Maybe I'm looking at the wrong places, but I can't find a definition of the coboundary map on the cochain complex of abelian cosimplicial groups.
What I have in mind is something similar to the "Moore construction in the alternating face map complex " for abelian simplicial groups, for example given in http://ncatlab.org/nlab/show/Moore+complex but this time obviously as a (co)boundary operator on a cochain complex.
Is there an obstruction to a definition? (I guess not)
If not how is it done? (Definition of the coboundary operator)