### Nonabelian Weak Deligne Hypercohomology

#### Posted by Urs Schreiber

What I described last time is really best thought of in the context of what I propose to call *nonabelian weak Deligne hypercohomolgy*.

Unless I am hallucinating the following is the correct formalism to generalize the well-known Deligne hypercohomology formulation of strict *abelian* $p$-gerbes to weak and nonabelian $p$-gerbes.

Consider two $p$-algebroids represented by dg-algebras $({d}^{A},{\bigwedge}^{\u2022}{A}^{*})$ and $({d}^{B},{\bigwedge}^{\u2022}{B}^{*})$ and let $I$ be some countable set.

Recall that the **Čech complex**

is the free abelian group generated by all tuples of elements of I,

equipped with the boundary operator

Let

be the space of maps from ${\bigwedge}^{\u2022}{B}^{*}$ to ${\bigwedge}^{\u2022}{A}^{*}$ of degree $-n$.

Let

be the space of linear maps from the Čech complex to dg-algebra maps.

On $\Omega $ the Čech boundary operator acts as

Also the operator $Q$ which I introduced last time acts as

Both these operators make $\Omega $ into a complex. On the resulting double complex we have the total differential

I believe that the infinitesimal cocycle laws and gauge transformations of a nonabelian weak $p$-bundle with $p$-connection (or equivalently a nonabelian weak $p-1$-gerbe) are specified by the cohomology class of a cocycle of $D$, i.e. that a $D$-closed element

with

specifies the infinitesimal cocycle relations of such a $p$-bundle/$(p-1)$-gerbe and that the shift

specifies an infinitesimal gauge transformation.

As a first sanity check, note that if ${d}^{B}=0$, i.e. when the target $p$-algebroid is strict and abelian, the above $D$ reduces to the usual abelian Deligne coboundary operator (up to signs that I mixed up, possibly). Also, the content of the last entry can be seen to be a special case of this.

A more detailed discussion follows.

(A little more details are given in section 3.3 and section 3.5.4 here.)