The n-Category Café

General remark: I’ll say $n$-group and $n$-bundle throughout, but all statements in the world of Lie $n$-groupoids I can and have made precise so far only up to $n=2$ with everything strict. But not so in the differential picture, which is mentioned in the remarks at the end: the great advantage of that differential picture is that we obtain it from the integral picture and then extend it to arbitrary $n$, since Lie $n$-algebras are handled so much more easily than Lie $n$-groups.

a) Obstruction theory.

To measure the obstruction to lifting a $G_{(n)}$-transport $$\mathrm{tra} : P \to \Sigma G_{(n)}$$ through a sequence

$$K \to B \to G_{(n)}$$

we form the weak cokernel

$$\mathrm{wcoker}(K \to B) \simeq G_{(n)}$$

to first puff up the $n$-group $G_{(n)}$ to an equivalent $(n+1)$-group $\mathrm{wcoker}(K \to B)$, such that the projection $B \to G_{(n)}$ becomes an inclusion $$B \stackrel{i}{\hookrightarrow} \mathrm{wcoker}(K \to B) \simeq G_{(n)}$$

which then allows us to form, in turn, the cokernel of that inclusion

$$\array{ &&&&&& \mathrm{coker}(i) \\ &&&&&\nearrow \\ &&&& \mathrm{wcoker}(K \to B) \\ &&&\multiscripts{^i}{\nearrow}{} & \uparrow^{\simeq} \\ K &\to& B &\to& G_{(n)} } \,.$$

The composition $$P \stackrel{\mathrm{tra}}{\to} \Sigma G_{(n)} \stackrel{\simeq}{\to} \mathrm{wcoker}(K \to B) \to \mathrm{coker}(i)$$ then gives a $\mathrm{coker}(i)$-transport which measures the failure of $\mathrm{tra}$ to lift to a $B$-transport.

(More details and examples are discussed in String- and Chern-Simons $n$-Transport – see the section of the same name – and in the BIG diagram. )

b) Curvature is the obstruction to lifting a trivial transport to a flat transport

A $G_{(n)}$-$n$-bundle without connection on $X$ is a transport $n$-functor $$P : \Pi_0(X) \to \Sigma G_{(n)}$$ equipped with a smooth local $G_{(n)}$-trivialization. (See The first Edge of the Cube if that sounds strange.)

A $G_{(n)}$-$n$-bundle on $X$ with flat connection is a transport $n$-functor $$\mathrn{tra} : \Pi_n(X) \to \Sigma G_{(n)} \,.$$

Given a $G_{(n)}$-bundle with connection, we may ask if we can extend it to a $G_{(n)}$-bundle with flat connection

$$\array{ \Pi_0(X) &\hookrightarrow& \Pi_n(X) \\ \downarrow^* & \swarrow_{\mathrm{tra}_{\mathrm{flat}}} \\ \Sigma G_{(n)} } \,.$$

In general we cannot. The obstruction is given by a $\mathrm{wcoker}(\mathrm{Id}_{G_{(n)}})$-transport.

To see this more clearly, we need a little bit of local data:

A possibly nontrivial $G_{(n)}$-bundle without connection on $X$ is a surjective submersion $F \to Y \to X$ with connected fibers, together with a flat $\Sigma G_{(n)}$-transport on the fibers

$$P : \Pi_n(F) \to \Sigma G_{(n)} \,.$$

A flat $G_{(n)}$-connection, on this, is an extension of this to a functor on all of $Y$:

$$\mathrm{tra}_{\mathrm{flat}} : \Pi_n(Y) \to \Sigma G_{(n)} \,.$$

In general, this does not exist. What always exists, though, is the completely trivial bundle with connection

$$\mathrm{tra}_0 : \Pi_n(Y) \to \{\bullet\} \,,$$

i.e. the principal bundle for the trivial structure group.

Hence the question that we are asking when asking for curvature is:

Can we lift the connection for the trivial group through the exact sequence $$G_{(n)} \stackrel{\mathrm{Id}}{\to} G_{(n)} \to \{\bullet\}$$ ?

Curvature is a very degenerate case of general obstruction theory: we are asking for obstructions to extending the trivial structure group.

More precisely, we want to find a lift of $\mathrm{tra}$ which does restrict to the fixed functor $P : \Pi_n(F) \to \Sigma G_{(n)}$ on the fibers of the surjective submersion, meaning we want to lift to

$$\array{ \Pi_n(F) &\hookrightarrow& \Pi_n(Y) \\ \downarrow && \downarrow^{\mathrm{tra}} \\ \Sigma G_{(n)} &\stackrel{\mathrm{Id}}{\to}& \Sigma G_{(n)} } \,.$$

In general this will not work. But we have obstruction theory as above to figure out what the obstructing $(n+1)$-bundle with connection will be: it will be an $(n+1)$-transport with values in $$\mathrm{wcoker}(i)$$ obtained by first lifting the $\{\bullet\}$-transport $\mathrm{tra}_0$ to an equivalent $\mathrm{wcoker}(\mathrm{Id}_{G_{(n)}})$-transport and then checking which mistake in $\mathrm{wcoker}(i)$ we make thereby:

$$\array{ \Pi_n(F) &\hookrightarrow& \Pi_n(Y) &\to& \Pi_n(X) \\ \downarrow && \downarrow^{\mathrm{tra}} && \downarrow^{\mathrm{curv}} \\ \Sigma G_{(n)} &\stackrel{\mathrm{Id}}{\to}& \mathrm{wcoker}(\mathrm{Id}_{G_{(n)}}) &\stackrel{\mathrm{Id}}{\to}& \mathrm{wcoker}(i) \\ \\ cocycle && attempted flat lift && failure of lift \\ \\ cocycle && n-Ehresmann connection && curvature } \,.$$

It remains to compute these weak cokernels.

c) Inner automorphisms and weak cokernels of identities on $n$-groups

As I reported in Detecting higher order necklaces Enrico Vitale discusses that the inner automorphism $(n+1)$-group on the $n$-group $G_{(n)}$ that I discussed with David Roberts is the weak cokernel of the identity on $G_{(n)}$: $$\mathrm{wcoker}(\mathrm{Id}_{(G_{(n)})}) = \mathrm{INN}(G_{(n)}) \,.$$

So that’s why we see these inner automorphism $(n+1)$-groups appearing in the theory of $n$-curvature.

This is the story behind the slogan

$(n+1)$-Curvature is the obstruction to $n$-flatness.

Remarks.

a) Notice how the obstructions here are not just classes, but really full-fledged $(n+1)$-transports. I am claiming that this is related to the phenomenon called holography, which also exhibits That shift in dimension – essentially the image of the shift we see here under quantization. There would be more to say about this. But it is, unfortunately, top secret.

b) Everything I said applies directly also to the differential Lie-$n$-algebraic picture. There the inner automorphism $(n+1)$-group $\mathrm{INN}(G_{(n)})$ becomes the inner derivation Lie $(n+1)$-algebra $\mathrm{inn}(g_{(n)})$, as described in section: Bundles with Lie $n$-algebra connection (see also, for instance, Lie $n$-algebra cohomology).

Recall from More on tangent categories that forming the inner derivation Lie $(n+1)$-algebra corresponds, Koszul-dually, to forming the shifted tangent bundle of the dg-manifold corresponding to $g_{(n)}$.

c) There is more going on than meets the eye. See the blockbuster diagram movie (original with subtitles) in section: $n$-Categorical background, subsection $G_{(n)}$-bundles with connection, based on the bestselling novel Tangent categories to get an impression.