2-Metric Geometry

May 25, 2006

Posted by Urs Schreiber

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Here is a reaction to a discussion taking place over at David Corfield’s weblog.

Don’t read this if you are not into this kind of game.

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Exercise (Corfield): Define and study a good notion of categorified euclidean geometry.

Different, but equivalent, concepts may yield in general different and non-equivalent concepts when categorified (which shouldn’t come as a surprise).

So what notion of metric geometry do we want to base our exercise on?

David Corfield and John Baez are currently concentrating on categorifying Klein’s Erlangen Program, where geometric objects are encoded in terms of their stabilizer groups.

Tom Leinster, on the other hand, made a comment suggesting that Lawvere’s categorical conception of metric spaces might be a promising starting point.

Here, I want to concentrate on this second idea, simply because it fits nicely into the context of some other things that I was thinking about lately.

As Tom Leinster remarked, there is this beautiful text:

F. W. Lawvere
Metric Spaces, Generalized Logic, and Closed Categories
Reprints in Theory and Applications of Categories, 1 (2002) pp 1-37
(pdf).

Therein Lawvere points out that several fundamental structures appearing in mathematics, and in particular the concept of a metric, are special cases of enriched categories, and that, furthermore, looking at them from this point of view makes a host of subsequent constructions and observations, even some theorems, an automatic consequence of general abstract nonsense.

Recall what a $V$ -enriched category is. Let $V$ be some monoidal category with all the additional properties that we need in the following. Essentially, a $V$ -enriched category has morphism spaces being objects in $V$ and composition of morphisms being a morphism in $V$ .

Nobody reading this needs to be told what a $V$ -enriched category is, so I use this opportunity to tell you all how I think personally about enriched categories, just so you have something to comment on.

Let’s see, either this is stupid or boring: I like to think about a $V$ -enriched category as a lax functor with target $\Sigma(V)$ .

Here $\Sigma(V)$ is supposed to denote the suspension of the monoidal category $V$ , i.e. the 2-category with a single object, with morphisms being the objects of $V$ and 2-morphisms being the morphisms of $V$ .

I like this lax-functor way of looking at enriched categories, because it reduces the amount of thinking required when I need to figure out what a $V_2$ -enriched 2-category might be, where now $V_2$ is a 2-category (bicategory for those who count uni, bi, tri, …).

Hm, I should then probably hasten to add that I am aware that we may simply enrich in 2-categories as we did before with 1-categories. After all, 2-categories are themselves nothing but categories enriched in the 2-category $\mathrm{Cat}$ .

But what I have in mind here is slightly different. I really want to start with a 2-category and then add labels in another 2-category to it. This is at least what I will want to use in a proposed attempt to categorify Lawvere’s concept of a metric space to something like a 2-metric space.

This is related to the fact that a lax functor to $\Sigma(V)$ is really something possibly more general than a $V$ -category. It reduces to the ordinary concept iff the functor’s domain has at most one morphism between any given objects.

All right. Next I recall what Lawvere says about metric spaces. As before, I slightly reformulate it in order to better suit my categorification needs. (More precisely, where Lawvere has addition I will use mutiplication of exponentials.)

Let us write $R = [0,\infty]$ for the non-negative real numbers. I will write all elements of $R$ in the form $\exp(\ell)$ .

Among other things, these form a poset with respect to $\geq$ . Any poset can be regarded as a category with precisely one morphism for every $\geq$ relation, so we’ll do this.

But, $R$ is also a monoid with respect to, for instace, the multiplication in $R$ . And it is so in a way that is compatible with $\geq$ .

In other words, $R$ has the structure of a monoidal category, with the tensor product being the ordinary product of numbers.

As Lawvere explains, we even have internal Homs in $R$ , so that $R$ is something we may enrich with.

The nice thing is, that a metric on some set $S$ is nothing but a category $C$ with $\mathrm{Obj}(C) = S$ , which is enriched in $R$ .

More precisely, this gives the concept of metric with some of the more irrelevant axioms - as Lawvere points out - omitted.

First of all, the objects of morphisms

(1)

\mathrm{Hom}(a,b) = \exp(\ell(a,b))

are nothing but real numbers that we can think of as encoding the distance between the source and target object.

