Noninvertibility, Semisupermanifolds and Categories Regularization

Abstract

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. * Invited talk given at the NATO Advanced Research Workshop "NONCOMMU-

Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence

“…We then generalize the correspondence thus introduced to the polyadic case and thereby obtain higher degree (in our definition) analogs of the former. The higher (degree) regular semigroups obtained in this way have appeared previously in semisupermanifold theory [6] and higher regular categories in Topological Quantum Field Theory [7]. The representations of the higher braid relations in vector spaces coincide with the higher braid equation and corresponding generalized R-matrix obtained in Reference [8], as do the ordinary braid group and the Yang-Baxter equation [9,10].…”

“…Definition 3. We say that the set of idempotent ternary matrices M g id (2) (6) is in ternary-binary correspondence with the regular (binary) semigroup G reg and write this as…”

“…The higher regularity conditions ( 23)-( 25) obtained above from the idempotence of polyadic matrices using the polyadic-binary correspondence, appeared first in Reference [20] and were then used for transition functions in the investigation of semisupermanifolds [6] and higher regular categories in TQFT [7,21]. Now, we turn to the second line of (2), and in the same way as above introduce higher degree braid groups.…”