  Central to the theory of special cube complexes is Haglund and Wise's
construction of the canonical completion and retraction, which enables one to
build finite covers of special cube complexes in a highly controlled manner. In
this paper we give a new interpretation of this construction using what we call
imitator homomorphisms. This provides fresh insight into the construction and
enables us to prove various new results about finite covers of special cube
complexes -- most of which generalise existing theorems of Haglund--Wise to the
non-hyperbolic setting. In particular, we prove a convex version of omnipotence
for virtually special cubulated groups.
