  We develop a theory of multigraded (i.e., $N^l$-graded) combinatorial Hopf
algebras modeled on the theory of graded combinatorial Hopf algebras developed
by Aguiar, Bergeron, and Sottile [Compos. Math. 142 (2006), 1--30]. In
particular we introduce the notion of canonical $k$-odd and $k$-even
subalgebras associated with any multigraded combinatorial Hopf algebra,
extending simultaneously the work of Aguiar et al. and Ehrenborg. Among our
results are specific categorical results for higher level quasisymmetric
functions, several basis change formulas, and a generalization of the
descents-to-peaks map.
