pair of pants’ (or, as Terry Pratchett likes to call them, the ‘trousers of time’) with the three holes capped off with $2$-cells.

This is how I did it, in case you want to check:

I started by hanging the trousers upside-down, and drew a seam down the outside of the left leg all the way to the waist. I put vertices at the top (on the hem of the trouser-leg), at the bottom (on the waistband) and halfway down. That gives two edges, which I directed to point down. I did the same on the right side. Then I put a seam down the inside of each leg, meeting in a vertex at the crotch, and put another vertex at the top of each of them. Again, the edges are directed downward. At the end of each leg, we now have two vertices, and I put two edges in, directed from the outside vertex to the inside vertex, one around each side of the hem. I ran two edges around the waistward end of each trouser leg from the outside vertex (the one halfway down the outer seam) to the vertex at the crotch, again one edge on the front of each trouser leg and one edge around the back. All these edges I directed from left to right. I also put left-to-right-directed edges around the front and back of the waistband. I closed off the waist end ends of the trouser-legs with $2$-cell caps.

The $2$-cells consist of three bigons (the leg and waist caps), four rectangles, (the front and back panels of each leg), and two pentagons (the front and back of the waist).

Then I assigned edge weights and vertex weights.

The edge weights were again a bit opaque but the vertex weights were: $1$ on the outer vertex on the hem of each trouser leg—i.e. the vertices with only outgoing edges; $1$ on the right vertex on the waistband—i.e. the vertex with only incoming edges; $-1$ on the vertex at the crotch; $0$ on all other vertices. This correctly gives a total of $2.$ Interestingly, the picture was essentially unchanged when I reversed the direction of all the $1$-edges (while keeping the orientation of the $2$-edges the same). Or rather, unchanged in an interesting way: the edge weights changed a lot, but the vertices still got assigned weights of $1$ if their edges were all incoming or all outgoing, $-1$ at the crotch, and $0$ elsewhere.

This selection doesn’t involve much variety (only spheres! Oops!) but it does tend to confirm that the Leinster weights (or coweights) and the partial ordering are doing something analogous the work of Morse functions on a manifold, i.e. detecting singular behaviour or something like that, and picking out odd and even indices. Except we have something a bit more general: not only analogies to the singular behaviour on slices through smooth manifolds (as determined by contours of a Morse function), but also handling manifolds with boundaries, and spaces made of manifolds glued together along boundaries. Since triangulations, or other ways of chopping spaces into contractible polytopic pieces, are of exactly this sort, the Leinster approach seems to mediate between the Morse function approach and the triangulation approach to calculating Euler characteristics.

I guess it’s likely this stuff is already well-known; but not to me …

Posted by: Tim Silverman on March 4, 2007 9:08 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

I was wrong with an interpretation of the Leinster-weighting before, and I may be again. But for the moment I will pretend the idea I have is good and try to point out why.

Second: in an $n$-diamond space, $n \gt 1$, why would we include only the underlying graph of 0-cells and 1-cells, and leave out the faces, 3-cells, etc?

We (I at least) would want an $n$-category as the “target space” for an $n$-particle. At the moment I guess we should be concentrating on ordinary particle physics, so I want target space to be a 1-category.

In the continuum I know this should be a category of paths in target space.

In the discrete version it should be some category over a finite set.

Actually, our concentration on “diamond” graphs is a little bit of a red herring: the results about the Leinster measure that we were talking about hold for every category generated from a finite directed graph without any loops (unless I am mistaken, of course):

Proposition: The Leinster measure $d\mu : V \to \mathbb{R}$ on the category of a finite directed graph $(V,E)$ without loops is $d\mu(x) = 1 - |s^{-1}(x)| \,,$ where the second term is supposed to denote the number of edges (in the graph) emanating from $x$.

Anyway, the target for a 1-particle should be a 1-category.

First: if the weighting or coweighting is supposed to be some sort of ‘curvature’ which one integrates over the space to get the Euler characteristic as a total curvature, one would expect the weighting or coweighting of an $n$-diamond space to be 0 everywhere (except perhaps at boundaries), seeing as these are surely supposed to be flat spaces.

I think there are really two aspects to this: first, Tom’s formula $|\mathrm{colim}_X F| = \sum_{x \in X} w(x) |F(x)|$ tells us (it is not at us to choose this interpretation) that the weighting behaves like a measure with respect to the natural summation operation on set-valued functors (of “good” set-valued functors, anyway).

So how do we interpret this, given that we also know that this weighting is about curvature?

Well, I guess the point is that it is not correct to assume that the graph we are talking about is, as a model of some space, “filled in” with pieces of $\mathbb{R}^n$, as your comment suggests. It is really that graph. As such, it has lots of curvature and holes.

For instance, this graph $\array{ x &\to& y \\ \downarrow && \downarrow \\ z &\to& w }$ is really not to be thought of as $[0,1]^2$, but as a circle!

