  We prove a generalization of the twisted geometric Satake equivalence of
Finkelberg--Lysenko in the context of the factorizable grassmannian of a
reductive group G relative to a smooth curve X, similar to Gaitsgory's
generalization in "On de Jong's conjecture" of the Satake equivalence itself.
In order to express the dependence of this result on factorizability, we
consider a certain 2-category of gerbes on this grassmannian, which we call "sf
gerbes"; the theorem then concerns perverse sheaves twisted by such gerbes.
This notion of twisting differs from, but in special cases is equivalent to,
that considered by Finkelberg--Lysenko. We give a classification of these
gerbes in terms of quadratic forms on the coweight lattice of G and gerbes on
X. In addition, we introduce the technical device of universal local acyclicity
(ULA) into the proof of the Satake equivalence, using it in combination with
Beilinson's nearby cycles gluing of perverse sheaves to reduce all of the main
constructions in the Mirkovic--Vilonen proof of the equivalence to the ULA
case, where they have simple descriptions whose properties have simple proofs.
We also give a geometric proof that the category of spherical perverse sheaves
on the ordinary affine grassmannian is semisimple.
