The n-Category Café

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

Regarding section 5.2:

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

This is my motivation for trying to understand $G$ as an $A_\infty$ simplicial space (or whatever). I can see easily how $\mathbf{E}G$ 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.

I would like to relate the discussion so far, which is based on Ross Street’s descent of $\omega$-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 $\omega$-categories, making use of the model category structure on $\omega Cat$:

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

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

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

On $\Sigma$-models and nonabelian differential cohomology

Abstract:

A “$\Sigma$-model” can be thought of as a quantum field theory (QFT) which is determined by pulling back $n$-bundles with connection (aka ($n-1$)-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 $\Sigma$-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 $\omega$-category-valued presheaves [Street] of parallel transport $\omega$-functors from $\omega$-paths to a given structure $\omega$-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 $L_\infty$-algebra cocycle there are Chern-Simons Lie $\infty$-connections arising as obstructions to lifts of $L_\infty$-connections through String-like extensions of $L_\infty$-algebras. Integrating these to differential cocycles yields general Chern-Simons $n$-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 $\omega\mathrm{Cat}$ 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 $\mathbf{B}\mathcal{C}$-valued differential cocycles, for $\mathcal{C}$ the corresponding modular tensor category.