  In this paper, a new higher Hochschild Complex is defined with an Iterated
Integral map to locally model differential forms on the space of bigons on $M$.
In particular, given the local data for a gerbe with structure 2-group given by
a crossed module of matrix-groups, there is an element in our curved zigzag
Hochschild complex associated to the local 2-holonomy given by such a gerbe.
This paper introduces an initial construction central to the author's PhD
Thesis.
