Noninvertibility and ``Semi-'' Analogs of (Super) Manifolds, Fiber   Bundles and Homotopies
  Supersymmetry contains initially noninvertible objects, but it is common to deal with the invertible ones only, factorizing former in some extent. We propose to reconsider this ansatz and try to redefine such fundamental notions as supermanifolds, fiber bundles and homotopies using some weakening invertibility conditions. The prefix semi- reflects the fact that the underlying morphisms form corresponding semigroups consisting of a known group part and a new ideal noninvertible part. We found that the absence of invertibility gives us the generalization of the cocycle conditions for transition functions of supermanifolds and fiber bundles in a natural way, which can lead to construction of noninvertible analogs of \v{C}ech cocycles. We define semi-homotopies, which can be noninvertible and describe mappings into the semi-supermanifolds introduced. 
Steven Duplij