  We introduce the category Pstem[n] of n-stems, with a functor P[n] from
spaces to Pstem[n]. This can be thought of as the n-th order homotopy groups of
a space. We show how to associate to each simplicial n-stem Q an
(n+1)-truncated spectral sequence. Moreover, if Q=P[n]X is the Postnikov n-stem
of a simplicial space X, the truncated spectral sequence for Q is the
truncation of the usual homotopy spectral sequence of X. Similar results are
also proven for cosimplicial n-stems. They are helpful for computations, since
n-stems in low degrees have good algebraic models.
