### 2-Metric Geometry

#### Posted by urs

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.

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

Among other things, these form a poset with respect to $\ge $. Any poset can be regarded as a category with precisely one morphism for every $\ge $ 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 $\ge $.

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

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

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

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

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

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 $\ge $. 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 $\ge $.

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?

## 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.