  Localisation is an important technique in ring theory and yields the
construction of various rings of quotients. Colocalisation in comodule
categories has been investigated by some authors where the colocalised
coalgebra turned out to be a suitable subcoalgebra. Rather then aiming at a
subcoalgebra we look at possible coalgebra covers p:D->>C that could play the
role of a dual quotient object. Codense covers will dualise dense (or rational)
extensions; a maximal codense cover construction for coalgebras with projective
covers is proposed. We also look at a dual non-singularity concept for modules
which turns out to be the comodule-theoretic property that turns the dual
algebra of a coalgebra into a non-singular ring. As a corollary we deduce that
hereditary coalgebras and hence path coalgebras are non-singular in the above
sense. We also look at coprime coalgebras and Hopf algebras which are
non-singular as coalgebras.
