If a connection on a principal -bundle is locally represented by the 1-form with values in the Lie algebra , then the connection is flat if and only if the curvature 2-form vanishes, that is, if is a solution of the Maurer-Cartan equation.
Notice that one can see curvature only on 2-dimensional paths (since it is a 2-form): if one restricts to an infinitesimal 1-simplex in , then the restriction of is clearly a solution to the Maurer-Cartan equation. One can say that is 1-flat.
Then, moving to an infinitesimal 2-simplex one sees that the connection is (generally) not 2-flat: holonomy along two sides of the 2-simplex is not the same thing as holonomy along the third side. Not the same, but in a very precise way: the curvature exactly measures the gap to go from a horn of the 2-simplex to the third edge. This is very 2-categorical, and suggests one can cure the lack of flatness of the original connection by adding a copy of in degree -1. More precisely this amounts to consider the 2-Lie algebra given by the cone of the identity .
The Maurer-Cartan equation for coincides with the original one on the 1-simplex (since only degree 1 elements are 1-forms with values in . But on the 2-simplex we would have, in addition to these elements, also 2-forms with coefficients in , and the Maurer-Cartan equation on the 2-simplex takes the form . So the original equation telling that had curvature is now equivalent to say that is flat.
In other words, what seemed a non-flat connection was so since we were not seeing the 2-bundle, but only a 1-bundle approximation. And on a 1-bundle one can only clearly see up to 1-simplices, where the original connection was actually flat. Once curvature has come in, we can repeat the argument: now we have a 2-flat connection and can test it on the 3-simplex. If it has 3-curvature, that will presumibly be because we are not seeing the 3-bundle, yet. So I find it natural to conjecture that any connection on a principal bundle (and more generally any -flat connection on an -bundle) can be seen as a flat connection on a suitable -bundle.
For a Lie algebra, a flat -valued connection (on a trivial bundle) is a morphism of Lie algebroids
This can be integrated to a morphism of Lie groupoids, where it becomes
where is the Lie group that integrates .
An extremely fruitful point of view on the algebroid morphism above is to look at it in the opposite direction, i.e. as a morphism of (sheaves of) differential graded commutative algebras
where the algebra on the left-hand side is the Chevalley-Eilenberg algebra of and is the de Rham algebra of .
An arbitrary (i.e. non necessarily flat) -connection on can still be seen as a morphism of graded commutative algebras , but this will in general not be compatible with the differentials. Obstruction to this compatibillity is precisely the curvature of the connection. See details at curvature of β-Lie algebroid valued differential forms in Ursβ personal area.
However, if we postcompone with the projection
then the map
is a morphism of differential graded commutative algebras. This because curvature is a 2-form. So we can look at curvature as the obstruction to lifting in the following diagram of dg-commutative algebras
This lifting problem can be factorized into lifting problems as follows:
And what we said about curvature can be rephrased as follows: once the first obstruction vanishes, all higher obstruction vanish. This happens because is a 1-Lie algebra. Moving to -Lie algebras nontrivial higher obstructions appear.
While on the base of a principal -bundle there is genrally no flat -connection, on each fiber of there is a canonical one. More precisely, let be the subgroupoid of consisting of vertical paths (i.e., of paths whose tangent vector at each point is a vertical vector with respect to the projection ). By the very definition of -principal bundle, there is a tautological morphism
Differentiating this, we get a Lie algebroid morphism
the canonical vertical flat connection on . In the dual picture we have a canonical dg-commutative algebra morphism
where the dg-algebra of vertical differential forms on is the quotient of by the sub-dg-algebra of differential form vanishing on vertical multivectors.
A first appearance of higher Lie algebras comes from noticing that the lower corner of the above diagram can be completed to a (homotopy) commutative diagram
where is the 2-Lie algebra given by the cone over the identity of . Indeed (by definition of cone) the 2-Lie algebra is contractible, and therefore so is its Chevalley-Eilenberg algebra. Itβs worth giving this dg-algebra a name:
is called the Weil algebra of . Our lifting problem has now been translated to a (homotopy) factorization problem:
To shed a bit of light on this problem, recall that the Chevalley-Eilenberg construction induces an opposite equivalence of categories between (higher) Lie algebras and dg-commutative algebras. Therefore to the fibration sequence
it corresponds the cofibration sequence
where is the dg-algebra of invariant polynomials on . This way we see that the diagram of our lifting problem is a part of a larger homotopy commutative diagram of dg-commutative algebras:
and the universal property of push-out tells us that the seeked lifting exists if and only if the composite map
is homotopy equivalent to the zero map. Since is weakly equivalent to , this is surely so, that is, up homotopy, the looked for lifting exists. This is nothing but the classical statement that any -connection on a trivial principal bundle is gauge-equivalent to the trivial connection. But seeing things from the abstract homotopy nonsense above clearly tells us which is the right setup if we want to consider curvature, and not only curvature up to homotopy: twisted cohomology.