Next, the main point to notice here is that the composition morphism of $V$ -enriched categories

(2)

\mathrm{Hom}(a,b)\otimes \mathrm{Hom}(b,c) \overset{\circ}{\to} \mathrm{Hom}(a,c)

for the case that $V=R$ is nothing but the triangle inequality

(3)

\overset{V=R}{\Leftrightarrow} \ell(a,b) + \ell(b,c) \geq \ell(a,c)

satisfied by that distance function.

That’s about it. As Lawvere emphasizes, this way of looking at metrics is quite fruitful.

Right, so how about David Corfield’s exercise, then?

Let’s think of the set of objects of the category we are dealing with as formig some space. We have somehow identified unique edges between any ordered pair of points in that space with straight (later, when we do 2-Riemannian geometry we will have to think straight = geodesic, locally) paths, and have assigned non-negative real numbers to these paths, their length.

Composition of straight paths is the unique straight path obtained by straightening the composite path. (Sorry for this sentence…)

So, we have a notion of metric measuring lengths. When categorifying this, I would like to obtain something which tells me how to measure surfaces.

Maybe I want to find a good general abstract nonsense way of thinking about the area metrics that appear in string theory.

In order to do so, I will now switch to the way of thinking of $V$ -enriched categories as lax functors to $V$ .

In the above setup, we’d be dealing with a category $P_1(S)$ that is akin to the fundamental groupoid of a space $(S)$ . It contains precisely one morphism for every ordered pair of points in $S$ (to be thought of as the unique straight path between the two points).

A lax functor

(4)

P_1 \to \Sigma(V=R)

associates a length $\ell(a,b)$ to every morphism $a \to b$ , such that the triangle inequality is satisfied.

Naturally, I now want to have a 2-category $P_2(S)$ , to be thought of as some sort of 2-category of “straight” (what is the right word here, minimal, extremal?) surfaces in $S$ .

The objects are the points of $S$ , the morphisms now are those freely generated from the elementary straight paths, and there is a unique 2-morphism filling every oriented triangle of 1-morphisms.

(This is the sort of geometric 2-path 2-category as appears all over the place in “transport theory” ( $\to$ ).)

What we are after is a “decoration” or “labeling” of this 2-path 2-category with some categorified notion of distances.

We need a good 2-categorical version of the category $R$ . Once we have found one we’ll call it $R_2$ . Then our 2-metric geometry on $S$ shall be a lax 2-functor

(5)

P_2(S) \to \Sigma(R_2) \,.

Let’s see. Maybe I am getting tired already. I’ll finish this entry by just noting some obvious first guesses.

My first guess is to build $R_2$ by replacing the monoid of non-negative real numbers by the monoidal category of real vector spaces. We need to build a monoidal 2-category from that by throwing in 2-morphisms that categorify the relation $\geq$ . Then we suspend the result to obtain a 3-category with a single object.

When everything is said and done, this should result in a labelling of 2-distances as follows.

1-paths in $S$ should be labelled by vector spaces. 2-paths (surfaces) by linear maps between these. Horizontal composition of surfaces will yield some sort of straightened-out version of the composite of the two surfaces. The details of that depend on what we find as the analogue of $\geq$ .

I’m tired and will maybe spell out more details later. In finishing, I just want to remark that the notion of 2-metric that appears here seems to nicely match the expectations of the area metrics that I would like to see.

Namely, these should contain a component given by the surface integral of some 2-form. More generally, this should really be the surface holonomy of some abelian gerbe. Now, the surface transport of an abelian 1-gerbe is precisely a 2-functor that assigns (1-dimensional, complex) vector spaces to paths and linear maps between these to cobounding surfaces ( $\to$ ).

So this looks reasonably promising. But did I mention that I am getting tired?

Posted at May 25, 2006 7:00 PM UTC

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Re: 2-Metric Geometry

I keep meaning to have a look at this, as I was thinking about homotopy theory as a form of path integral in the rig of truth values, and Grandis’s paper seems to use other rigs for similar ends. Might be relevant to what you’re doing.

Posted by: David Corfield on May 25, 2006 9:57 PM | Permalink | Reply to this

Integration

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Thanks for the link to Grandis’ paper.

His remarks on lengths of path in section 1.8 made me think.