This is consistent with the fact that the Leinster characteristic of a category coincides with the Euler characteristic of the classifying space of that category, when it exists, and for categories generated from oriented graphs without loops, I’d think these classifying spaces are given by (the union of the edges of ) these graphs (forgetting the direction on the edges).

I made a comment on that here.

So I think that does give a consistent picture of sorts, if we accept that for this to model what we want it to model, this is telling us that what looks like Riemannian volume on large scales is the accumulation of lots of topological defects on tiny scales.

Posted by: urs on March 4, 2007 9:53 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

Urs, thanks for your thoughts. (I’m afraid I have a bad habit of using other people’s posts to leap off in my own odd directions, so it’s nice you detected something relevant in what I said.)

You said:

So I think that does give a consistent picture of sorts, if we accept that for this to model what we want it to model, this is telling us that what looks like Riemannian volume on large scales is the accumulation of lots of topological defects on tiny scales.

This is a very nice idea—in principle. The problem, in this particular case, is that the only flat target space is a line! Trying to model higher-dimensional spaces automatically introduces curvature! This is really what set me off wondering about higher-dimensional cells in the diamond lattice in the first place.

Have you thought how this might relate to the dynamical triangulations way of looking at things?

On the subject of motivation—you said something about how the Leinster curvature gives the right measure on configurations in Dijkgraaf-Witten theories. That’s very, very intriguing. Could you briefly summarise how it works?

Posted by: Tim Silverman on March 5, 2007 7:24 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

Urs, thanks for your thoughts. (I’m afraid I have a bad habit of using other people’s posts to leap off in my own odd directions, so it’s nice you detected something relevant in what I said.)

It’s good, I would like to think of this blog as a place where people also exchange ideas that do not quite fit on the back of one single envelope!

The problem, in this particular case, is that the only flat target space is a line! Trying to model higher-dimensional spaces automatically introduces curvature!

Oh, that’s right, but just on “microscopic scales”!

On “macroscopic scales”, as we do something or other that makes sums of the sort $\frac{1}{|\mathrm{Obj}(X)|}\sum_{x \in \mathrm{Obj}(X)} w(x) |F(x)|$ tend to honest integrals, it does look like flat pseudo-Riemannian space in a precise sense:

the direction of the arrows do induce a light-cone structure on the space of objects.

The weighting does provide a measure.

But light cone structure and measure gives us a pseudo-Riemannian metric. Which is flat if we started from a perfect diamond graph.

See, I am in a way spelling out an old quantum-gravitist’s dream: realize flat space(time) on small scales as a very non-classical object which lots of curvature and lots of topological funny stuff going on, such that on sufficiently large scales it looks just like ordinary space.

The fun thing is: I am not dreaming this up! I am just pointing out that given the single one proviso that we accept that it is good to think of integer-valued functions on a graph as actually set-valued functors, then this interpretation sort of forces itself upon us. This is just how such large diamond graphs then do appear to us on small and on large scales.

So that’s why I find it interesting to ponder the question what natural measure such a graph would actually induce on the category of maps of 1-dimensional graphs into it…

Posted by: urs on March 5, 2007 10:34 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

On the subject of motivation—you said something about how the Leinster curvature gives the right measure on configurations in Dijkgraaf-Witten theories. That’s very, very intriguing. Could you briefly summarise how it works?

Sorry, for some reason I almost forgot to reply to this part. But that’s really a very important point.

In Dijkgraaf-Witten theory, parameter space is the fundamental groupoid $\mathrm{par} = \Pi_1(M) \,,$ of some manifold $M$.

Target space is a finite group $G$, regarded as a category $\mathrm{tar} = \Sigma G$ with a single object.

Configuration space $\mathrm{conf} = [\mathrm{par},\mathrm{tar}]$ is the category of functors from the fundamental groupoid into the gauge group. Each such functor can be thought of as representing a $G$-bundle with connection on $M$. Since, for $G$ finite, thre is really only a unique connection on every such bundle, this is the same here as just $G$ bundles on $M$.

The morphisms in $\mathrm{conf}$ are natural transformations between these functors. These describe gauge transformations relating the respective bundles.

In particular, configuration space $\mathrm{conf}$ here happens to be a groupoid!

And it is equivalent to a finite groupoid. For instance, we can simply replace the fundamental groupoid of $M$ with its skeleton, which is then nothing but a disjoint union of (suspended) fundamental (finite) groups, one for each connected component. (This works here, but not in general gauge theory, because there is no smoothness condition or anything like that. All categories and functors are internal just to $\mathrm{Set}$ and so every connected groupoid is equivalent to any of its vertex groups.)

As a finite groupoid, its Leinster measure is defined and coincides with the Baez-Dolan prescription: $d\mu : x \mapsto \frac{1}{|\mathrm{Aut}(x)|} \,.$

This measure is indeed precisely the measure that enters the path integral (which here is just a finite sum) in Dijkgraaf-Witten theory.

You can find this, for instance, in equation (2.1) on p. 9 of

Daniel S. Freed, Frank Quinn, Chern-Simons Theory with Finite Gauge Group

or in equation (4.25) and (4.31) on p. 68 of

Bruce H. Bartlett, Categorical Aspects of Topological Quantum Field Theories.

