The n-Category Café

The First Edge of the Cube

Posted by Urs Schreiber

One knows one is getting to the heart of the matter when the definitions in terms of which one conceives the objects under consideration categorify effortlessly.

Take the ordinary definition of a smooth fibre bundle with connection. Write it out in full detail. Stare at that page for a while and try to categorify everything in sight.

Then, when sufficiently frustrated – to increase the effect – notice that the connection on the bundle gives rise to a parallel transport functor from paths in base space, $\mathcal{P}_1(X)$, to the category of fibers.

For a principal bundle this is $$\mathrm{tra} : \mathcal{P}_1(X) \to G\mathrm{Tor}$$ for a vector bundle this is $$\mathrm{tra} : \mathcal{P}_1(X) \to \mathrm{Vect} \,.$$ This are nothing but representations of the path groupoid.

Now categorify this. This is so trivial that trying to avoid it is like trying to avoid to breathe. Here is a parallel $n$-transport on a principal $n$-bundle with connection: $$\mathrm{tra} : \mathcal{P}_n(X) \to G_n\mathrm{Tor} \,.$$ Here is a parallel transport on an $n$-vector bundle with connection $$\mathrm{tra} : \mathcal{P}_n(X) \to n\mathrm{Vect} \,.$$ Here is a parallel transport on an $n$-bundle whose fibers are objects in the $n$-category $T$ of your choice: $$\mathrm{tra} : \mathcal{P}_n(X) \to T \,.$$ We are just looking at representations of the path $n$-groupoid of a space $X$.

Not all of these $n$-bundles with connection will be useful in applications. Most will be badly pathological. The useful ones will be locally trivial with respect to a predefined local structure.

Again, we want to say this in a way that it’s a breeze to categorify. This means we draw arrows.

A little contemplation reveals that a local structure on an $n$-functor is a square

where $\pi$ is epi and $i$ is mono.

If we can complete the $n$-functor to such a square, it makes sense to say that ($\pi$-)locally the functor has $i$-structure.

All that remains is to choose the $i$ that describes your application.

In fact, it was the observation that plenty of intricate-looking structures that people consider in the literature on gerbes and other higher gauge theory (notice that the term is catching on slowly but steadily) all follow from nothing but a square as above, for various choices of $i$.

In particular, every such square naturally produces an object in a category of descent data, just by itself. All that is required is playing Lego with these squares: two of them paste together to give a transition function. Six to give a triangle. Twelve a tetrahedron – these shapes are the differential cocycle representing the differential cohomology class of the $n$-transport.

And a morphism of squares gives a morphism of differential cocycles in the obvious way, so we have an $n$-functor $$\mathrm{Ex}_\pi : \mathrm{Triv}^n_\pi(i) \to \mathrm{Desc}_\pi^n(i)$$ which extracts from an $n$-transport with $\pi$-local $i$-trivialization $t$ this cocycle information.

All this is god-given. The first point where one actually has to do work comes when we ask if $\mathrm{Ex}_\pi$ may be inverted.

Notice that two, six, twelve, etc. are even numbers. So the descent data $\mathrm{Desc}_\pi^n(i)$ can only know about pairs of squares (1). Can we take the square root?

If we can, and in examples we can, we get a weak inverse $n$-functor $$\mathrm{Rec}_\pi : \mathrm{Desc}_\pi^n(i) \to \mathrm{Triv}^n_\pi(i) \,.$$ which reconstructs an $n$-bundle with connection from its differential cocycle data.

Thus showing that locally $i$-trivializable $n$-functors form an $n$-stack, as they should.

One aspect of the power of this formulation of differential cohomology is that I haven’t even had to mention the word differentiable yet. Nor the word continuous.

While nice, this is a tad more general than what one would be willing to deal with on a normal day: we will want to be able to say that an $n$-transport functor is not just locally trivializable – but also that its descent data has extra structure, like topological, or smooth structure.

For satisfying this demand, essentially no additional work is required beyond translating this demand into arrow-theory. It will take care of itself then.

As John Baez teaches (and as his students have written up here and here – the lore of stuff, structure and property) extra structure (on sets, here) is a faithful functor $$f : C \to \mathrm{Set} \,.$$ Extra structure on our $n$-categories requires that $C$ admits enough pullbacks.

So if we want continuous transport, we choose $C = \mathrm{Top}$. If we want smooth transport, we choose $C = S^\infty$, the category of Chen-smooth spaces.

