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March 22, 2008

Nonabelian Differential Cohomology in Street’s Descent Theory

Posted by Urs Schreiber

As a followup to our recent discussion #:

Nonabelian differential cohomology
in Street’s descent theory
(pdf, 20 pages)

Abstract: The general notion of cohomology, as formalized \infty-categorically by Ross Street, makes sense for coefficient objects which are \infty-category valued presheaves. For the special case that the coefficient object is just an \infty-category, the corresponding cocycles characterize higher fiber bundles. This is usually addressed as nonabelian cohomology. If instead the coefficient object is refined to presheaves of \infty-functors from \infty-paths to the given \infty-category, then one obtains the cocycles discussed in [BS, SWI, SWII, SWIII] which characterize higher bundles with connection and hence live in what deserves to be addressed as nonabelian differential cohomology. We concentrate here on ω\omega-categorical models (strict globular \infty-categories) and discuss nonabelian differential cohomology with values in ω\omega-groups obtained from integrating L(ie)-\infty algebras.

Introduction

A principal GG-bundle is given, with respect to a good cover by open sets of its base space, by a trivial GG-bundle on each open subset, together with an isomorphism of trivial GG-bundles on each double intersection, and an equation between these on each triple intersection. This is the archetypical example of what is called descent data, forming a cocycle in nonabelian cohomology. It can be vastly generalized by replacing the group GG appearing here by some \infty-category. For each cocycle obtained this way there should be a corresponding \infty-bundle whose local trivialization it describes.

The crucial basic idea of [BS, SWI, SWII, SWIII] is to describe \infty-bundles \emph{with connection} by cocycles which have

- a (“transport”) functor from paths to GG on each patch;
- an equivalence between such functors on double overlaps
- and so on.

The cocycles thus obtained deserve to be addressed as cocycles in differential nonabelian cohomology.

Forming the collection of ω\omega-functors from paths in a patch to some codomain provides a functor from “spaces” to ω\omega-categories: an ω\omega-category valued presheaf.

In [Street] Ross Street descibes a very general formalization for cohomology taking values in ω\omega-category valued presheaves. We recall the basic ideas (subject to some slight modifications, a discussion of which is in 6) and describe how the differential cocoycles of [BS, SWI, SWII, SWIII] fit into that.

Of particular interest are differential cocycles which can be expressed differentially in terms of L(ie) \infty-algebras. Building on the discussion of [SSS] we give in characteristic forms a definition (def. 14) of non-flat non-abelian differential cocycles and their characteristic classes.

Posted at March 22, 2008 7:33 PM UTC

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8 Comments & 4 Trackbacks

Re: Nonabelian Differential Cohomology in Street’s Descent Theory

In the very last section there are some questions which I would like to hear your comments on.

Posted by: Urs Schreiber on March 22, 2008 8:02 PM | Permalink | Reply to this

Re: Nonabelian Differential Cohomology in Street’s Descent Theory

Regarding section 5.2:

EG=INN 0G\mathbf{E}G = \mathrm{INN}_0G is most certainly not literally Cone(BGidBG)Cone(\mathbf{B}G \stackrel{id}{\to} \mathbf{B}G)! Perhaps Cone(BGidBG)Cone(\mathbf{B}G \stackrel{id}{\to} \mathbf{B}G) could be used as a shorthand for the one-object \infty-category corresponding to the algebraic construction arising from Cone(GidG)Cone(G \stackrel{id}{\to} G), where we don’t really know what GG is, only BG\mathbf{B}G.

This is my motivation for trying to understand GG as an A A_\infty simplicial space (or whatever). I can see easily how EG\mathbf{E}G can be considered as an \infty-group - as a thing with a product and coherences - but to get the one-object \infty-category associated to it is then hard from a simplicial point of view.

Posted by: David Roberts on March 23, 2008 7:57 AM | Permalink | Reply to this

Re: Nonabelian Differential Cohomology in Street’s Descent Theory

Thanks, David.

On p. 15 this was a typo. It should read BEG=Cone(BGIdBG)\mathbf{B E}G = Cone(\mathbf{B}G \stackrel{Id}{\to}\mathbf{B} G), corresponding to the integration of

inn(g):=Cone(gIdg) inn(g) := Cone(g \stackrel{Id}{\to} g)

as it appears a little above in def 13.

