Noncommutative Differential Forms and Quantization of the Odd Symplectic Category

There is a simple and natural quantization of differential forms on odd Poisson supermanifolds, given by the relation [f,dg]={f,g} for all functions f and g. We notice that this noncommutative differential algebra has a geometrical realization as a convolution algebra of the symplectic groupoid integrating the Poisson manifold. This quantization is just a part of a quantization of the odd symplectic category (where objects are odd symplectic supermanifolds and morphisms are Lagrangian relations) in terms of ₂-graded chain complexes. It is a straightforward consequence of the theory of BV operator acting on semidensities, due to H. Khudaverdian.