On comonadicity of the extension-of-scalars functors

A criterion for comonadicity of the extension-of- scalars functor associated to an extension of (not necessarily commutative) rings is given. As an application of this criterion, some known results on the comonadicity of such functors are obtained

Pure morphisms are effective for modules

Yet another proof of the result asserting that a morphism of commutative rings is an effective descent morphism for modules if and only if it is pure is given. Moreover, it is shown that this result cannot be derived from Moerdijk's descent criterion

Entwining Structures in Monoidal Catrgories

Interpreting entwining structures as special instances of J. Beck's distributive law, the concept of entwining module can be generalized for the setting of arbitrary monoidal category. In this paper, we use the distributive law formalism to extend in this setting basic properties of entwining modules

Descent in ∗ -autonomous categories

AbstractWe extend the result of Joyal and Tierney asserting that a morphism of commutative algebras in the ∗-autonomous category of sup-lattices is an effective descent morphism for modules if and only if it is pure, to an arbitrary ∗-autonomous category V (in which the tensor unit is projective) by showing that any V-functor out of V is precomonadic if and only if it is comonadic

Galois functors and entwining structures

{\em Galois comodules} over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of {\em Galois functors} over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injectives (projectives) in this context. Then we generalise the notion of corings (derived from an entwining of an algebra and a coalgebra) to the entwining of a monad and a comonad. Hereby a key role is played by the notion of a {\em grouplike natural transformation} $g:I\to G$ generalising the grouplike elements in corings. We apply the evolving theory to Hopf monads on arbitrary categories, and to comonoidal functors on monoidal categories in the sense of A. Brugui\`{e}res and A. Virelizier. As well-know, for any set $G$ the product $G\times-$ defines an endofunctor on the category of sets and this is a Hopf monad if and only if $G$ allows for a group structure. In the final section the elements of this case are generalised to arbitrary categories with finite products leading to {\em Galois objects} in the sense of Chase and Sweedler

Descent in ∗ -autonomous categories

AbstractWe extend the result of Joyal and Tierney asserting that a morphism of commutative algebras in the ∗-autonomous category of sup-lattices is an effective descent morphism for modules if and only if it is pure, to an arbitrary ∗-autonomous category V (in which the tensor unit is projective) by showing that any V-functor out of V is precomonadic if and only if it is comonadic

On Rational Pairings of Functors

In the theory of coalgebras $C$ over a ring $R$, the rational functor relates the category of modules over the algebra $C^*$ (with convolution product) with the category of comodules over $C$. It is based on the pairing of the algebra $C^*$ with the coalgebra $C$ provided by the evaluation map \ev:C^*\ot_R C\to R. We generalise this situation by defining a {\em pairing} between endofunctors $T$ and $G$ on any category \A as a map, natural in a,b\in \A, \beta_{a,b}:\A(a, G(b)) \to \A(T(a),b), and we call it {\em rational} if these all are injective. In case \bT=(T,m_T,e_T) is a monad and \bG=(G,\delta_G,\ve_G) is a comonad on \A, additional compatibility conditions are imposed on a pairing between \bT and \bG. If such a pairing is given and is rational, and \bT has a right adjoint monad \bT^\di, we construct a {\em rational functor} as the functor-part of an idempotent comonad on the \bT-modules \A_{\rT} which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories