### Obstructions, Tangent Categories and Lie *N*-tegration

#### Posted by Urs Schreiber

Last week I visited Yale, where I talked with Hisham Sati about supergravity, Chern-Simons 3-bundles and other higher structures in String theory, with Mikhail Kapranov about parallel surface transport, with Todd Trimble about Hecke algebras, groupoidification and the sociology of quantum gravity, and with Anton Zeitlin about BV formulation of Yang-Mills theory. With a short stop at home I went from there straight to Croatia, where I am currently attending Categories in Geometry and Physics.

The first day here was quite remarkable, with an astonishing amount of Croatian media attention for us followed by a couple of rather impressive talks. After Pavol Ševera’s talk I had the chance to chat with with him about differentiation and integration of Lie $n$-algebras, much to my delight.

Last night at about 3 am I learned that I am supposed to give a public talk to high school kids and journalists next Friday. And sure enough, later towards the pm part of the day I found myself being interviewed by some TV team on how I plan to convey the ideas of Categories in Geometry and Physics to the broad public.

Trying to think of something which is both interesting and relevant, as well as easy to understand and expressible in terms of lots of pictures, I thought I’d give a talk revolving around the Eckman-Hilton argument, illustrated by two party-balloons joined at their tip and labeled “A” and “B”.

(The very moment that I am posting this, a journalist sends me the following two questions:

Q1. Your field of interest in mathematics is the relation between categories and quantum physics. During the Split categories conference you will be giving a popular lecture on this topic to the high school students, so can you briefly summarize that lecture for our readers?

Q2. You are also one of the admins at $n$-category cafe blog. As it has become an interesting and influential internet based communication and discussion tool for resarchers in category theory, can you tell me a bit more about it - how did the idea start, how it developed, and what is its influence on the category maths community professional work?

)

Hence there’d be many fun things to write about here. And I should. But right now I will instead talk more about two of the issues I had mentioned recently in Obstructions for $n$-bundle lifts and related.

I’ll say a couple of things that need to be said. But even more needs to be said. Some of which can already be found in the incomplete

String and Chern-Simons $n$-transport.

The big obstruction diagram presented below exists as a nice big hand-drawn picture which includes a copy of the diagram that spells out the explicit meaning of each and every arrow in full concrete detail. I am hoping to be able to provide an electronic copy of that in a while.

**
Curvature of parallel $n$-transport
**

For $G_{(n)}$ a Lie $n$-group, $X$ a smooth space and $\Pi_{(n+1)}(X)$ the fundamental $(n+1)$-groupoid of $X$, a (general, non-fake flat) $G_{(n)}$-$n$-bundle with connection is a pullback diagram of the suspended universal $G_{(n)}$-$n$-bundle

$\array{ T \leftarrow \Sigma \mathrm{INN}(G_{(n)}) \leftarrow \Sigma G_{(n)} }$

(where $T$ should have a universal realization but is for the moment part of the concrete data)

to

$\array{ \Pi_{(n)}(X) &\leftarrow& \Pi_{(n+1)}(P) &\leftarrow& \Sigma G_{(n)} \\ \downarrow &\Downarrow& \downarrow^{(\mathrm{tra},\mathrm{curv})} &\Downarrow& \downarrow \\ T &\leftarrow& \Sigma \mathrm{INN}(G_{(n)}) &\leftarrow& \Sigma G_{(n)} } \,,$

with everything demanded to be smooth, in a straightforward generalization of the magic square for fake-flat parallel $n$-transport.

Here the leftmost functor is the “classifying map”, the one in the middle is the parallel $n$-transport with its $(n+1)$-curvature and the rightmost one is the $G_{(n)}$-$n$-cocycle corresponding to this differential $G_{(n)}$-$n$-cocycle.

**Mapping cones and tangent categories**

Recall from the discussion at The inner automorphism group of a strict 2-group that the mapping cone of the identity 2-functor $C \to C$, in the world of strict 2-groupoids, amounts to forming what I called the “tangent category” $T C$, which is obtained by mapping the interval $\bullet \stackrel{\sim}{\to} \circ$ into $C$ and admitting only those homotopies between such maps which fix the left endpoint.

Now, you won’t be surprised to learn that the mapping cone of an arbitrary injection $t : C_0 \hookrightarrow C_1$ turns out to correspond to maps of the interval into $C_1$, whose homotopies are resticted to fix the left endpoint and to have a right endpoint transerve a 2-path in the image of $t$. But I mention it nevertheless. Since I think it is true and useful.

