  Classically, there are two model category structures on coalgebras in the
category of chain complexes over a field. In one, the weak equivalences are
maps which induce an isomorphism on homology. In the other, the weak
equivalences are maps which induce a weak equivalence of algebras under the
cobar functor. We unify these two approaches, realizing them as the two
extremes of a partially ordered set of model category structures on coalgebras
over a cooperad satisfying mild conditions.
