We consider the local deformation problem of coisotropic submanifolds inside symplectic
or Poisson manifolds. To this end the groupoid of coisotropic sections (with respect to some
tubular neighbourhood) is introduced. Although the geometric content of this groupoid is evident,
it is usually a very intricate object.
We provide a description of the groupoid of coisotropic sections in terms of a differential graded
Poisson algebra, called the BFV-complex. This description is achieved by constructing a groupoid
from the BFV-complex and a surjective morphism from this groupoid to the groupoid of coisotropic
sections. The kernel of this morphism can be easily chracterized.
As a corollary we obtain an isomorphism between the moduli space of coisotropic sections and
the moduli space of geometric Maurer–Cartan elements of the BFV-complex. In turn, this also sheds
new light on the geometric content of the BFV-complex.