  A coring (A,C) consists of an algebra A and a coalgebra C in the monoidal
category of A-bimodules. Corings and their comodules arise naturally in the
study of Hopf-Galois extensions and descent theory, as well as in the study of
Hopf algebroids. In this paper, we address the question of when two corings in
a symmetric monoidal model category V are homotopically Morita equivalent,
i.e., when their respective categories of comodules are Quillen equivalent.
  The category of comodules over the trivial coring (A,A) is isomorphic to the
category of A-modules, so the question above englobes that of when two algebras
are homotopically Morita equivalent. We discuss this special case in the first
part of the paper, extending previously known results. To approach the general
question, we introduce the notion of a 'braided bimodule' and show that
adjunctions between A-Mod and B-Mod that lift to adjunctions between
(A,C)-Comod and (B,D)-Comod correspond precisely to braided bimodules between
(A,C) and (B,D). We then give criteria, in terms of homotopic descent, for when
a braided bimodule induces a Quillen equivalence. In particular, we obtain
criteria for when a morphism of corings induces a Quillen equivalence,
providing a homotopic generalization of results by Hovey and Strickland on
Morita equivalences of Hopf algebroids. To illustrate the general theory, we
examine homotopical Morita theory for corings in the category of chain
complexes over a commutative ring.
