26 research outputs found
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 β -autonomous categories
AbstractWe extend the result of Joyal and Tierney asserting that a morphism of commutative algebras in the β-autonomous category of sup-lattices is an effective descent morphism for modules if and only if it is pure, to an arbitrary β-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}
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 the product defines an
endofunctor on the category of sets and this is a Hopf monad if and only if
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 β -autonomous categories
AbstractWe extend the result of Joyal and Tierney asserting that a morphism of commutative algebras in the β-autonomous category of sup-lattices is an effective descent morphism for modules if and only if it is pure, to an arbitrary β-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 over a ring , the rational functor relates
the category of modules over the algebra (with convolution product) with
the category of comodules over . It is based on the pairing of the algebra
with the coalgebra provided by the evaluation map \ev:C^*\ot_R C\to
R. We generalise this situation by defining a {\em pairing} between
endofunctors and 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