The n-Category Café

For the present purpose we are interested in $n$-connections on trivial $n$-bundles. This means that we may choose our domain path category to be

the fundamental groupoid of $X$, based on a couple of chosen points in $X$, at least one for each connected component. (All choices will lead to equivalent groupoids.)

An $n$-connection on a flat $n$-bundle over $X$ is then a pseudofunctor

where $T$ is some codomain $n$-category. (Equivalently, we could turn $P_.(X)$ into an $n$-category by filling in higher simplices and look at proper $n$-functors.)

In the present case the codomain $T$ will be nothing but

the $n$-fold suspension of the group $U(1)$. This has $U(1)$-worth of $n$-morphisms with everything else being trivial. So this describes flat principal $\Sigma^n(U(1))$-$n$-bundles ($(n-1)$-gerbes) on $X$.

Using the nature of pseudofunctors, it is now easy to see that such $n$-functor $\mathrm{tra}$ is nothing but a groupoid $n$-cocycle on $\Pi(X)$. We have talked about that rather recently here. It is a special case of how $p$-functors with values in $\Sigma^p(U(1))$ are a generalization of differential $p$-forms #.

A groupoid cocycle is defined in straightforward analogy to a group cocycle. For instance a group 3-cocycle on a finite group $G$ governs Dijkgraaf-Witten theory. Since $G= \Pi_1(B G)$ we may, from the above point of view, regard such a 3-cocycle on $G$ as a flat 2-gerbe with connection on $B G$. This is precisely the 2-gerbe on $B G$ which Carey-Johnson-Murray-Stevenson-Wang construct in terms of transition bundle gerbes, as discussed here.

Let’s try to say what a section and in particular a flat section of an $n$-bundle with connection is in this language.

First, to get a feeling for what is going on, consider the familiar case $n=1$.

Since our $(n=1)$-$U(1)$-bundle is flat, it is trivializable and we may assume it to be in fact trivial. Then, if we consider the associated complex line bundle, every fiber is precisely the standard copy of $\mathbb{C}$.

A section is hence a map

which assigns a complex number to every point in $P_.(X)$.

The section is flat (covariantly constant) if for any two points $x$ and $y$ connected by a morphism $x \stackrel{\gamma}{\to} y$ the value over $y$ is the same as that over $x$ parallel transported along $\gamma$:

(For convenience I am not distinguishing notationally between the principal transport $\mathrm{tra}$ and its associated vector transport $\rho_* \mathrm{tra}$ induced by the standard rep $\rho : \Sigma(U(1)) \to \mathrm{Vect}$).

We can say this in a more natural way: denote by

the constant functor which sends sends every object to $\mathbb{C}$ and every morphism to the identity morphism. Then a flat section is precisely a natural transformation

That’s because the corresponding naturality squares read

for all $\gamma \in \mathrm{Mor}(P_.(X))$.

(More generally we also want to understand non-flat sections this way. This then requires that we move from codomain $T$ to codomain $\mathrm{inn}(T)$ such that these naturality squares can be filled with something - namely with the covariant derivative of the section #.)

A little reflection # shows that we actually want to think of $P_.(X)$ not as what physicist call target space, but as what they call configuration space. In the case $p=1$ it is the configuration space of the 1-particle

for which, by coincidence, target space $P_.(X)$ and configuration space $\mathrm{Hom}(d^1,P_.(X))$ happen to coincide:

But this is no longer true for $p \gt 1$. So we might want to equivalently reformulate the above by saying that a flat section is a natural transformation

Now move on to $n=2$. Simon Willerton gives a couple of definitions of what he wants to understand under a section of a flat $U(1)$-gerbe defined in terms of a groupoid 2-cocycle on $P_.(X)$ on p.17 and following of his paper, showing that they are all equivalent.

One definition goes like this: a flat section associated to the 2-cocycle $\tau$ on the groupoid $P_.(X)$ is an assignment of vector spaces to objects of $P_.(X)$ and of linear maps

between these, such that for all composable $\gamma$ and $\gamma'$ we have

Whenever you see an apparently twisted functor like that, chances are you are actually looking at a pseudonatural transformation between 2-functors.

Same here:

By the argument given here, a connection on a line 2-bundle (line bundle gerbe) is a 2-functor locally with values in bimodules which sends all points to the $\mathbb{C}$-algebra $\mathbb{C}$ (representing the 2-vector space $\mathrm{Mod}_\mathbb{C} = \mathrm{Vect}$), sends all morphisms to $\mathbb{C}$, regarded as a $\mathbb{C}$-$\mathbb{C}$-bimodule, and finally sends 2-morphisms (here: triangles spanned by pairs of composable morphisms) to elements of $U(1)$ (regarded as bimodule homomorphisms).

In the present context this means that we should think of the groupoid 2-cocycle $\tau$ on $P_.(X)$ as defining a pseudofunctor whose nontrivial compositor is

Even though the flatness condition on our transport makes most everything rather trivial, it is crucial that the above is really considered as taking values in bimodules

because this affects the properties of pseudonatural transformations

Here $d^2$ is the 2-particle (the “open string”), i.e. the category

with two objects and a single nontrivial morphism. Accordingly, $[d^2, P_.(X)]$ is the configuration space of this 2-particle and $[d^2, \mathrm{Bim}]$ its space of phases.

By the general reasoning #, we would like to address such an $e$ as a covariantly constant section of our flat 2-bundle with connection $\mathrm{tra}$.

Does that notion coincide with Simon Willerton’s definition?

To answer that, we need to write down the naturality tin can diagram in $\mathrm{Bim}$ which defines the above pseudonatural transformation. It reads

for all composable $\gamma$, $\gamma'$.

Notice that, since this diagram lives in $\mathrm{Bim}$, all the $e_x$, $e_y$, etc must be $\mathbb{C}$-bimodules - hence vector spaces - and all the $e(\gamma)$, $e(\gamma')$ must be morphisms of $\mathbb{C}$-bimodules - hence linear maps between vector spaces.

So we find that a pseudonatural transformation $e$ like this is precisely the same as the twisted groupoid representation from Simon Willerton’s paper.

Connaisseurs of gerbes will notice that the above is pretty similar to the situation encountered in the study of modules and trivializations of gerbes. A gerbe with connection may be trivialized by a vector bundle with connection $\nabla$, such that the curving $B$ of the gerbe equals the curvature $F_\nabla$ of the bundle as

roughly. This equation is nothing but the infinitesimal version of the twisted groupoid rep condition

Of course all this completely harmonizes with the general picture advertized in the paper. All I have would have to add here to that is the observation that the general way to say what a (flat) section of an $n$-bundle with connection is seems to be:

a flat section over the $p$-point of an $n$-bundle with connection is a 2-morphism