So for the purposes of section 5.2 I am taking def 13 (BGBEGBBG):=Π ωS(CE(g)Cone(CE(gg))inv(g)) (\mathbf{B}G \to \mathbf{B E}G \to \mathbf{B B}G) := \Pi_\omega \circ S(CE(g) \leftarrow Cone(CE(g \to g)) \leftarrow inv(g))

as the definition of all the one-object ω\omega-groupoids on the left.

One reason is that this way also BEG\mathbf{B E} G is an ω\omega-groupoid (instead of ω\omega-Grpd-enriched). As I tried to indicate, this can be thought of as getting around the result that INN 0(G)INN_0(G) of a strict 2-group is just a Gray group by noticing that for gg an ordinary Lie algebra, for instance, BG:=Π omgea(S(CE(g)))\mathbf{B} G := \Pi_\omgea(S(CE(g))) is not equal to the 1-groupoid which you expect to see, but something ω\omega-categorical equivalent to it.

Have to run now. The easter bunny is calling. More later.

Posted by: Urs Schreiber on March 23, 2008 9:45 AM | Permalink | Reply to this

Re: Nonabelian Differential Cohomology in Street’s Descent Theory

The easter bunny is calling.

Or the Easter Bell of France?

Posted by: David Roberts on March 25, 2008 6:30 AM | Permalink | Reply to this

Re: Nonabelian Differential Cohomology in Street’s Descent Theory

Hard to tell, especially since we had snow on Easter after having had daffodils (German: “Easter Bells”) show up already shortly after a sunny Christmas.

In any case, as soon as I find the time I’ll try to follow the hints on BBG\mathbf{B B}G that Tim Porter provided in his latest comment here.

Posted by: Urs Schreiber on March 25, 2008 5:19 PM | Permalink | Reply to this

Re: Nonabelian Differential Cohomology in Street’s Descent Theory

I would like to relate the discussion so far, which is based on Ross Street’s descent of ω\omega-category valued presheaves to the discussion along the lines of

B. Toen, Stacks and nonabelian cohomology.

This has a similar starting point, but then a different strategy.

Toen notes that the descent condition for plain old set-valued sheaves can be defined entirely implicitly (i.e. without actually mentioning any descent condition) by noticing that the category of sheaves is equivalent to the homotopy category of presheaves.

He then uses this as the starting point for an “implicit” definition of \infty-stacks by defining these to be the homotopy category of presheaves with values in simplicial sets, making use, I think, of the model category structure on simplicial sets.

This gives him a very elegant and powerful, albeit somewhat non-concrete notion of \infty-descent and cohomology.

I am wondering if one couldn’t pretty much mimic this construction with simplicial sets replaced by ω\omega-categories, making use of the model category structure on ωCat\omega Cat:

Yves Lafont, Francois Metayer, Krzysztof Worytkiewicz: A folk model structure on omega-cat

(many thanks to Mike Shulman for pointing this out to me).

Posted by: Urs Schreiber on March 26, 2008 7:04 PM | Permalink | Reply to this

Re: Nonabelian Differential Cohomology in Street’s Descent Theory

More concretely, the category of simplicial presheaves (i.e. presheaves with values in simplicial sets)SPr(S) SPr(S) on a category (site) SS becomes enriched over simplicial sets by forming for any two simplicial presheaves XX and YY the functor

hom˜(X,Y):SSet opSet \tilde hom(X,Y) : SSet^{op} \to Set given by SHom SPr(S)(S×X,Y), S \mapsto Hom_{SPr(S)}(S \times X, Y) \,, where the product is pointwise the product of simplicial sets.

Apparently this functor is always representable, and the representing simplicial set is then taken to be the hom-object hom(X,Y)SSethom(X,Y) \in SSet:

hom˜(X,Y)Hom Set SSet op(,hom(X,Y)). \tilde hom(X,Y) \simeq Hom_{Set^{SSet^{op}}}(-,hom(X,Y)) \,.

I am wondering to which degree this depends on the particular nature of SSetSSet.

Suppose I have any cartesian closed category CC, and write CPr(S)C Pr(S) for the category of CC-valued presheaves on SS.