**Obstructions of $g$-bundles through String-like extensions**

Given a Lie $n+1$-cocycle $\mu$ on a Lie algebra $g$, we obtain an extension

$\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu \to g$

of $g$ by a $(n-1)$-fold shifted $\mathrm{Lie}(U(1))$ to a Lie $n$-algebra of Baez-Crans type. For $\mu$ the 3-cocycle on a semisimple Lie algebra, this is the extension which gives the (skeletal version of the) String Lie 2-algebra.

Whenever (see String and Chern-Simons Lie $n$-algebras) the cocycle $\mu$ here is related by transgression to an invariant polynomial $k$, the Lie $n$-algebra $g_\mu$ sits in a weakly exact sequence

$g_\mu \to (\mathrm{cs}_k(g) \simeq \mathrm{inn}(g_\mu)) \to ch_k(g)$

of Lie $(n+1)$-algebras.

Noticing that for any Lie $n$-algebra $g_{(n)}$ the sequence

$g_{(n)} \to \mathrm{inn}(g) \to (b g_{(n)} \simeq H^\bullet(BG))$ is the Lie version of the universal $g_{(n)}$-$n$-bundle we obtain in fact a sequence of sequences

$\array{ \mathrm{Lie}(\Sigma^{n} U(1)) &\leftarrow& \mathrm{inn}(\mathrm{Lie}(\Sigma^{n-1} U(1))) &\leftarrow& \mathrm{Lie}(\Sigma^{n-1} U(1)) \\ \downarrow && \downarrow && \downarrow \\ \mathrm{ch}_k(g) &\leftarrow& \mathrm{cs}_k(g) \simeq \mathrm{inn}(g) &\leftarrow& g_{\mu_k} \\ \downarrow && \downarrow && \downarrow \\ b g &\leftarrow& \mathrm{inn}(g) &\leftarrow& g }$

describing how the universal $g_{\mu}$-$n$-bundle is the extension by the universal $\Sigma^{n-1}U(1)$-bundle of the universal $g$-bundle.

Any concrete $g$-bundle with a classifying map

$T X \to b g_{(n)}$

and differential parallel $n$-transport

$T P \to \mathrm{inn}(g)$

completed to a diagram

$\array{ T_{li} (|G_{(n)}|) &\stackrel{\simeq}{\to}& g \\ \downarrow &\Downarrow& \downarrow \\ T P &\stackrel{(A,F_A)}{\to}& \mathrm{inn}(g) \\ \downarrow &\Downarrow& \downarrow \\ T X &\to& b g_{(n)} }$

(which should be a pullback, but I need to clarify this. The 2-morphisms here have to respect the sequence property of the left and right vertical composites, meaning that they need to vanish after the corresponding whiskering. The space $|G_{(n)}|$ is discussed below.)

We are asking under which conditions this diagram factors through the String extension

$\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu \to g \,.$

To check this we first puff up $g$ as the mapping cone

$g \simeq (\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu)$

then look at the embedding of $g_\mu$ into this

$i : g_\mu \hookrightarrow (\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu)$

and consider the pushforward along the strict cokernel

$g_\mu \stackrel{i}{\hookrightarrow} (\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu) \to (\mathrm{coker}(i) = \mathrm{Lie}(\Sigma^n U(1)) \,.$

Doing this to the entire sequence of universal $n$-bundles above

$\array{ \mathrm{Lie}(\Sigma^{n} U(1)) &\leftarrow& \mathrm{inn}(\mathrm{Lie}(\Sigma^{n-1} U(1))) &\leftarrow& \mathrm{Lie}(\Sigma^{n-1} U(1)) \\ \downarrow && \downarrow && \downarrow \\ \mathrm{ch}_k(g) &\leftarrow& \mathrm{cs}_k(g) \simeq \mathrm{inn}(g) &\leftarrow& g_{\mu_k} \\ \downarrow && \downarrow && \downarrow \\ b g &\leftarrow& \mathrm{inn}(g) &\leftarrow& g \\ \downarrow && \downarrow && \downarrow \\ &\leftarrow& \mathrm{inn}(\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu) &\leftarrow& (\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu) \\ \downarrow && \downarrow && \downarrow \\ \mathrm{Lie}(\Sigma^{n+1} U(1)) &\leftarrow& \mathrm{inn}(\mathrm{Lie}(\Sigma^{n} U(1))) &\leftarrow& \mathrm{Lie}(\Sigma^{n} U(1)) }$

