The n-Category Café

I think it might equivalent (biequivalent) to $H \to G$

Yes, right. I should have commented on that. We had run into this issue before, for instance here (and also by private email), going back to the considerations on $n$-curvature # and the description of $n$-connections in terms of higher Schreier theory #.

Let’s look at the analogous situation one level down.

Say we have a central extension

of an ordinary group $H$.

Say we want to understand lifts of principal $H$-bundles to $\hat H$-bundles. These lifts are obstructed by the corresponding $U(1)\to 1$ lifting gerbes. If this lifting gerbe is nontrivial, the lift does not exist.

Notice this:

These $(U(1)\to 1)$-lifting gerbes together with the $H$-bundles they correspond to, provide exactly the data of a trivializable nonabelian gerbe with 2-group

If you write it down, you see immediately that the 3-cocycle of a $(U(1) \to \hat H)$-gerbe is the same as that of the (possibly twisted) bundle obtained by lifting a 2-cocycle for an $H$-bundle to $\hat H$.

So this is the same sort of situation as with the 3-group above:

the 2-group

should be equivalent to

Correspondingly, all $(U(1)\to \hat H)$-gerbes are trivializable.

In fact, they are trivializable because they are all manifestly trivialized by a twisted $\hat H$-bundle.

But, if you take the $(U(1)\to \hat H)$ 3-cocycle and forget the $\hat H$-part, just remembering the $U(1)$-part, you do get a $(U(1)\to 1)$ 3-cocycle, which is (in general) non-trivializable as a $(U(1)\to 1)$-cocycle.

I think this is what is going on in the Chern-Simons theory, too.

As supportive evidence, consider this: Stolz and Teichner emphasize that a string bundle, i.e. a $(\hat \Omega G \to P G)$-gerbe, provides a trivialization for Chern-Simons theory (pp. 79-80).

For these reasons I think it does make sense to consider these “blown up” $n$-groups such as $U(1)\to \hat H$ and $\mathrm{ker}(t) \to H \stackrel{t}{\to} G$.

While the $n$-bundles with these structure groups will be trivializable, there is interesting information in how they are trivialized, i.e. which morphism (twisted $(n-1)$-bundle) trivializes them.

It’s been a while, but what I said in 2006 was wrong. Writing $t\colon \widehat{K} \to L$ for the crossed module, $A = ker(t)$ and $K = \widehat{K}/A \trianglelefteq L$, I believe we have weak equivalences $$(A \hookrightarrow \widehat{K} \to L)\quad \xrightarrow{\sim}\quad (1 \to K \hookrightarrow L)\quad \xrightarrow{\sim}\quad (1 \to 1 \to L/K).$$ Hence the 3-group Urs mentions in his post is up to equivalence of smooth group stacks, just an ordinary Lie group $G = L/K$. What this does, however, is give us a representative $$G \xleftarrow{\sim} (A \hookrightarrow \widehat{K} \to L) \to (A \to 1 \to 1)$$ for the map of 3-group stacks $G \to \mathbf{B}^2 A$ and hence, after delooping, the classifying map $\mathbf{B}G \to \mathbf{B}^3 A$ for the multiplicative $A$-gerbe on $G$.

As the pieces of the puzzle fall into place, have you gained a better view of what the picure is?

What made puzzle pieces fall - and apparently even into place - was, for me, the observation (stated here and supported - or in fact explained - there) that the right notion of $n$-transport with values in $T$ is not, in general, an $n$-functor with values in $T$, but an $(n+1)$-functor with values in $\mathrm{Aut}(T)$ (or, more generally, $\mathrm{End}(T)$).

For the relevant applications, we may find $T$ sitting inside $\mathrm{End}(T)$ as part of the “inner” endomorphisms. Let me call these $\mathrm{Inn}(T)$.

But even if we restrict to these inner morphisms, there are more degrees of freedom in $\mathrm{Inn}(T)$ than in $T$. There are injections $$T \stackrel{\subset}{\to} \mathrm{Inn}(T) \stackrel{\subset}{\to} \mathrm{End}(T) \,.$$

More precisely - for $T = \Sigma(C)$ the suspension of a monoidal category - $\mathrm{Inn}(\Sigma(T))$ looks essentially like $\Sigma(T)$, only that 2-morphisms of $\Sigma(T)$, which are 2-globes $$\array{ \bullet &\stackrel{R}{\to}& \bullet \\ &\;\Downarrow f& \\ \bullet &\stackrel{R'}{\to}& \bullet }$$ are replaced by squares $$\array{ \bullet &\stackrel{R}{\to}& \bullet \\ v_f \downarrow\;\; &\;\Downarrow f& \;\;\downarrow u_f \\ \bullet &\stackrel{R'}{\to}& \bullet } \,,$$ and that there are now 3-morphisms going between the vertical sides of these squares.

Now, unless I am mixed up, the new freedom provided by these vertical arrows $v_f$ and $u_f$ accounts in particular for the following structures:

• For 2-functors into $T = \Sigma(G_2)$, for $G_2$ a strict 2-group, we have $v_f = \mathrm{Id}$ and $u_f$ provides # the fake curvature #.
• In particular, for 2-functors into $T = \Sigma(H \stackrel{t}{\to} G)$, we have $(\mathrm{ker}(t) \to H \stackrel{t}{\to} G) \subset \mathrm{Aut}(H \to G)$, with $u_f = \mathrm{Id}$ and the $\mathrm{ker}(t)$-degree of freedom in the morphisms $u_f \to u'_f$ which run “perpendicular to $(H \to G)$”.
• For lax 2-functors into $T = \Sigma(C_2)$, for $C_2$ a monoidal category with duals on objects, $v_f$ and $u_f$ seem to provide # the “Wilson lines” running perpendicular to the boundary in Chern-Simons 3D TFT.

The third point can be understood as the image under a 3-representation of the second point.

Could you describe it in terms of the “X is a Y-structure in the context Z” story you told us about here?

Very roughly, my impression is this:

The “Y-structure” we are dealing with is:

- an $n$-transport pseudofunctor $$\mathrm{tra} : P_n \to \mathrm{End}(T)$$ with associated curvature $$\mathrm{curv}_\mathrm{tra} : P_{n+1} \to \mathrm{End}(T) \,.$$

Then

• A non-fake-flat principal $(H \to G)$-gerbe with connection is a Y-structure of the above sort for $T$ a 2-groupoid with vertex 2-group equivalent to $\Sigma(H \to G)$

• Chern-Simons theory with gauge group $G$ is the quantum theory of # a Y-structure as above, which is associated to a principal $\mathrm{String}_G = (\hat \Omega G \to P G)$-gerbe example above by a canonical 3-rep.

  In particular, the relation $$\text{2D CFT} \leftrightarrow \text{3D TFT}$$ is induced by $$\mathrm{tra} \leftrightarrow \mathrm{curv}_\mathrm{tra}$$

The first of these two items I think I understand sufficiently. For the second item I have so far no full description. But I do have the evidence provided in the entry above. There are probably refinements necessary in my statement of this second item.

But that’s the picture that I thought I see emerging.