Then form hom˜(X,Y):C opSet \tilde hom(X,Y) : C^{op} \to Set given by cHom CPr(S)(c×X,Y) c \mapsto Hom_{C Pr(S)}(c \times X, Y) as above.

Will that be representable?

Concretely, I’d like to know this for (C,×)=(ωCat, Gray)(C,\times) = (\omega Cat, \otimes_{Gray}).

If this works, it’s the answer to my last remark in 8.2.

Posted by: Urs Schreiber on March 27, 2008 9:42 PM | Permalink | Reply to this
Read the post What has happened so far
Weblog: The n-Category Café
Excerpt: A review of one of the main topics discussed at the Cafe: Sigma-models as the pull-push quantization of nonabelian differential cocycles.
Tracked: March 27, 2008 6:40 PM

Re: Nonabelian Differential Cohomology in Street’s Descent Theory

A discussion on nonabelian differential cohomology in the context of quantization of Σ\Sigma-models is developing here:

On Σ\Sigma-models and nonabelian differential cohomology

Abstract:

A “Σ\Sigma-model” can be thought of as a quantum field theory (QFT) which is determined by pulling back nn-bundles with connection (aka (n1n-1)-gerbes with connection, aka nonabelian differential cocycles) along all possible maps (the “fields”) from a “parameter space” to the given base space.

If formulated suitably, such Σ\Sigma-models include gauge theories such as notably (higher) Chern-Simons theory. If the resulting QFT is considered as an “extended” QFT, it should itself be a nonabelian differential cocycle on parameter space whose parallel transport along pieces of parameter space encodes the QFT propagation and correlators.

We are after a conception of nonabelian differential cocycles and their quantization which captures this. Our main motivation is the quantization of differential Chern-Simons cocycles to extended Chern-Simons QFT and its boundary conformal QFT, reproducing the cocycle structure implicit in [FFRS].

Classical

We conceive nonabelian differential cohomology in terms of cohomology with coefficients in ω\omega-category-valued presheaves [Street] of parallel transport ω\omega-functors from ω\omega-paths to a given structure ω\omega-group [BS, SW], discuss curvature and characteristic forms.

We describe Lie \infty-algeraic Cartan-Ehresmann connections [SSS] and integrate these, following [Getzler,Henriques], to nonabelian differential cocycles whenever certain connectedness and integrality conditions are met.

- For each transgressive L L_\infty-algebra cocycle there are Chern-Simons Lie \infty-connections arising as obstructions to lifts of L L_\infty-connections through String-like extensions of L L_\infty-algebras. Integrating these to differential cocycles yields general Chern-Simons nn-bundles with connection, reproducing in particular the known cocycles for Pontryagin classes [BrylinskiMcLaughlin].

Quantum

Our aim is to quantize such differential cocycles.

- We observe that for simple cases such as finite group and finite 2-group Chern-Simons theory (the Dijkgraaf-Witten and the Yetter model) the usual path integral is a decategorified categorical colimit over the given transport functor.

We interpret the holographic relation between Chern-Simons theory and its boundary conformal field theory as essentially being the hom-adjunction in ωCat\omega\mathrm{Cat} applied to a morphism between the two chiral copies of the Chern-Simons transport. This allows to conceive the Reshetikhin-Turaev description of the Chern-Simons 3d TFT together with the corresponding Frobenius-algebraic description of the boundary conformal QFT [FFRS] in terms of B𝒞\mathbf{B}\mathcal{C}-valued differential cocycles, for 𝒞\mathcal{C} the corresponding modular tensor category.

Posted by: Urs Schreiber on April 8, 2008 9:49 AM | Permalink | Reply to this
Read the post Connections on Nonabelian Gerbes and their Holonomy
Weblog: The n-Category Café
Excerpt: An article on transport 2-functors.
Tracked: August 15, 2008 8:15 PM
Read the post Waldorf on Transport Functors and Connections on Gerbes
Weblog: The n-Category Café
Excerpt: A talk on parallel 2-transport.
Tracked: September 6, 2008 4:39 PM
Read the post Codescent and the van Kampen Theorem
Weblog: The n-Category Café
Excerpt: On codescent, infinity-co-stacks, fundamental infinity-groupoids, natural differential geometry and the van Kampen theorem
Tracked: October 21, 2008 9:29 PM

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