Stems and spectral sequences

We introduce the category Pstem.[n] of n-stems, with a functor P [n] from spaces to Pste m[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.