The n-Category Café

Danny’s concept of connection is based on a fundamental idea called Schreier theory, which is about the classification of fibrations.

You can get a good idea of what this is about by looking at

John Baez
TWF 223

and following the references given there.

Danny starts his discussion with the following standard observation.

Given any principal $G$-bundle $B \to X$, we obtain an exact sequence of vector bundles from it

called the Atiyah sequence.

Here $\mathrm{ad}(B)$ is the vector bundle associated to $B$ by using the adjoint action of $G$ on its Lie algebra.

I believe this sequence actually extends to a sequence of Lie algebroids ($\to$), all with anchor maps to $T X$.

This is important for what I would like to discuss here, since I would like to integrate these Lie algebroids to Lie groupoids.

It is a well-known standard fact, that a splitting

of the Atiyah sequence is the same as a connection on $B$. In general, this splitting is just a splitting at the level of morphisms of vector bundles, not at the level of Lie algebroids. The failure of $\nabla$ to actually be a a morphism of Lie algebroids is measured by its curvature 2-form.

That should make us wonder. If everything here lives in the world of Lie algebroids, we do expect connections to be expressible in terms of Lie algebroid morphisms.

Danny explains what is going on by comparing with the general idea of higher Schreier theory.

There, too, we are dealing with splittings of short exact sequences

which fail to respect the available structure. But there it turns out that the structure is in fact respected one level higher. The splitting

actually extends to a homomorphism

where $\mathrm{AUT}(K)$ is a $(n+1)$-categorical structure if $K$ is an $n$-categorical structure.

In terms of the concrete example we are dealing with here, this means the following.

The algebroids we are talking about involve the Lie algebras of sections of the bundles that appear in the Atiyah sequence

The “automorphism Lie 2-algebra” of the Lie algebra $\Gamma(\mathrm{ad}(B))$ is usually called the Lie 2-algebra of autoderivations. Danny writes

He notes that combining the splitting $\nabla : TM \to TB/G$ with its curvature, regarded as a linear map $\bigwedge^2 T X \to \mathrm{ad}(B)$ does yield a morphism of Lie 2-algebras

This now is indeed a homomorphism (though of 2-algebras instead of 1-algebras). In analogy to the former situation, this property of being a homomorphism again is equivalent to a condition which says that this linear map is “flat” in some sense.

But now this flatness is something desireable. It encodes precisely the Bianchi identity satisfied by the curvature.

From a different point of view I had described this idea of forming a flat “curvature $n+1$-gerbe” from a given $n$-gerbe with connection and parallel transport here.

But I would like to now understand this more systematically - by “integrating” Danny’s theory to a theory of sequences of Lie groupoids and their splittings.

My intention here is not to present a fully worked-out idea, but to start by discussing some first observations.

I believe it is known what the Lie groupoids corresponding to the three Lie algebroids appearing in the Atiyah sequence are. They should be the following.

• The Lie algebroid $T M \stackrel{\mathrm{Id}}{\to} T M$ should be the differential version of the fundamental groupoid $P(X)$ of $X$, whose objects are points of $X$ and whose morphisms are homotopy classes of paths in $X$. (This is at least true when $X$ is simply connected.)
• The Lie algebroid $T B/G \to T X$ should be the differential version of the transport groupoid $\mathrm{Trans}(B) = B \times B / G \to X$, whose objects are the fibers of $B$ and whose morphisms are the torsor morphisms between these.
• The Lie algebroid $\mathrm{ad}(B) \to T X$ should be the differential version of the skeletal groupoid $\mathrm{Ad}(B) \to X$, which I guess should be called the endomorphism groupoid of $B$. It is just a bundle of groups over $X$ obtained by associating $G$ by the adjoint action of $G$ on itself to $B$.

Assuming this is true, the integrated version of the Atiyah sequence of $B$ would be

Here the morphisms are supposed to be the obvious smooth functors.

The first one takes a group element in a fiber $\mathrm{Ad}(B)_x$ of $\mathrm{Ad}(B)$ and interprets as a an torsor morphism $B_x \to B_x$.

The second functor takes a torsor morphism $B_x \to B_y$ and sends it to the corresponding class of paths $x \to y$. (Here in my notation I am assuming that $X$ is simply connected. This should generalize to the general case in the obvious way.)

Clearly, the kernel of the second functor is precisely the image of the first one. Moreover, the first one is monic, the second one is epi, so we do have an exact sequence.

I conjecture that differentiating this sequence of morphisms of groupoids yieds precisely the Atiyah sequence of algebroids. But I haven’t tried to write down a rigorous proof for this.

Now, with a fibration of groupoids in hand, we need to know Schreier theory for groupoids in order to have a chance to translate Danny’s concepts to the world of groupoids.

Luckily this is discussed in this nice paper:

V. Blanco, M. Bullejos, E. Faro
Categorical non abelian cohomology, and the Schreier theory of groupoids
math.CT/0410202.

Even more luckily, these authors find that to discuss a sequence of groupoids

we want to assume that $K$ is skeletal, i.e. that it is just a bundle of groups! That’s precisely the situation we found above, so we can apply Schreier theory of groupoids to our integrated Atiyah sequence

According to the results of this paper, now, the analog of Danny’s algebroid morphism

is now a pseudofunctor

where (this definition is hidden on p. 4 of the above paper, penultimate paragraph)

is the 2-groupoid whose

• objects are the fibers of $\mathrm{Ad}(G)$, which we may identitfy with the points $x\in X$
• 1-morphisms are group isomorphisms $\mathrm{Ad}(G)_x \to \mathrm{Ad}(G)_y$
• 2- morphisms are natural isomorphisms between these.

I expect that

is the right notion of parallel transport whose differential version yields Danny’s conception of connection in terms of algebroid morphisms.

We can make some quick consistency checks of this claim.

Assume that $B$ is trivial. Then the above says a connection on $B$ is a pseudofunctor

And that’s true. Using a connection 1-form, we pick any representatives of paths between given pairs of points and associate to these paths the group element $P \exp(\int_\path A)$. This won’t respect the composition of the pair groupoid $P(X)$, unless $A$ is flat. But the failure of composition to be respected strictly is given by a nontrivial compositor, which precisely encodes the curvature of $A$. Together, this does give the required pseudofunctor.

(I should draw a simple diagram to illustrate this. Maybe I’ll type one into an extra pdf.)

Similarly in the categorified case. Locally, or equivalently for trivial 2-bundles, the above says that a $G_2$-2-connection with parallel transport is a pseudo-2-functor to the 3-group $\mathrm{AUT}(G_2)$. That this does in fact yield the expected result is part of what I checked in my $n$-curvature entry ($\to$).

So it seems that the above is on the right track.

Addendum: When comparing these consistency checks with the above discussion, one should note that there is a natural way to pass between $n$-functors on cubical $n$-paths which satisfy a certain flatness constraint and pseudofunctors on the pair groupoid.

The latter associate something to 1-simplices, such that composition is respected up to something involving 2-simplicies, which satisfy something involving 3-simplices, and so on. At the highest level there is just an equation and no more data is associated to higher-dimensional simplices. That is what yields the flatness constraint.

We pass between these two pictures by slicing $n$-cubes into $n$-simplies or gluing $n$-simplices to $n$-cubes.