and then composing with the pullback

$\array{ T_{li} (|G_{(n)}|) &\to& g \\ \downarrow &\Downarrow& \downarrow \\ T P &\to& \mathrm{inn}(g) \\ \downarrow &\Downarrow& \downarrow \\ T X &\to& b g_{(n)} }$

yields the obstructing $\Sigma^{n} U(1)$-$(n+1)$-bundle

$\array{ T X &\leftarrow& T P &\leftarrow& g \\ \downarrow && \downarrow^{(A,F_A)} && \downarrow \\ b g &\leftarrow& \mathrm{inn}(g) &\leftarrow& g \\ \downarrow && \downarrow && \downarrow \\ &\leftarrow& \mathrm{inn}(\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu) &\leftarrow& (\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu) \\ \downarrow && \downarrow && \downarrow \\ \mathrm{Lie}(\Sigma^{n+1} U(1)) &\leftarrow& \mathrm{inn}(\mathrm{Lie}(\Sigma^{n} U(1))) &\leftarrow& \mathrm{Lie}(\Sigma^{n} U(1)) } \,.$

All this can be spelled out very explicitly and one finds that it indeed correctly describes how the obstruction is the ($\Sigma^n U(1)$-($n+1$)-bundle classified by) the characteristic class $k(F_A)$.

**Integration of Lie $n$-algebras to Lie $n$-groupoids**

For $g_{(n)}$ any Lie $n$-algebroid, let

$B G_{(n)} := (X \mapsto \mathrm{Hom}_{n\mathrm{Lie}}(T X , g_{(n)}))$

be the corresponding presheaf on smooth manifolds.

This is not quite a Chen-smooth structure, but it is useful to think of it as a slight generalization of Chen-smooth structures. We may work entirely inside smooth spaces realized by presheaves over smooth manifolds. So for now such a presheaf is a “smooth space” and an $n$-category internal to such presheaves is a smooth $n$-category, an $n$-group internal to these presheaves a Lie $n$-group.

Using this setup, the Lie $n$-group(oid) integrating $g_{(n)}$ should be

$G_{(n)} := \Pi_n( B G_{(n)}) \,,$

where, recall, $B G_{(n)}$ is the *smooth* space from above, and where $\Pi_n(\cdot)$ forms the smooth internal fundamental $n$-groupoid.

I am proposing that *this* is the right way to think of the process sketched at the beginning of Pavol Ševera’s Some title containing the words “homotopy” and “symplectic”, e.g. this one, which we had discussed here recently

The fun thing is that with this definition one easily finds that there is canonically $p$-form data on $B G_{(n)}$ which plays the role of left-invariant $p$-forms and obeys the differential graded algebra rulese that define $g_{(n)}$ in the Koszul dual picture.

This quasi-tautological (but also quasi-tautological things need to be figured out first…) construction is pretty useful. It gives for instance a 1-line proof that $\mathrm{String}_k(G)$ is something like 3-connected: $B G_{(n)}$ tautologically carries a flat $g$-valued 1-form such that $\langle A \wedge [A \wedge A]\rangle$ is exact. This should say that its third cohomology is trivial.

## Re: Obstructions, tangent categories and Lie n-tegration

Have fun in Split! I hope your high-school talk gets all those kids excited about $n$-categories!

Regarding this talk: when introducing people to the Eckmann–Hilton argument, I like to emphasize that this argument is really how we prove to kids that addition is commutative. Take 3 rocks and set them on the table next to 2 rocks:

$\bullet \bullet \bullet \quad \bullet \bullet$

This is $3+2$ rocks, by definition.

Now, slide the 3 rocks around the 2 rocks, so they trade places:

$\bullet \bullet \quad \bullet \bullet \bullet$

Now we have $2+3$ rocks, by definition. We have the same number of rocks — more precisely, the process of sliding the rocks around is an isomorphism of sets. Therefore, we have an equation

$3 + 2 = 2 + 3$

or more precisely, a braiding isomorphism

$B_{3,2} : 3 + 2 \stackrel{\sim}{\to} 2 + 3$

The fun thing is that with the Eckmann–Hilton argument, this proof becomes completely rigorous! The process of carrying out the proof becomes a braid. And, we see why we need at least 2 dimensions of space to carry out this argument!

If you’ve ever read

Flatland, you may remember the scene in ‘Lineland’, where space is 1-dimensional and everyone is stuck talking to their two neighbors: their neighbor to the left and their neighbor to the right. There’s not enough room for them to trade places. For such people, the commutative law might seem counterintuitive — something only high-powered mathematicians could understand.