(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 *braidedbimodules*. I will describe descent-type criteria for when a

*braided*

bimoduleinduces a Quillen equivalence between $\mathcal V_{A}^{C}$ and

bimodule

$\mathcal V_{B}^{D}$. 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.

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