  This paper is a sequel to [Ga1]. We study the semi-infinite category on the
Ran version of the affine Grassmannian, and study a particular object in it
that we call the semi-infinite intersection cohomology sheaf.
  Unlike the situation of [Ga1], this version of the semi-infinite intersection
IC sheaf is defined as the middle of extension of the constant (more precisely,
dualizing) sheaf on the basic stratum, in a certain t-structure. We give
several explicit description and characterizations of our semi-infinite
intersection IC sheaf: we describe its !- and *- stalks; we present it
explicitly as a colimit; we relate it to the IC sheaf of Drinfeld's relative
compactification; we describe it via the Drinfeld-Plucker formalism.
