  We give a useful new characterization of the set of all completely positive,
trace-preserving (i.e., stochastic) maps from 2x2 matrices to 2x2 matrices.
These conditions allow one to easily check any trace-preserving map for
complete positivity. We also determine explicitly all extreme points of this
set, and give a useful parameterization after reduction to a certain canonical
form. This allows a detailed examination of an important class of non-unital
extreme points which can be characterized as having exactly two images on the
Bloch sphere.
  We also discuss a number of related issues about the images and the geometry
of the set of stochastic maps, and show that any stochastic map on 2x2 matrices
can be written as a convex combination of two "generalized" extreme points.
