  We propose to extend ``invertibility'' to ``regularity'' for categories in
general abstract algebraic manner. Higher regularity conditions and
``semicommutative'' diagrams are introduced. Distinction between commutative
and ``semicommutative'' cases is measured by non-zero obstruction proportional
to the difference of some self-mappings (obstructors) $e^{(n)}$ from the
identity. This allows us to generalize the notion of functor and to
``regularize'' braidings and related structures in monoidal categories. A
``noninvertible'' analog of the Yang-Baxter equation is proposed.
