*n*-Curvature, Part III

#### Posted by Urs Schreiber

The context of weak cokernels within obstruction theory seems to be the best way to think of $n$-curvature (I II, III).

Consider the statement

Curvature is the obstruction to flatness.

For $n=1$ this sounds pretty obvious and trivial. But I claim that we should really read it as

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

This is now a statement about $n$-bundles with connection and $(n+1)$-bundles with connection (or rather about the correspodinng $(n+1)$-transport and $n$-transport). And it is not all that trivial anymore. There is a general notion of obstruction theory for $n$-bundles with connection, I think, and it applies here and produces a statement about $(n+1)$-curvature which is at least non-obvious enough to have occupied me for quite a while.

But what was non-obvious once may become obvious as we refine our senses.

There is something non-trivial to be understood here, but we want to understand it in a natural way.

The main thing to be understood is why $(n+1)$-curvature of a $G_{(n)}$-valued $n$-transport takes values in the $(n+1)$-group of inner automporphisms $\mathrm{INN}_0(G_{(n)}) \,.$

The full answer to this involves three main insights:

a) **Obstruction theory**.

b) **$(n+1)$-Curvature is the obstruction to lifting a trivial $n$-transport to a flat $n$-transport**

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

**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.

## Re: n-Curvature, Part III

This was of course my main motivation to do n-gauge theory all along. On the lattice, 1-flatness means that horizontal and vertical links commute (if the corresponding matrices only depend on orientation and not on position). Similarly, 2-flatness becomes the Yang-Baxter equation, and n-flatness the n-simplex equation, which is necessary condition for lattice integrability in n dimensions.

This is the useful definition of (n+1)-curvature on the lattice, because flatness reduces to an important condition. Unfortunately, no good solutions to the n-simplex equations beyond n=2 appear to be known, so maybe (n>2)-flatness is impossible.