Π-twist and orientability in braided categories

We introduce the notion of orientability in a strict braided tensor category and define (pre)orientable and orientable objects, using the previous notion of Π-twist formulated exclusively in terms of adjunctions. We obtain a ribbon category of oriented objects (where the 2π-twist is like Π2) by giving an appropriate definition of compatible morphisms between these objects. For a stable object we introduce the abstract Möbius band and obtain a strictly algebraic proof for some classical properties of the topological Möbius band. Finally a topological visualization is given.