### Nonabelian Differential Cohomology in Street’s Descent Theory

#### Posted by Urs Schreiber

As a followup to our recent discussion #:

Nonabelian differential cohomologyin 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 $\omega$-categorical models
(strict globular $\infty$-categories) and discuss
nonabelian differential cohomology with values in
$\omega$-groups obtained from integrating L(ie)-$\infty$
algebras.

**Introduction**

A principal $G$-bundle is given, with respect
to a good cover by open sets of its base space, by
a *trivial* $G$-bundle on each open subset, together
with an isomorphism of trivial $G$-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 $G$ 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 $G$ 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 $\omega$-functors from paths in a patch to some codomain provides a functor from “spaces” to $\omega$-categories: an $\omega$-category valued presheaf.

In [Street] Ross Street descibes a very general formalization for cohomology taking values in $\omega$-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.

## 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.