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September 5, 2006

n-Transport and Higher Schreier Theory

Posted by Urs Schreiber

We are interested in categorifying the notion of parallel transport in a fiber bundle with connection.

There are several ways to define an ordinary connection on an ordinary bundle. Depending on which of these we start with, we end up with categorifications that may differ.

One definition goes like this:

Given a principal GG-bundle BXB \to X, let

  • Trans(B)=B×B/G\mathrm{Trans}(B) = B\times B/G be the transport groupoid of BB, whose objects are the fibers of BB and whose morphisms are the torsor morphisms between these;
  • P(X)P(X) be the groupoid of thin homotopy classes of paths in XX (meaning that we divide out by orientation-preserving diffeomorphisms and let orientation-reversing diffeos send a path to its inverse class).

Then a connection on BB is a smooth functor

(1)tra:P(X)Trans(B). \mathrm{tra} : P(X) \to \mathrm{Trans}(B) \,.

This definition has an obvious categorification. Working it out (\to, \to), one finds a notion of 2-connection with a special property that has been termed “fake flatness”.

There are a couple of applications where precisely this fake flatness is required (\to). For others, however, fake flatness is too restrictive (\to, \to).

Now, there have been several indications that in order to get a slightly more general categorification we need a definition of connection with parallel transport which somehow involves not just the gauge group, but its automorphism 2-group (\to).

In fact, Danny Stevenson has developed a rather beautiful theory of connections - without an explicit description of parallel transport - and their categorification, by using not transport along finite paths, but infinitesimal/differential transport. He sees essentially this automorphism-extension appearing there and does get around fake flatness.

Danny Stevenson
Lie 2-Algebras and the Geometry of Gerbes
Chicago Lectures on Higher Gauge Theory, April 7-11, 2006

This is directly inspired by

Lawrence Breen
Théorie de Schreier supérieure
Annales Scientifiques de l’École Normale Supérieure Sér. 4, 25 no. 5 (1992), p. 465-514

In this entry here I want to understand the integrated, finite version of Danny’s theory. Where he uses morphisms of Lie algebroids, I would like to see morphisms of Lie groupoids (smooth functors between smooth groupoids) along the lines of the first definition of connection with parallel transport stated above.

I had begun making comments on that over in the comment section of the 10D supergravity thread (\to). But it does deserve an entry of its own.

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 GG-bundle BXB \to X, we obtain an exact sequence of vector bundles from it

(1)0ad(B)TB/GTX0, 0 \to \mathrm{ad}(B) \to T B/G \to T X \to 0 \,,

called the Atiyah sequence.

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

I believe this sequence actually extends to a sequence of Lie algebroids (\to), all with anchor maps to TXT 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

(2):TXTB/G \nabla : T X \to T B/G

of the Atiyah sequence is the same as a connection on BB. 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

(3)1KGB1 1 \to K \to G \to B \to 1

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

(4)BG B \to G

actually extends to a homomorphism

(5)BAUT(K), B \to \mathrm{AUT}(K) \,,

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

(6)0Γ(ad(B))Γ(TB/G)Γ(TX)0. 0 \to \Gamma(\mathrm{ad}(B)) \to \Gamma(T B/G) \to \Gamma(T X) \to 0 \,.

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

(7)DER(Γ(ad(B))). \mathrm{DER}(\Gamma(\mathrm{ad}(B))) \,.

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

(8)(,F ):Γ(TX)DER(Γ(ad(B))). (\nabla, F_\nabla) : \Gamma(T X ) \to \mathrm{DER}(\Gamma(\mathrm{ad}(B))) \,.

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+1n+1-gerbe” from a given nn-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 TMIdTMT M \stackrel{\mathrm{Id}}{\to} T M should be the differential version of the fundamental groupoid P(X)P(X) of XX, whose objects are points of XX and whose morphisms are homotopy classes of paths in XX. (This is at least true when XX is simply connected.)
  • The Lie algebroid TB/GTXT B/G \to T X should be the differential version of the transport groupoid Trans(B)=B×B/GX\mathrm{Trans}(B) = B \times B / G \to X, whose objects are the fibers of BB and whose morphisms are the torsor morphisms between these.
  • The Lie algebroid ad(B)TX\mathrm{ad}(B) \to T X should be the differential version of the skeletal groupoid Ad(B)X\mathrm{Ad}(B) \to X, which I guess should be called the endomorphism groupoid of BB. It is just a bundle of groups over XX obtained by associating GG by the adjoint action of GG on itself to BB.

