Homotopical Morita theory for corings

TL;DR

This paper develops a homotopical Morita theory for corings in monoidal model categories, introducing braided bimodules and criteria for Quillen equivalences, extending classical results to a homotopical setting.

Contribution

It introduces the notion of braided bimodules and provides criteria for homotopical Morita equivalences of corings, generalizing known algebraic results to a homotopical context.

Findings

Criteria for when a morphism of corings induces a Quillen equivalence.

Extension of Morita theory to corings in chain complexes over rings.

Homotopical generalization of Morita equivalences for Hopf algebroids.

Abstract

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.