  We make and generalize the observation that summing of probability amplitudes
of a discrete-time quantum walk over partitions of the walking graph consistent
with the step operator results in a unitary evolution on the reduced graph
which is also a quantum walk. Since the effective walking graph of the
projected walk is not necessarily simpler than the original, this may bring new
insights into the dynamics of some kinds of quantum walks using known results
from thoroughly studied cases like Euclidean lattices. We use abstract
treatment of the walking space and walker displacements in aim for a generality
of the presented statements. Using this approach we also identify some
pathological cases in which the projection mapping breaks down. For walks on
lattices, the operation typically results in quantum walks with
hyper-dimensional coin spaces. Such walks can, conversely, be viewed as
projections of walks on inaccessible, larger spaces, and their properties can
be inferred from the parental walk. We show that this is is the case for a lazy
quantum walk, a walk with large coherent jumps and a walk on a circle with a
twisted boundary condition. We also discuss the relation of this theory to the
time-multiplexing optical implementations of quantum walks. Moreover, this
manifestly irreversible operation can, in some cases and with a minor
adjustment, be undone, and a quantum walk can be reconstructed from a set of
its projections.