From the above discussion we learnt two things:
This motivates the following definition:
A (local) -valued connection over is a dg-commutative algebra morphism
Equivalently, it is an -Lie groupoid morphism
where is the -Lie groupoid integrating . Curvatures of the connection are the datum of on higher morphisms in . On the dg-algebra side, they correspond to the datum of curvature forms; since by definition curvature is related to higher morphisms, one has curvature forms in degree or higher. Note that if is an -Lie algebroid, then one has curvature forms only in degrees from to . The connection is flat if all of its curvature forms vanish.
(Deatils to be added, here, to show how to express curvatures in term of the dg-algebra morphism .)
With this definition, what is called a -valued connection in classic differential geometry (with a Lie algebra), is the partial datum for a true -connection . Since is a 2-Lie algebra when is a Lie algebra, the connection has exactly one curvature form concentrated in degree . This -form is nothing but the usual curvature 2-form of from classic differential geometry.
The notion of -valued connection is too well rooted in classic differential geometry, to just repalce it with another notion as above and forget about it. So it is convenient and reasonable to keep the classical notion, slightly changing its name in order to stress that it only gives a partial picture. So, let us call preconnection a dg-algebra morphism
and, more in general, -preconnection a dg-algebra morphism .
In the dual (integrated) picture, a -preconnection is a morphism of -groupoids
where is the sub--groupoid of whose -morphisms are smooth maps which kill all differential forms on of degree greater than .
In particular, ordinary -connections of differential geometry are 1-preconnections. Other examples of 1-preconnections can be found in the literature on βabelian gerbes with connection but without curvingβ, where -functors , for the 2-group are considered. Also the truncated versions, i.e., involving the path n-groupoids are often met in existing literature. Note that for .
Let us go back to the original problem, and describe the integration problem in terms of Maurer-Cartan functors. If
is a Lie algebra, and is a Lie group integrating it, then a valued connection on a trivial principal -bundle on a manifold is a 1-form on with values in , i.e. a degree 1 element in the dgla , where is the de Rham algebra of . Integrating gives a functor
This can be seen as an -functor from the nerve of to the nerve of . It is a remarkable result by Hinich that this latter nerve is homotopy equivalent to the Kan complex , where MC denotes the set of solutions of the Maurer-Cartan equation in a dgla. Using this model for , the functor is straightforwardly described: a path is mapped to the degree 1 element in . This element is automatically a Maurer-Cartan element, since is a 2-form and so it vanishes when pulled back via . By the very same reason, is a Maurer-Cartan element in for every thin map (by definition of thin map). Hence we have the seeked functor.
Trying to lift to a functor stemming from we meet the curvature obstruction: for a generic map , the element is not a Maurer-Cartan element in unless is flat. Once we require to be flat, we are done: what we are asking is that \omega is a Maurer-Cartan element in , and so it will produce Maurer-Cartan elements wherever we pull it back. In particular the functor can be lifted to for any , and so to a functor from .
This is very much a 0-1 situation: either we are blocked on or we have vanishing curvature. To cook something intersting with nontrivial curvature, let us move from Lie algebras to -Lie algebras, which will be convenient to think as -algebras concentrated in degrees . Getzler shows in his work on Lie theory for nilpotent algebras, that the Maurer-Cartan construction sketched above verbatim generalizes from Lie algebras to -Lie algebras (it is possible some version of this result predates Getzler, it seems Iβm quite unable to give proper credits for results..). So let us take for instance a 2-Lie algebra , where is the subspace of degree elements. Since the higher brackets
have degree , for a 2-Lie algebra only the following brackets survive:
Now, consider a manifold and the (it is an -algebra concentrated in degrees ). A flat connection on a trivial principal -bundle on (where is the 2-group integrating ) is a solution of the Maurer-Cartan equation
on . The 2-bracket and the 3-barcket come from the 2- and 3-bracket in ; the differential has a contribution from the 1-bracket of and one from the de Rham differential . A particularly simple and interesting case is when is the Lie 2-algebra , i.e., the Lie 2-algebra coming from the crossed module , for a Lie algebra . In this case, , , the 3-bracket vanishes, the 2-brackets are just the 2-brackets of , and the differential is (up to a sign) the identity of . A flat -connection on has the form , with a 1-form on with coefficients in , and a 2-form on with coefficients in (and so with some abuse we can think of as a 2-form with coefficients in ; we should avoid doing so, since degrees are relevant, but let us note this in order to make contact with a familiar picture).
Then the Maurer-Cartan equation for has the form
(recall that the 3-bracket of vanishes). The -degree makes a set of several equations out of the single equation above (one for each degree). So we actually end up with two equations, one for a 2-form, and one for a 3-form. Namely,
i.e., is the curvature of , and and
i.e., the Bianchi identity.
(to be continued)
(to be continued)
Discussion goes here.