Semisupermanifolds and regularization of categories, modules, algebras andYang-Baxter equation

The concepts of regular n-cocycles, obstruction and the regularization proc edure are developed as the category theory analogy of the semisupermanifold s with noninvertible transition functions. It is shown that the regularizat ion of a category with nonivertible morphisms and obstruction form a 2-cate gory. The generalization of some related structures to the regular case is considered.