  A logic has uniform interpolation if its formulas can be projected down to
given subsignatures, preserving all logical consequences that do not mention
the removed symbols; the weaker property of (Craig) interpolation allows the
projected formula -- the interpolant -- to be different for each logical
consequence of the original formula. These properties are of importance, e.g.,
in the modularization of logical theories. We study interpolation in the
context of coalgebraic modal logics, i.e. modal logics axiomatized in rank 1,
restricting for clarity to the case with finitely many modalities. Examples of
such logics include the modal logics K and KD, neighbourhood logic and its
monotone variant, finite-monoid-weighted logics, and coalition logic. We
introduce a notion of one-step (uniform) interpolation, which refers only to a
restricted logic without nesting of modalities, and show that a coalgebraic
modal logic has uniform interpolation if it has one-step interpolation.
Moreover, we identify preservation of finite surjective weak pullbacks as a
sufficient, and in the monotone case necessary, condition for one-step
interpolation. We thus prove or reprove uniform interpolation for most of the
examples listed above.