Given that, realize the local domain $\mathcal{P}_n(X)$ and the local codomain $T^\prime$ internal to $C$. (For instance, let $T^\prime = \mathrm{Gr}$ be a Lie groupoid, a groupoid internal to smooth spaces. This will enforce that $i$-trivial transport functors locally take values in objects and morphisms of that Lie groupoid.) Then denote by $${\mathrm{Desc}^n_\pi(i)}^C$$ the $n$-category of descent data as before, now with everything internal to $C$. Define $${\mathrm{Triv}_\pi^n(i)}^C$$ to be the pre-image of that under $\mathrm{Ex}_\pi$: this are those $n$-transport functors whose local can be lifted through $f : C \to \mathrm{Set}$ $C$, i.e. those whose descent data has $C$-structure. Check that this restricts to an equivalence $$\mathrm{Ex}_\pi^C : {\mathrm{Triv}_\pi^n(i)}^C \to {\mathrm{Desc}^n_\pi(i)}^C$$ and thus obtain a notion of $n$-transport $n$-functor equipped with $t$ a local-$C$ $i$-trivialization. Such a gadget is what really deserves to be called “transport” (as opposed to being an arbitrary functor). And if so, we have the urge to forget the choice of local trivialization $t$ and call the image of the forgetful morphism $${\mathrm{Triv}_\pi^n(i)}^C \to \mathrm{Funct}_n(\mathcal{P}_n(X),T)$$ the $n$-category of transport functors (locally $i$-trivializable with $C$-structure) $${\mathrm{Trans}^n(i)}^C \subset \mathrm{Funct}_n(\mathcal{P}_n(X),T) \,.$$

While this, again, is all given to us, the second point where we may want to do real work is in checking if an $n$-functor in ${\mathrm{Tra}^n(i)}^C$ admits a unique (up to equivalence) $i$-trivialization with $C$-structure.

If so, we arrive at one of the main goals of the entire exercise: an equivalence $${\mathrm{Trans}^n(i)}^C \simeq {\mathrm{Desc}^n_\pi(i)}^C$$ which tells us that given any old $n$-functor $$\mathrm{tra} : \mathcal{P}_n(X) \to T$$ all we need is the guarantee that there is some locally-$C$ $i$-trivialization of it – but we don’t need to carry it around.

All you ever want to do with such an $n$-functor, like taking its sections $$e : \mathrm{tra}_o \to \mathrm{tra}$$ or forming its curvature $$\mathrm{curv}(\mathrm{tra}) : \Pi_{n+1}(X) \to T_{n+1}$$ (which corresponds to things like finding gerbe modules or quantizing the $n$-particle coupled to this ) you’ll do as for plain ordinary $n$-functors $\mathcal{P}_n(X)\to T$.

This is, all in all, what the first edge of the cube is about: local trivialization of $n$-functors.

And for $n=1$ it has now finally been written up:

K. Waldorf & U. S.
Parallel Transport and Functors
arXiv:0705.0452

Some relation to other work:

For $n=2$, and $i : \Sigma G \stackrel{\mathrm{Id}}{\to} \Sigma G_2$ the identity of a strict 2-group, and $C = S^\infty$, the descent category $${\mathrm{Desc}_\pi^2(i)}^C$$ which turns out to be equivalent to differential $G_2$-cocycles subject to the constraint of fake flatness, is what I worked out with John Baez.

(The generalization to non-fake-flat $G_2$-cocycles will actually involve 3-transport.)

A couple of things at that time were not as cleanly worked out as they eventually should be. This includes the relation between smooth 1-functors $\mathcal{P}_1(X) \to \Sigma G$ and $\mathrm{Lie}(G)$-valued differential forms. In fact, one finds an equivalence of the category of these smooth 1-functors with the category of global differential $G$-1-cocycles: $$\mathrm{Funct}(\mathcal{P}_1(X))^\infty \simeq \bar Z^1_{\mathrm{id}_X}(G) \,.$$

The category of local such cocycles $$\bar Z^1_\pi(G)$$ is in fact canonically equivalent to that of suitable smooth anafunctors. A full proof of this equivalence didn’t make it into the paper.

These constructions all involve crucially the category of paths in the transition groupoid.

In fact, much of the structure one encounters here can be traced back to the fact that the functor $$\mathcal{P}_1 : S^\infty \to \mathrm{Cat}_{S^\infty}$$ which sends a space to its path groupoid preserves the square $$\array{ Y^{[2]} &\to& Y \\ \downarrow && \downarrow \\ Y &\to& X }$$ as a pullback square, but not as a pushout square. Instead, the pushout after applying $\mathcal{P}_1$ is that category of paths in the transition groupoid.

This discussion didn’t make it into the paper.

Another equivalence which didn’t make it into this paper is that with the notion of smooth functors used by Stolz and Teichner.

All this is obtained by choosing suitable $i$ in the general theory of locally trivializable 1-transport.

Curiously, it does make good sense to consider $i = \mathrm{Id}$. For instance, we might start with a principal transport $$\mathrm{tra} : \mathcal{P}_1(X) \to G\mathrm{Tor}$$ and then postcompose this with the morphism $$f : G\mathrm{Tor} \to \Sigma G$$ which is – at the level of categories internal to $\mathrm{Set}$ an equivalence that identifies the groupoid with its vertex group – here by choosing for each $G$-torsor in the world an isomorphism to $G$ itself.

This is of course not smooth. In fact $f \circ \mathrm{tra}$ will be badly pathological in general.

Such arbitrary functors from paths to the group are called generalized connections in parts of the literature. Applying our theorem to such pathological functors we find: either a generalized connection comes from a smooth connection by applying an $f$ as above, or it doesn’t. But if it does, then this smooth bundle with connection that it comes from is unique up to equivalence.

We give a condition to check this explicitly: a generalized connection can be equipped with a smooth local structure if and only if all its “Wilson lines” are smooth.

Update: It turns out that some discussion of this paper is taking place in the thread Quantization and Cohomology (Week 23).