  Recently, a theory has been proposed that classifies cyclic processes of
symmetry protected topological (SPT) quantum states. For the case of spin
chains, i.e.\ one-dimensional bosonic SPT's, this theory implies that cyclic
processes are classified by zero-dimensional SPT's. This is often described as
a generalization of Thouless pumps, with the original Thouless pump
corresponding to the case where the symmetry group is $U(1)$ and pumps are
classified by an integer that corresponds to the charge pumped per cycle. In
this paper, we review this one-dimensional theory in an explicit and rigorous
setting and we provide a proof for the completeness of the proposed
classification for compact symmetry groups $G$.