It would be good if there were a more functorial way for doing this, if we want to to lift it to 2-metric geometry.

So I began thinking about how to formulate integration in the spirit of Lawvere’s approach to metric geometry.

Yesterday I had posted two comments on that issue. But I realized that they contained some serious flaws, so I removed them again.

Friday night on the train I then found the time to think about all this more carefully.

Now I believe there is a nice way to define integration in the present context, one that seamlessly fits into Lawvere’s approach and that categorifies nicely.

In the following I try to sketch this.

So let $X$ be some (1-)categroy which models some space. We equip it with a generalized metric $d_X$ by specifying a lax functor

(1)

d_X : X \to S \,,

where $S$ is any 2-category.

As discusses in the entry above, this reduces to Lawvere’s concept of a metric space in the case that $X$ is the pair groupoid of $\mathrm{Obj}(X)$ and that

(2)

S = \Sigma(R)

is the suspension of the monoidal category of non-negative real numbers

(3)

R = \{ a \geq b \} \,.

But I would like to have a concept of integration that does not depend on the special nature of $R$ . It should apply to all generalized metric spaces $d_X : X \to S$ and reduce to the ordinary notion of integration for the spacial case $d_X : P(\mathrm{Obj}(X)) \to \Sigma(R)$ .

More precisely, when I say integration I have in mind measuring the length of paths.

Let $I$ be some category modelling the interval, possibly equipped with some metric itself.

Let

(4)

\gamma : I \to X

be a functor that models the notion of a path in $X$ .

In general we want that functor to have some nice properties. For instance that it be smooth. If $I$ carries a metric

(5)

d_I : I \to S

itself (which we can think of as inducing a topology on $I$ ), we might for instance demand $\gamma$ to extend to a morphism of metric spaces

(6)

\array{ I &\overset{\gamma}{\to}& X \\ d_I \downarrow\;\; && \;\;\downarrow d_X \\ S &= & S } \,.

What I am interested in is computing the length of $I$ as measured by the metric on $X$ .

There is an obvious way how to pull the metric on $X$ back to $I$ . Since for us metrics are functors, we simply set

(7)

\gamma^* d_X : I \overset{\gamma}{\to} X \overset{d_X}{\to} \,.

For the motivating case where $S = \Sigma(R)$ this reproduces the expected notion of pullback metric.

So the remaining task is to define the length of the metric space

(8)

(I, \gamma^* d_X) \,.

We know how to do this in the ordinary case that $S = \Sigma(R)$ .

In that case we choose approximations to our path $\gamma$ by piecewise straight lines. Their length can be measured by $\gamma^* d_X$ . The length of the path that we are after is the supremum of these approximated lengths, taken over all possible piecewise linear approximations.

This is of course nothing but what Marco Grandis is talking about on his page 7.

But let me try to reformulate this procedure in a more categorical language.

What is a piecewise linear approximation to our path? If you think about it, it is nothing but a tuple of $n$ -composable morphisms in $I$ .

So what we want to take a sumpremum over is actually the nerve of $I$ , the simplicial set whose objects are all $n$ -tuples of composable morphisms in $I$ .

This simplicial set has all its face and degenerace maps. We can think of them as morphisms between the $n$ -tuples of morphisms of $I$ , hence as a category. I write $|I|$ for the nerve of $I$ , but regarded as a catgeory whose morphisms are face and degeneracy maps and their composites.

Now, unless I am overlooking something, there is this cute fact:

Any lax functor

(9)

d: I \to \Sigma(C)

naturally induces a functor

(10)

|d| : |I| \to C \,.

$|d|$ acts on the objects of $|I|$ by applying $d$ to every single morphism in the given $n$ -tuple of morphisms.

Moreover, to the face maps in $|I|$ the functor $|d|$ associates the respective “compositor” and to the degeneracy maps it assigns the “unitor”.

Unless I am making a mistake, the simplicial identities controlling the composition of face and degeneracy maps correspond precisely to the coherence laws for the compositor and unitor of the lax functor $d$ . Hence $|d|$ is indeed functorial on $|I|$ .

Next, recall, as Lawvere explains, that limits in $R$ are nothing but suprema.

