Take the 2-minute tour ×  In the category of rational spaces,
loop spaces split as products of Eilenberg-Mac Lane spaces and
SUSPENSIONS split as wedges of (rational) spheres. I wonder if
anything of the following form is true:  (*) Any functor $F$
from spaces to spaces which splits suspensions and loop spaces
as above must factor through the rationalization.  EDIT 1: Greg
raises some fine questions, but I stand by my wording. This is a
question that arises from curiosity, not because I need it for
anything, so I'd be happy with "anything like" the given
statement.  EDIT 2: At least for simply-connected spaces,
rationalization commutes with loop and suspension. But, it seems
to me that the power of the property is that the suspension of
any F-space splits and the loops of any F-space splits. So I
would go with: the suspension of any rational space splits as a
wedge of rational spheres and the loops of any rational space
splits as a product of rational Eilenberg-Mac Lanes spaces.
Thus, we'd be looking for fun