2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/56705The 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.19 pages, Latex 2e (amsmath,amsfonts,amssymb,amsthm). Invited talk given at the NATO Advanced Research Workshop "Noncommutative Structures In Mathematics And Physics" held in Kiev, 24-28 September 2000. To be published in the Proceedings by Kluwer Academic Publishers, 2001Mathematical PhysicsHigh Energy Physics - Theorytext