Condensation inversion and Witt equivalencevia generalised orbifolds

In [MR1] it was shown how a so-called orbifold datum A in a given modular fusion category (MFC) C produces a new MFC C A .Examples of these associated MFCs include condensations, i.e. the categories C B of local modules of a separable commutative algebra B C. In this paper we prove that the relation C C A on MFCs is the same as Witt equivalence.This is achieved in part by providing one with an explicit construction for inverting condensations, i.e. finding an orbifold datum A in C B whose associated MFC is equivalent to C. As a tool used in this construction we also explore what kinds of functors F : C D between MFCs preserve orbifold data.It turns out that F need not necessarily be strong monoidal, but rather a 'ribbon Frobenius' functor, which has weak monoidal and weak comonoidal structures, related by a Frobenius-like property.