Taken all this together, it should follow that the length of the path $\gamma : I \to X$ with respect to $d_X$ is nothing but the categorical limit over $|I|$ of the functor $|d| = |\gamma^* d_X|$ :

(11)

\int_I \gamma^* d_X := \mathrm{lim}_{|I|} |\gamma^* d_X| \,.

This is nice (if correct) for two reasons.

1) The right hand side is well defined not just for our motivating example but for the general case where

(12)

d_X : X \to S

is any lax functor to any target 2-category $S$ .

This allows to extend the notion of integration to generalized 1-transport ( $\to$ ). I’ll sketch an example for that below.

2) Given the categorical nature of the above definition, the categorification becomes straightforward.

For 2-metric geometry we’d be considering lax 2-functors (this are really 3-functors) $d_X$ instead of 1-functors. I have made some indications in that direction at the end of the above entry.

Again, in this case one can try to understand how we would measure the surface area of a surface measured by some 2-metric. Again, we would approximate the surface by lots of small plane polygons, whose size we can obtain from $d_X$ . The surface integral would again be the limit over all these polygonal approximations.

But this is now nothing but the limit over the nerve of a 2-category. The above formula for the integral should apply without modification to integral in 2-metric geometry.

Finally, I give an example for a generalized metric that is completely unlike a functor to $\Sigma(R)$ , but still yields useful and interesting information when inserted into the above integration formula.

Namely, consider the case where we have some monoidal category $C$ as we need for describing topological or rational conformal 2D field theories ( $\to$ ).

Moreover, let $X$ be the category obtained from the graph which has one vertex for every $D$ -brane present in a background of that field theory, and an edge between any two vertices whenever there is a species of open strigs stretching between the corresponding two D-branes.

A background of such a 2D field theory is in fact nothing but a lax functor

(13)

d_X : X \to \Sigma(C) \,.

This is just a categorical reformulation of a well known fact about 2D field theories, as I have indicated before ( $\to$ ).

Interestingly, the notion of TQFT/RCFT background in this incarnation seamlessly fits into the above concept of a generalized metric on the “quiver” $X$ .

So what would it mean to have a path

(14)

\gamma : I \to X

in $X$ and to measure its length with respect to $d_X$ ?

Well, plugging all this in the formula

(15)

\mathrm{lim}_{|I|} |\gamma^* d_X|

we find (unless I am wrong) that this limit is nothing but the functor from 1D cobordisms to $C$ , which characterizes the corresponding 2D QFT.

Namely, $I$ here plays the role of a concatenation of “incoming” open strings. The image $\gamma(I)$ in $X$ assigns to each piece $J\subset I$ of $I$ the algebra of string states that stretch between the branes that the endpoints $\partial J$ get assigned to.

The morphisms that come with a categorical limit enocde all the interactions between all these open strings. Being universal, the limit encodes in the end the universal endproduct of the reaction of all the incoming string species. Being universal here means nothing but that this result is independent of the order in which we compute this reaction.

This is all for open strings.

I believe it is relatively easy to see that we similarly obtain reaction process of closed topological strings by suitably applying the 2-metric geometry integral which I talked about above.

But maybe this deserves a more detailed discussion.

Posted by: urs on May 27, 2006 3:19 PM | Permalink | Reply to this

Re: 2-Metric Geometry

Hi Urs,

I could probably be accused of having only one idea. That is probably true, but since you were a part of that idea, I am sometimes confused why you don’t verbalize it before I do (even though I know you are probably thinking it!) :)

We found that most concepts turn out to be much more transparent on a diamond graph than in the continuum. We also know that we can construct a continuum limit, so most if not all, results carry over directly from a diamond graph to Riemannian/Lorentzian manifolds.

If I were to try to construct 2-metric geometry, the obvious starting point would be with diamonds. Is that a totally misguided idea?

Eric

PS: DC, if you haven’t seen it yet, I get the impression you might enjoy our paper.

Posted by: Eric on May 26, 2006 3:19 PM | Permalink | Reply to this

Re: 2-Metric Geometry

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If I were to try to construct 2-metric geometry, the obvious starting point would be with diamonds.

Yes, that should be one nice way to define a domain 2-category $X$ , as we have discussed here.

In the present context, we want to equip such a 2-complex $X$ with some lax 2-functor

(1)

d_X : X \to S

which we want to regard as a generalized (2-)metric on $X$ .

