Homotopical Morita theory for corings

(Joint work with Alexander Berglund)

A coring $(A,C)$
consists of an algebra $A$ in a symmetric monoidal category 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
lecture, I will address the question of when two corings $(A,C)$ and
$(B,D)$ in a symmetric monoidal model category $\mathcal V$ are
homotopically Morita equivalent, i.e., when their respective categories
of comodules $\mathcal V_{A}^{C}$ and $\mathcal V_{B}^{D}$ are Quillen
equivalent.

The category of comodules over the trivial coring $(A,A)$ is isomorphic
to the category $\mathcal V_{A}$ of $A$-modules, so the question
englobes that of when two algebras are homotopically Morita equivalent.
I will begin by discussing this special case, extending previously known
results.

To approach the general question, I will introduce the notion of a
braided bimodule and show that adjunctions between $\mathcal
V_{A}$ and $\mathcal V_{B}$ that

lift to adjunctions between $\mathcal
V_{A}^{C}$ and $\mathcal V_{B}^{D}$ correspond precisely to braided
bimodules. I will describe descent-type criteria for when a braided
bimodule induces a Quillen equivalence between $\mathcal V_{A}^{C}$ and
$\mathcal V_{B}^{D}$. In particular, I will provide conditions under
which a morphism of corings induces a Quillen equivalence, providing a
homotopic generalization of results by Hovey and Strickland on Morita
equivalences of Hopf algebroids. As an illustration of the general
theory, I will describe in detail the homotopical Morita theory of
corings in the category of chain complexes over a commutative ring.