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

(9)Ad(B)Trans(B)P(X). \mathrm{Ad}(B) \to \mathrm{Trans}(B) \to P(X) \,.

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

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

The second functor takes a torsor morphism B xB yB_x \to B_y and sends it to the corresponding class of paths xyx \to y. (Here in my notation I am assuming that XX 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

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

(10)KEG K \to E \to G

we want to assume that KK 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

(11)Ad(B)Trans(B)P(X). \mathrm{Ad}(B) \to \mathrm{Trans}(B) \to P(X) \,.

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

(12)(,F ):TXDER(ad(B)) (\nabla,F_\nabla) : T X \to \mathrm{DER}(\mathrm{ad}(B))

is now a pseudofunctor

(13)(tra,curv tra):P(X)AUT(Ad(G)), (\mathrm{tra},\mathrm{curv}_\mathrm{tra}) : P(X) \to \mathrm{AUT}(\mathrm{Ad}(G)) \,,

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

(14)AUT(Ad(G)) \mathrm{AUT}(\mathrm{Ad}(G))

is the 2-groupoid whose

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

I expect that

(15)(tra,curv tra):P(X)AUT(Ad(G)) (\mathrm{tra},\mathrm{curv}_\mathrm{tra}) : P(X) \to \mathrm{AUT}(\mathrm{Ad}(G))

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 BB is trivial. Then the above says a connection on BB is a pseudofunctor

(16)P(X)AUT(G). P(X) \to \mathrm{AUT}(G) \,.

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 Pexp( pathA)P \exp(\int_\path A). This won’t respect the composition of the pair groupoid P(X)P(X), unless AA is flat. But the failure of composition to be respected strictly is given by a nontrivial compositor, which precisely encodes the curvature of AA. 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 2G_2-2-connection with parallel transport is a pseudo-2-functor to the 3-group AUT(G 2)\mathrm{AUT}(G_2). That this does in fact yield the expected result is part of what I checked in my nn-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 nn-functors on cubical nn-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 nn-cubes into nn-simplies or gluing nn-simplices to nn-cubes.

Posted at September 5, 2006 1:53 PM UTC

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2 Comments & 11 Trackbacks

Read the post Kock on 1-Transport
Weblog: The n-Category Café
Excerpt: A new preprint by Anders Kock on the synthetic formulation of the notion of parallel transport.
Tracked: September 8, 2006 5:47 PM

Re: n-Transport and Higher Schreier Theory

Beware the term ‘fibration’. To a homotopy theorist it’s likely to mean the homotopy lifting property or locally fibre homotopy trivial. See posting on the arXiv Sept 11 by Wirth and Stasheff. `Just like bundles’ EXCEPT everything up to strong homotopy e.g. homotopy coherent functor tra:P x 0(X)H(F) tra: P_{x_0}(X) \to H(F) and 1-cocycles up to strong homotopy coherence.

Posted by: jim stasheff on September 10, 2006 8:13 PM | Permalink | Reply to this

Re: n-Transport and Higher Schreier Theory

Beware the term ‘fibration’.

Okay. What would be a better term to use where I used “fibration”?

See posting on the arXiv Sept 11
by Wirth and Stasheff.

Thanks! I’ll have a look at that.

Posted by: urs on September 11, 2006 5:37 PM | Permalink | Reply to this
Read the post Wirth and Stasheff on Homotopy Transition Cocycles
Weblog: The n-Category Café
Excerpt: Stasheff recalls an old result by Wirth on passing between fibrations and their homotopy transition cocycles.
Tracked: September 11, 2006 7:55 PM
Read the post Quantum n-Transport
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Excerpt: An attempt to understand the path integral for an n-dimensional field theory as a coproduct operation over transport n-functors.
Tracked: September 14, 2006 2:11 PM
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Weblog: The n-Category Café
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Tracked: November 3, 2006 2:16 PM
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Weblog: The n-Category Café
Excerpt: On the category of paths whose canonical Leinster measure reproduces the path integral measure appearing in the quantization of the charged particle.
Tracked: March 19, 2007 9:01 PM
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