The n-Category Café

`jumping on my horse and riding off in all directions’

that’s known as an explosive departure ;-)

for classifying spaces in hte Lie case at least in characteristic 0 we have the Cartan or Weil models, though not having read your
notes past the intro I’m not sure what you want a `classifying space’ to do

and for the assoc alg case, the bar construction

how can one do the associative algebra and Lie algebra analogues of the use of classifying spaces

Apart from the discussion of that which we have in $L_\infty$-connections # I can offer the following integration procedure (as described at the end of Space and Quantity (blog, pdf)):

Let’s say a smooth space is a sheaf on the site whose objects are natural numbers and whose morphisms are smooth maps $\mathbb{R}^n \to \mathbb{R}^m$.

Write $C$ for the category of smooth spaces.

For every smooth space we get a family of $\omega$-groupoids

$$\Pi_n : C^{\mathrm{op}} \to \omega\mathrm{Cat}$$

whose $(k \lt n)$-morphisms are thin-homotopy classes of globular $k$-paths and whose $n$-morphisms are full homotopy classes of globular $n$-paths.

Write DGCA for the category of differential $\mathbb{N}_+$-graded commutative algebras. There is a contravariant functor

$$S : DGCA \to C$$

(part of an adjunction) which sends each DGCA $A$ to the space

$$S(A) : U \mapsto Hom_{DGCA}(A,\Omega^\bullet(U)) \,.$$

For every finite dimensional $L_\infty$-algebra on a $\mathbb{N}_+$-graded vector space $g$ we get its Chevalley-Eilenberg DGCA

$$CE(g) := (\wedge^\bullet g^*, d_g)$$

with $d_g$ of degree +1 and $(d_g)^2 = 0$. In fact, this defines the $L_\infty$-structure on $g$.

Then we can define the strict $n$-group integrating a given $L_\infty$-algebra $g$ as

$$\mathbf{B} G := \Pi_n(S(CE(g))) \,.$$

The left hand side means that we have a 1-object $\omega$-groupoid. Indeed, the right hand side always has a single object, as one easily sees (since on the point we only have the 0 differential $(p\gt 0)$-form).

Let $g$ be an ordinary Lie algebra. Then

$$\mathbf{B} G := \Pi_1(S(CE(g)))$$

is indeed the simply connected Lie group integrating it. The right hand side boils down to nothing but the familiar (in some circles) integration of Lie algebras in terms of classes of Lie algebra valued forms on the interval.

We also have for each $g$ its Weil DGCA $$W(g) := CE(inn(g)) := CE(Cone(g \stackrel{Id}{\to} g))$$ as well as the DGCA of basic forms $$inv(g) := W(g)_{basic}$$ those invariant under inner derivations along the fiber of the inclusion $$g \hookrightarrow inn(g) \,.$$

I think that, for $g$ still an ordinary Lie algebra, the 2-groupoid integrating $inn(g)$ is

$$\Pi_2(S(W(g))) \simeq \mathbf{B} (G \stackrel{Id}{\to} G)$$

namely the one-object grouopoid version of the strict 2-group which comes from the crossed module coming from the identity on $G$.

This is what I usually denote

$$(G \to G) := INN(G) := \mathbf{E}G \,.$$

So:

- $g$ integrates to $G$

- $inn(g) = Cone(g \to g)$ integrates to $\mathbf{E} G$.

I think this goes through for arbitrary $L_\infty$-algebras, but then things begin to depend a little more crucially on the model for $\infty$-groupoids which one uses.

For instance, if we stay withing strict $\omega$-groupoids then the general $\omega$-groupoid integrating any $L_\infty$-algebra would be

$$\mathbf{B} G = \Pi_\omega(S(CE(g)))$$

and we’d get the universal $G$-bundle in its $\omega$-groupoid incarnatation as $$\array{ & \mathbf{B}G &\to& \mathbf{B E}G &\to& \mathbf{B B }G \\ := \Pi_\omega\circ S( & CE(g) &\leftarrow& W(g) &\leftarrow& inv(g) & ) } \,.$$

But one needs to notice that when doing it this way for $g$ an ordinary Lie algebra, the $\omega$-groupoid $\mathbf{B}G$ won’t be equal to a 1-groupoid, but just equivalent. But the benefit is that then $\mathbf{B E}G$ exists as an $\omega$-groupoid, too.

I am in the process of preparing notes with more on this.

It has just occured to me (so perhaps partially baked not fully formed) that the theory of crossed $n$-cubes and the corresponding $n$-cube complexes may have some useful insights to shed on this problem. There is a filtration of the category of crossed $\mathbb{N}$-cubes by reflexive subcategories, and this is reflected (pun intended) back in the category of crossed $n$-complexes. With crossed $\mathbb{N}$-cubes, some of the Puppe stuff looks FUN as it is combinatorial and algebraic rather than homotopy theoretic. This may lead nowhere but I have a feeling it may yield other ideas.

(Crossed $n$-cubes are the analogue of cat$^n$-groups i.e. of $n$-fold categories in the category of groups.)

I should also point out that my notes are not full enough when describing the work, in this area, of the Granada research team and Jack Duskin.

In a later part of the notes, I started trying to unravel what would happen if I started with a non-strict 3-group

I need to read the later parts. I was seeing if you do something like a cofiber Puppe sequence in the context of $n$-groups, not just spaces. When I try to do that, I run into the problem that $X \stackrel{f}{\to} Y \to C_f$ may not be normal in $C_f$.

So what’s a good approximation to “$|\mathbf{B B}G|$”?

We know that whatever it is it needs to be the home of the characteristic classes. So the rational approximation I had looked allright. But of course it does lack information.

Tim, thanks!

As you probably have seen, i was a bit distracted by other things. Am hoping to get back to this here tonight.

Actually, I came to think that there must be a way to do this entirely in terms of crossed complexes (so in strict $\infty$-groups), if we allow ourselves to encode a group, say, $G$, not by the obvious crossed complex conctentrated in the lowest degree, but one equivalent to that but possibly with stuff in higher degrees.

The reason is that for the case that our $n$-group is “higher simply connected” this is what I get from integrating Lie $\infty$-algebras:

Every $L_\infty$-algebra $g$ I can integrate to a crossed complex by forming the $\omega$-groupoid of thin-homotopy classes of globular paths in the classifying space of $g$-valued forms: $$\mathbf{B} G := \Pi_\omega(S(CE(g))) \,.$$ This is a one-object $\omega$-groupoid and hence encodes a crossed complex of groups.

The point is that if $g$ is an ordinary Lie algebra, this crossed complex will not be the obvious one $$\cdots 1 \to 1 \to G$$ with $G$ the simply connected Lie group integrating $g$ but one equivalent to that (I think). (The standard one would be obtained by using $\Pi_1$ instead of $\Pi_\omega$, where we divide out full homotopy at level 1 already.)

The good thing about this is that at the level of $L_\infty$-algebras I know the sequence that I am looking for, it is:

$$CE(g) \leftarrow CE(Cone(g \to g)) = W(g) \leftarrow W(g)_basic$$

and the entire sequence integrates to a sequence of $\omega$-groups:

$$\array{ \mathbf{B} G &\to& \mathbf{B E}G &\to & \mathbf{B B}G \\ \Pi_\omega(S(CE(g))) &\to& \Pi_\omega(S(W(g))) &\to& \Pi_\omega(S(inv(g))) } \,.$$

This suggests that given any crossed complex/strict one-object $\infty$-groupoid $\mathbf{B} G$, there one equivalent to it such that $\mathbf{B E}G$ and “$\mathbf{B B}G$” exist as crossed complexes.