Posted by: urs on March 7, 2007 10:58 AM | Permalink | Reply to this

Re: Isham on Arrow Fields

Starting by very naively applying this philosophy to 2-diamond spaces, we say we have a vertex, contributing 1 to its local Euler characteristic (or ‘curvature’), 2 incoming (or outgoing) edges contributing −1 each, and one incoming face giving a contribution of 1. Totalling 1−1−1+1 we get 0 as expected.

Okay. Maybe before trying to generalize the weighting procedure to 2-categories, let’s see what the above means on the level of ordinar 1-categories, to maybe better see what is going on.

The situation you describe would for instance be something like this:

$\array{ x &\to& y \\ \downarrow && \downarrow \\ z &\to& w } \,.$

We want to know the weighting on $x$, which has two outgoing edges.

If we take the category freely generated from this graph, then

$x \to y \to w$ is different from $x \to z \to w \,.$

As we form the classifying space of the category, there would be no face filling in the above. Hence we really have something of the topology of a circle.

But take instead the category generated from these morphisms, but subject to the relation that $(x \to y \to w) = (x \to z \to w) \,.$

This models the fact that now there is a bridge between the one and the other (and the only 2-bridges here are identities). Accordingly, the weighting on $x$ now is indeed 0.

Posted by: urs on March 4, 2007 10:19 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

Urs wrote:

But take instead the category generated from these morphisms, but subject to the relation that $(x\rightarrow y\rightarrow w) = (x\rightarrow z\rightarrow w)$

But that’s—gasp!—decategorification! That “=” sign is the mark of evil!

Or, in other words, yes, indeed, that was kinda my starting point for my attempt at categorification, to get the effect of having non-identity 2-morphisms. I needed the 2-cells to sort of ‘cancel out’ one of the 1-cells. This turned out to be reasonably straightforward to do, because it’s extremely similar to what happens one level down, with edges ‘cancelling’ vertices.

(Hmm, not much content in this post, except to say that yes, what you said makes perfect sense to me. Phew!)

Posted by: Tim Silverman on March 5, 2007 7:44 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

“=” is not evil when used to say something about top level morphisms. What else could a relation be between such things? Either they’re the same or they’re different.

Posted by: David Corfield on March 5, 2007 8:35 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

It was a sort of joke—not a very successful one. Urs wants to keep everything at the level of 1-graphs and 1-categories, and I keep wanting to go up to 2-graphs and 2-categories (because those lattices look so darn 2-dimensional). So from that perspective, equating the paths is decategorification.

Fortunately things work out the same either way.

Posted by: Tim Silverman on March 5, 2007 10:17 PM | Permalink | Reply to this

Re: Isham on Arrow Fields

the graphics capabilities of this interface are worse than ASCII-art

You can use $$\array{…}$$ to open an array environment and then use all kinds of arrows to get pretty much everything you’d want to do in ASCII-art, like

$$\array{ \nearrow \searrow \\ \searrow \nearrow }$$

Posted by: urs on March 4, 2007 10:34 PM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: Disk Path Integral for String in trivial KR Field
Weblog: The n-Category Café
Excerpt: The arrow-theoretic perspective on the path integral for the disk diagram of the open string.
Tracked: March 5, 2007 4:24 PM
Read the post Canonical Measures on Configuration Spaces
Weblog: The n-Category Café
Excerpt: On how the Leinster weighting on a category might provide path integral measures in physics.
Tracked: March 8, 2007 9:41 AM
Read the post What is a Lie derivative, really?
Weblog: The n-Category Café
Excerpt: On the arrow-theory behind Lie derivatives.
Tracked: May 31, 2007 11:26 AM
Read the post Physical Systems as Topoi, Part III
Weblog: The n-Category Café
Excerpt: The third part of the talk.
Tracked: July 23, 2007 8:30 PM
Read the post Arrow-Theoretic Differential Theory
Weblog: The n-Category Café
Excerpt: We propose and study a notion of a tangent (n+1)-bundle to an arbitrary n-category. Despite its simplicity, this notion turns out to be useful, as we shall indicate.
Tracked: July 27, 2007 5:27 PM
Read the post Arrow-Theoretic Differential Theory, Part II
Weblog: The n-Category Café
Excerpt: A remark on maps of categorical vector fields, inner derivations and higher homotopies of L-infinity algebras.
Tracked: August 8, 2007 10:52 PM
Read the post On BV Quantization. Part I.
Weblog: The n-Category Café
Excerpt: On BV-formalism applied to Chern-Simons theory and its apparent relation to 3-functorial extentended QFT.
Tracked: August 17, 2007 10:04 PM
Read the post Eli Hawkins on Geometric Quantization, I
Weblog: The n-Category Café
Excerpt: Some basics and some aspects of geometric quantization. With an emphasis on the geometric quantization of duals of Lie algebras and duals of Lie algebroids.
Tracked: June 20, 2008 5:12 PM

Post a New Comment