  The classification of subfactors of small index revealed several new
subfactors. The first subfactor above index 4, the Haagerup subfactor, is
increasingly well understood and appears to lie in a (discrete) infinite family
of subfactors where the Z/3Z symmetry is replaced by other finite Abelian
groups. The goal of this paper is to give a similarly good description of the
Asaeda-Haagerup subfactor which emerged from our study of its Brauer-Picard
groupoid. More specifically, we construct a new subfactor S which is a Z/4Z x
Z/2Z analogue of the Haagerup subfactor and we show that the even parts of the
Asaeda-Haagerup subfactor are higher Morita equivalent to an orbifold quotient
of S. This gives a new construction of the Asaeda-Haagerup subfactor which is
much more symmetric and easier to work with than the original construction. As
a consequence, we can settle many open questions about the Asaeda-Haagerup
subfactor: calculating its Drinfel'd center, classifying all extensions of the
Asaeda-Haagerup fusion categories, finding the full higher Morita equivalence
class of the Asaeda-Haagerup fusion categories, and finding intermediate
subfactor lattices for subfactors coming from the Asaeda-Haagerup categories.
The details of the applications will be given in subsequent papers.
