Monads and comonads in module categories

Let $A$ be a ring and \M_A the category of $A$-modules. It is well known in module theory that for any $A$-bimodule $B$, $B$ is an $A$-ring if and only if the functor -\otimes_A B: \M_A\to \M_A is a monad (or triple). Similarly, an $A$-bimodule \C is an $A$-coring provided the functor -\otimes_A\C:\M_A\to \M_A is a comonad (or cotriple). The related categories of modules (or algebras) of $-\otimes_A B$ and comodules (or coalgebras) of -\otimes_A\C are well studied in the literature. On the other hand, the right adjoint endofunctors \Hom_A(B,-) and \Hom_A(\C,-) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of \Hom_A(B,-)-comodules is isomorphic to the category of $B$-modules, while the category of \Hom_A(\C,-)-modules (called \C-contramodules by Eilenberg and Moore) need not be equivalent to the category of \C-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of \C-comodules and \Hom_A(\C,-)-modules are equivalent provided \C is a coseparable coring. Furthermore, a bialgebra $H$ over a commutative ring $R$ is a Hopf algebra if and only if \Hom_R(H-) is a Hopf bimonad on \M_R and in this case the categories of $H$-Hopf modules and mixed \Hom_R(H,-)-bimodules are both equivalent to \M_R.Comment: 35 pages, LaTe

A necessary and sufficient condition for induced model structures

A common technique for producing a new model category structure is to lift the fibrations and weak equivalences of an existing model structure along a right adjoint. Formally dual but technically much harder is to lift the cofibrations and weak equivalences along a left adjoint. For either technique to define a valid model category, there is a well-known necessary "acyclicity" condition. We show that for a broad class of "accessible model structures" - a generalization introduced here of the well-known combinatorial model structures - this necessary condition is also sufficient in both the right-induced and left-induced contexts, and the resulting model category is again accessible. We develop new and old techniques for proving the acyclity condition and apply these observations to construct several new model structures, in particular on categories of differential graded bialgebras, of differential graded comodule algebras, and of comodules over corings in both the differential graded and the spectral setting. We observe moreover that (generalized) Reedy model category structures can also be understood as model categories of "bialgebras" in the sense considered here.Comment: 49 pages; final journal version to appear in the Journal of Topolog