Modules and comodules for corings

A coring C over a ring A is an (A, A)-bimodule with a comultiplication Δ: C → C _⨂A C and a counit ε: C → A, both being left and right A-linear mappings satisfying additional conditions. The dual spaces C* = Hom _A (C, A) and *C = _A Hom(C, A) allow the ring structure, and the right (left) comodules over C can be considered as left (right) modules over *C (respectively, C*). In fact, under weak restrictions on the A-module properties of C, the category of right C-comodules can be identified with the subcategory σ[_*C C] of *C-Mod, i.e., the category subgenerated by the left *C-module C. This point of view allows one to apply results from module theory to the investigation of coalgebras and comodules. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 51–72, 2005.     

Equivalences between categories of modules and categories of comodules

We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gomez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, Algebr. Represent. Theory, 10 （2007）, 271 306] and [Wisbauer, On Galois comodules, Comm. Algebra 34 （2006）, 2683-2711]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.     

Monoidal categories of corings

We introduce a monoidal category of corings using two different notions of corings morphisms. The first one is the (right) coring extensions recently introduced by T. Brzeziński in [2], and the other is the usual notion of morphisms defined in [6] by J. Gómez-Torrecillas.

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Sunto Introduciamo una categoria monoidale di coanelli usando due diverse nozioni di morfismi di coanelli. La prima è l'estensione (destra) di coanelli recentemente introdotta da Brzeziński in [2], mentre la seconda è la nozione usuale di morfismo definita in [6] da J. Gómez-Torrecillas.

     

Fibre functors of finite dimensional comodules

Let A be a commutative Hopf algebra over a field k; the k-valued fibre functors on the category of finite dimensional A-comodules correspond to Spec(A)-torsors over k as was shown by Saavedra Rivano and Deligne-Milne. We prove a non-commutative version of this result by using methods developed in a previous paper [5] for the case of finite Hopf algebras over a commutative ring. We also exhibit right adjoints for fibre functors under the assumption that the antipode is bijective.     

Galois extensions as functors of comodules

Let A be a finite Hopf algebra over a commutative ring k. We show a one-to-one correspondence between the A-Galois extensions of k and certain functors from the category of A-comodules to the category of k-modules.     

The fiber of functors between categories of algebras

We investigate the fiber of a functor F:C→D between sketchable categories of algebras over an object D∈D from two points of view: characterizing its classifying space as a universal -space; and parametrizing its objects in cohomological terms.     

Quadratic functors on pointed categories

We study polynomial functors of degree 2, called quadratic, with values in the category of abelian groups Ab, and whose source category is an arbitrary category C with null object such that all objects are colimits of copies of a generating object E which is small and regular projective; this includes all categories of models V of a pointed theory T. More specifically, we are interested in such quadratic functors F from C to Ab which preserve filtered colimits and suitable coequalizers.A functorial equivalence is established between such functors F:C→Ab and certain minimal algebraic data which we call quadratic C-modules: these involve the values on E of the cross-effects of F and certain structure maps generalizing the second Hopf invariant and the Whitehead product.Applying this general result to the case where E is a cogroup these data take a particularly simple form. This application extends results of Baues and Pirashvili obtained for C being the category of groups or of modules over some ring; here quadratic C-modules are equivalent with abelian square groups or quadratic R-modules, respectively.     

A remark on fixed points of functors in topological categories

For a topological category over Set we prove that if a functor T: has a fixed cardinal (i.e. for each object K with card (UK)= we have card (UTK)), then T has a least fixed point, and if T has a successive pair of fixed cardinals and ⁺, then T has a greatest fixed point. This extends results of Adámek and Koubek.Partial financial support of the Grant Agency of the Czech Republic under Grant No. 201/93/0950 is gratefully acknowledged.     

A geometric construction for permutation equivariant categories from modular functors

Let G be a finite group. Given a finite G-set Xcal{X} and a modular tensor category Ccal{C}, we construct a weak G-equivariant fusion category C^Xcal{C}^{cal{X}}, called the permutation equivariant tensor category. The construction is geometric and uses the formalism of modular functors. As an application, we concretely work out a complete set of structure morphisms for mathbbZ/2mathbb{Z}/2-permutation equivariant categories, finishing thereby a program we initiated in an earlier paper.     

Calculus of functors and model categories

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large categories, preprint, 2005]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the n-homogeneous model structure, the nth derivative is a Quillen functor to the category of spectra with Σn-action. After taking into account only finitary functors—which may be done in two different ways—the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T.G. Goodwillie [T.G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645-711 (electronic)].     

Tannakian formalism for fiber functors over tensor categories

It is well known that the pseudovariety \(\mathbf {J}\) of all \(\mathscr {J}\)-trivial monoids is not local, which means that the pseudovariety \(g\mathbf {J}\) of categories generated by \(\mathbf {J}\) is a proper subpseudovariety of the pseudovariety \(\ell \mathbf {J}\) of categories all of whose local monoids belong to \(\mathbf {J}\). In this paper, it is proved that the pseudovariety \(\mathbf {J}\) enjoys the following weaker property. For every prime number p, the pseudovariety \(\ell \mathbf {J}\) is a subpseudovariety of the pseudovariety \(g(\mathbf {J}*\mathbf {Ab}_p)\), where \(\mathbf {Ab}_p\) is the pseudovariety of all elementary abelian p-groups and \(\mathbf {J}*\mathbf {Ab}_p\) is the pseudovariety of monoids generated by the class of all semidirect products of monoids from \(\mathbf {J}\) by groups from \(\mathbf {Ab}_p\). As an application, a new proof of the celebrated equality of pseudovarieties \(\mathbf {PG}=\mathbf {BG}\) is obtained, where \(\mathbf {PG}\) is the pseudovariety of monoids generated by the class of all power monoids of groups and \(\mathbf {BG}\) is the pseudovariety of all block groups.     

Covariant functors in categories of topological spaces

This survey is devoted to the properties of certain concrete covariant functors-normal and almost normal functors-in the category of compacta, as well as the algebraic theory of covariant functors, and the connections between the theory of functors with absolute extensors and manifolds.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 28, pp. 47–95, 1990.     

A note on K-theory and triangulated categories

We provide an example of two closed model categories having equivalent homotopy categories but different Waldhausen K-theories. We also show that there cannot exist a functor from small triangulated categories to spaces which recovers Quillen’s K-theory for exact categories and which satisfies localization. Oblatum 28-V-2001 & 7-III-2002?Published online: 17 June 2002