Theory of (k⊕l|q)-dimensional supermanifolds

We define a new object which we call a (k ⊕ l|q) -dimensional supermanifold or partially formal supermanifold. This manifold has q odd coordinates and k + l even coordinates with l of them taking only nilpotent values. We develop the theory of such supermanifolds using functorial approach. Such a theory is completely analogous to the theory of standard supermanifolds. Therefore the paper can be considered also as an introduction to supergeometry written in the language of functors. We list some cases when (k ⊕ l|q) -dimensional supermanifolds appear naturally in physics and mathematics.