Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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).


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

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


(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




specifies the infinitesimal cocycle relations of such a p-bundle/(p1 )-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.)

Posted at May 17, 2005 1:46 PM UTC

TrackBack URL for this Entry:

0 Comments & 0 Trackbacks

Post a New Comment