Noninvertibility, semisupermanifolds and categories regularization
  The categories with noninvertible morphisms are studied analogously to the semisupermanifolds with noninvertible transition functions. The concepts of regular n-cycles, obstruction and the regularization procedure are introduced and investigated. It is shown that the regularization of a category with nonivertible morphisms and obstruction form a 2-category. The generalization of functors, Yang-Baxter equation, (co-) algebras, (co-) modules and some related structures to the regular case is given. 
Steven Duplij (Kharkov National University), Wladyslaw Marcinek   (University of Wroclaw)