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May 17, 2005

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, A *) and (d B, B *) and let I be some countable set.

Recall that the Čech complex

(1)C=C(I)= n=1 C n(I)

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

(2)C n={(i 1 i 2 i n)i 1 ,i 2 ,,i nI},

equipped with the boundary operator

(3)δ:C n C n1 (i 1 i n) m=1 n(1 ) (m+1 )(i iî mi n).

Let

(4)T n={f: B * (n)A *}

be the space of maps from B * to A * of degree n.

Let

(5)Ω={ω n:C nT n}

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

On Ω the Čech boundary operator acts as

(6)(δ˜ω)(i 1 i n)=ω(δ(i 1 i n)).

Also the operator Q which I introduced last time acts as

(7)(Qω)(i 1 i n)=d Bω(i 1 i n)+(1 ) nω(i 1 i n)d A.

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

(8)D=δ˜+(1 ) nQ.

I believe that the infinitesimal cocycle laws and gauge transformations of a nonabelian weak p-bundle with p-connection (or equivalently a nonabelian weak p1 -gerbe) are specified by the cohomology class of a cocycle of D, i.e. that a D-closed element

(9)ωΩ

with

(10)Dω=0

specifies the infinitesimal cocycle relations of such a p-bundle/(p1 )-gerbe and that the shift

(11)ωω+Dλ

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.)

Posted at May 17, 2005 1:46 PM UTC

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