Posted by: urs on May 27, 2006 3:25 PM | Permalink | Reply to this

Re: 2-Metric Geometry

There are tons of ways of constructing geometries. Just curious, why are you concentrating on distance functions? Why not concentrate on a metric tensor? In that case, it would fall right out of a diamond and I would almost wager that if you constructed a discrete 2-geometry, the metric tensor that pops out would be a discrete version of the area metric. If I had a clue, I would try to do it myself :)

Posted by: Eric on May 27, 2006 5:26 PM | Permalink | Reply to this

Re: 2-Metric Geometry

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Just curious, why are you concentrating on distance functions?

The difference between distance functions and Riemannian metrics is small.

Currently I am just processing the nice insight of Lawvere’s seeing where this leads me.

Posted by: urs on May 27, 2006 8:54 PM | Permalink | Reply to this

Re: 2-Metric Geometry

I’ve resurrected the conversation I had with John Baez which concerned partly the remark I made in my first comment that “I was thinking about homotopy theory as a form of path integral in the rig of truth values”. It’s here.

I’m always surprised that there’s not more of this rethinking of basic ideas stuff going on. I think we continually underestimate how substantial is the part of mathematics which is what Lawvere calls *generalized logic*. In his 2002 commentary he notes the further scope he has discovered:

“Thus, contrary to the apology in the introduction of the 1973 paper, it appears that the unique role of the Pythagorean tensor does indeed have expression strictly in terms of the enriched category structure.”

Posted by: David Corfield on May 27, 2006 4:39 PM | Permalink | Reply to this

Re: 2-Metric Geometry

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I’ve resurrected the conversation

Thanks. Very interesting. I will have to think about this. At the moment, I am not sure that I am aware of all the details that the two of you are talking about.

I am wondering, though, if we can incorporate the generalized path integrals that you have in mind into something along the lines of integration in Lawvere’s spirit, the way I have tried to indicate above.

If instead of the semi-ring $R$ of non-negative integers which Lawvere uses to describe metric geometry we use the ring semi $R_\mathrm{max}$ , where addition is the supremum, then we can set up all the above generalized metric theory with this semi-ring, too.

And (unless I am making some mistake), it appears to me that when using that $R_\infty$ rig instead of $R$ , and plugging it into the formula for the categorical integral which I described in the precvious comment on integration ( $\to$ ) we do obtain precisely formula (1) in Litvinov’s paper.

Moreover, here is one observation possibly relating the notion of integral I talked about (using limits of distance funcors over nerve categories).

In my comment on integration I considered the case of integrals over categories modelling 1D spaces, only.

But, if you look at the construction, you see that the definition of the integral is much more general.

We can also compute that categorical limit over the nerve of some category $X$ (the way I described), but throwing out all $n$ -tuples of morphisms in the nerve which do not start and end at two specified points.

What the categorical integral spits out in this situation is precisely the path (of morphisms) in $X$ which maximises (if our ring $R$ or $R_\mathrm{max}$ , or whatever, is regarded as a monoidal category with $\geq$ as morphisms, otherwise maximization is replaced by something more general) - which maximizes the integral of the “distance function” $d_X : X \to \Sigma(R)$ over that path.

So, I think, this concept of integral does incorporate at least

1) the ordinary one (over 1D spaces)

2) any tropical generalizations of the ordinary one

3) classical mechanics, in the sense of producing paths extremalizing some “actions”.

What I cannot see right now (and I am a little bit in a hurry), if there is any lax functor $d_X : X \to S$ which would make that integral also incorporate path integrals as in statistical mechanics and quantum mechanics.

Gotta run now.

Posted by: urs on May 27, 2006 8:50 PM | Permalink | Reply to this

Re: 2-Metric Geometry

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I wrote:

What the categorical integral spits out in this situation is precisely the path (of morphisms) in $X$ which maximises

Sorry, this is not true in general. In fact, it makes sense only in very simple special cases.

Posted by: urs on May 28, 2006 6:47 PM | Permalink | Reply to this

Re: 2-Metric Geometry

A quick glance at Lawvere’s paper reminds me of the “Space-Time as a Causal Set” stuff.

I also wonder if 2-geometries would end up having anything to do with some stuff Jenny Harrison was working on.

