The n-Category Café

Of course, I am familiar with the $n=2$ case,

Okay, sorry.

I haven’t been reading all the posts at the cafe, so I must have missed the ones talking about strict 3-groups; could you provide some links?

I mostly talked about Gray groups in general, namely those arising from weak cokernels of 2-groups (here (inner automo. 3-group) and here (weak cokernels of 2-groups)), but in many special cases of interest, notably that one discussed here (Chern-Simons cocycles) (also slide 523) these become strict 3-groups.

For instance there is the strict 2-group $String(G)$ which comes from a crossed module of the form

$$(\hat \Omega G \to P G)$$

where the right group is paths in $G$, the left one centrally extended loops in $G$.

This sits in a short exact sequence

$$1 \to \mathbf{B} U(1) \to String(G) \to G \to 1$$

(for instance section 4.2 here)

and we can ask what the obstruction is to lifting a $G$-cocycle to a $String(G)$-2-cocycle.

To do so we notice that we can lift every $G$-cocycle to the a cocycle for the weak cokernel

$$(\mathbf{B} U(1) \to String(G)) ,円.$$

This is a strict 3-group (it happens to be strict here, since $\mathbf{B} U(1)$ is so boring) coming from a crossed complex of groups

$$(U(1) \hookrightarrow \hat \Omega G \to P G) ,円.$$

Then projecting out from this 2-cocycle the 3-coycle obtained under

$$(U(1) \hookrightarrow \hat \Omega G \to P G) \to (U(1) \to 1 \to 1)$$

yields the 3-cocycle obstructing the lift (a “Chern-Simons 3-cocycle”).

That’s maybe the most interesting example that I have talked about. But notice that one gets lots of examples of non-trivial crossed complexes from (filtered) spaces, by forming their “fundamental $\omega$-groupoids”.

I guess the best review for that is

Ronnie Brown, Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems.

I’m not sure that I’m ready to dive into $L_\infty$-connections

No need. You asked for the example I had in mind, and I tried to provide the context. But the only relevant point for the discussion here is that $L_\infty$-algebras are a source of spaces, and that spaces are a source for $\omega$-groupoids.

a brief glance at section 3 didn’t answer my main question

Oh, it wasn’t supposed to answer your question! It certainly does not and is not intended to. I just mentioned it to indicate where that example you asked me to describe arises from.

why is $FiveBrane(G)$ a strict 6-group?

I tried to say that in the second but last paragraph of my above comment: in the present setup, it’s strict by construction.

I am claiming that for any smooth space $X$ and any integer $n$, we can form a strict globular $n$-groupoid $\Pi_n(X)$ which has thin-homotopy classes of globular $k$-paths as $k$-morphisms for all $k \lt n$ and has ordinary homotopy classes of globular $n$-paths as $n$-morphisms.

This is just a slight variant of the fundamental $\omega$-groupoid of a space that is described in

Ronnie Brown, A new higher homotopy groupoid: the fundamental globular $\omega$-groupoid of a filtered space.

For low $n$ the $\Pi_n(X)$ that I have in mind is described in more detail in section 2.1 arXiv:0802.0663 (well, there it’s thin homotopy all the way up, so its called $P_2(X)$ instead of $\Pi_2(X)$, but it’s trivial to get from there to $\Pi_2(X)$).

It seems that this is saying something about the vanishing of various higher homotopy invariants of the space $X_{CE(g_{\mu_7})}$; is there some conceptual reason for that?

The thing is strict by construction. The question is how much information about the original space it retains. Probably the best general answer to that is the list of theorems summarized in Ronnie Brown’s diagram which I linked to above.

Specifically, in the special example of the String 2-group I mentioned, there is the space

$$X_{CE(g_{\mu_3})}$$

and we can wonder how its various flavours of path $n$-groupoids are related.

In math/0603563 André Henriques forms a weak path $\infty$-groupoid from it (essentially just the Kan complex of singular simplices in that space, but then also truncated at some point).

In that entry on Chern-Simons cocycles I mentioned #, I talk about the strict path 2-groupoid of that space. There I conclude that this strict path 2-groupoid is essentially $\mathbf{B} String(G)$, with $String(G)$ the strict version of the String 2-group.

And it is known that both version of the path groupoid of $X_{CE(g_{\mu_3})}$ have equivalent realizations.

(These two approaches are mentioned on slides 152-7).

But I wish I would understand this relation between weak and strict fundamental path $\infty$-groupoids of spaces better and more generally. I’d be glad if we discuss this further.

Re: Infinity-Groups with Specified Composition

I wrote:

But I wish I would understand this relation between weak and strict fundamental path $\infty$-groupoids of spaces better and more generally. I’d be glad if we discuss this further.

In particular, this here is puzzling me:

let $g$ be an ordinary Lie algebra. It sits inside its Lie 2-algebra of inner derivations

$$\array{ g \\ \downarrow \\ inn(g) } ,,円$$

passing to the dual CE-algebras gives the surjection of the Weil algebra onto the CE-algebra

$$\array{ CE(g) \\ \uparrow \\ CE(inn(g)) &=& W(g) }$$

then forming spaces from that (along these lines)

$$S \left( \array{ CE(g) \\ \uparrow \\ CE(inn(g)) &=& W(g) } \right) = \array{ X_{CE(g)} \\ \downarrow \\ X_{W(g)} }$$

and then forming the fundamental strict 2-groipoids of these yields

$$\array{ \mathbf{B} G \\ \downarrow \\ \mathbf{B} INN(G) &=& \mathbf{B} (G // G) } ,,円$$

where $G$ is the simply connected Lie group integrating $g$.

(For the top item this is standard, for the bottom item I think this is true, but I haven’t written up a full proof.)

So this yields two different ways to arrive at this universal $G$-bundle thing: either form inner derivations at the Lie algebra level and then integrate, or first integrate and then form inner automorphisms.

But it’s not quite as simple as we move up in dimension:

starting with a strict 2-group $G_2$ coming from a strict Lie 2-algebra $g_2$, we know that $INN(G_2)$ is not a strict 3-group in general, but a Gray group. On the other hand, that procedure of forming strict fundamental $n$-groupoids will integrate

$$\array{ g_2 \\ \downarrow \\ inn(g_2) }$$

to some morphism of strict Lie 3-groups,

$$\array{ \mathbf{B} G_2 \\ \downarrow \\ \Pi_3(X_{W(g_2)}) } ,円.$$

There should certainly be an equivalence

$$\mathbf{B} INN(G_2) \simeq \Pi_3(X_{W(g_2)}) ,,円$$

simply because $W(g_2)$ and $INN(G_2)$ are both “contractible”, but I need to eventually better understand how that goes along with the rest of the structure and what’s really going on here.

(I have recently talked about that point with David Roberts here.)