PS: While looking up the Bombelli paper, found this recent one.

Posted by: Eric on May 29, 2006 5:59 AM | Permalink | Reply to this

Re: 2-Metric Geometry

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A quick glance at Lawvere’s paper reminds me of the “Space-Time as a Causal Set” stuff.

Yes, that’s one fun thing about Lawvere’s very general concept.

His metrics are not necessarily symmetric. There may be edges which have finite length when you walk along them in one direction, but infinite length if you try to walk them the other way!

In other words, these edges are going just one way. They are directed, in a sense.

In an extreme case the cost of walking along such an edge in the admissable direction is 0 everywhere. In this case the non-symmetric metric reduces to a mere ordered set, where all we know is if we can reach some point from a given point or not (which Sorkin would think of as whether one point is in the timelike future of the other one or not).

Of course the really fun thing about Lawvere’s approach is that things are even more general than that. We can consider any (closed, etc.) monoidal category whatsoever as the decoration category whose objects we regard as labels for lengths of paths. These labels don’t have to be like numbers at all.

I made some remarks above on how we get interesting structures when we measure the “length” of an edge by an object in some general tensor category $C$ . Turns out that then our generalized metric associates internal bimodules in $C$ to edges.

These bimodules, on the other hand, we know how to interpret (for many cases of $C$ ) as spaces of states of strings stretching from the source to the target of the given edge.

That’s pretty neat, I think. It makes a couple of further bells ring, like how condensates of string stretched between branes determine some (noncommutative) distances between these branes.

And this fancy noncommutative D-brane geometry is all encoded (in some sense) already in the general abstract nonsense that Lawvere already knew in 1974! :-)

I also wonder if 2-geometries would end up having anything to do with some stuff Jenny Harrison was working on.

Hm, I have still no idea what Jenny Harrison actually did. Do you?

Posted by: urs on May 29, 2006 6:22 PM | Permalink | Reply to this

Re: 2-Metric Geometry

The way she explained it to me once is that she solved an apparently open question of how to stretch a minimal surface across a closed loop. Why I thought it might be relevant is that in 1-geometry, the distance metric finds the minimal length connecting two points. I thought that in 2-geometry, the distance metric might find the minimal surface area connecting two loops. Like a soap film. I do not know much more than that (which is not much already) :)

Posted by: Eric on May 29, 2006 7:44 PM | Permalink | Reply to this

Re: 2-Metric Geometry

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Like a soap film.

That’s interesting. I thought about something along these lines when I wrote the above comments on 2-metric geometry.

For the following reason.

There is a simple geometric picture behind all this abstract-sounding language. If I could quickly draw some skecthes here, they would be very helpful.

But let me try to describe it in words.

How do we measure the length of a path using Lawvere’s categories?

You know this, let me just make it explicit for the sake of being explicit.

So we have some curved path in some space $X$ . We chose a couple of points $p_i$ on $X$ . Then we connect these points by the shortest possible paths in $X$ .

This yields an approximation to our original path by piecewise “straight” lines.

In fact, these straight lines are nothing but the morphisms in the category describing the space $X$ .

That’s sort of the cool thing about Lawvere’s approach here. The categories know only about the most simple paths involved. Nothing infinitesimal has to be specified.

We get the true length of our curved path as the supremum of the length of all these approximating piecewise straight lines.

So its the categorical limit procedure which extracts the infinitesimal information from the rather coarse-grained information provided by the category $X$ itself.

That’s what is expressed by that fancy categorical integral that I enjoyed talking about above.

Now, let’s categorify. We will replace $X$ by a 2-category. Pick your favorite construction (diamonds ;-). So now we have objects being points in $X$ and 1-morphisms being straight lines.

Given a couple of 1-morphisms, we now need to define the “straight” surface between them and let that be one of our 2-morphisms.

For just a pair of straight lines with a common origin this is simple, since these span a simplex.

But, we might be in need of “straight” surfaces which interpolate between more than 2-edges.

Possibly one would need to contemplate some minimal surface geometry in this contect, for that reason.

I gather the answer is in Jenny Harrison’s work…

Posted by: urs on May 29, 2006 8:16 PM | Permalink | Reply to this

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May 25, 2006