Corings and Comodules Brzezinski, Tomasz; Wisbauer, Robert
2003., Volume:
v.Series Number 309
eBook
Corings and comodules are fundamental algebraic structures, which can be thought of as both dualisations and generalisations of rings and modules. Introduced by Sweedler in 1975, only recently they ...have been shown to have far reaching applications ranging from the category theory including differential graded categories through classical and Hopf-type module theory to non-commutative geometry and mathematical physics. This is the first extensive treatment of the theory of corings and their comodules. In the first part, the module-theoretic aspects of coalgebras over commutative rings are described. Corings are then defined as coalgebras over non-commutative rings. Topics covered include module-theoretic aspects of corings, such as the relation of comodules to special subcategories of the category of modules (sigma-type categories), connections between corings and extensions of rings, properties of new examples of corings associated to entwining structures, generalisations of bialgebras such as bialgebroids and weak bialgebras, and the appearance of corings in non-commutative geometry.
Azumaya Monads and Comonads Mesablishvili, Bachuki; Wisbauer, Robert
Axioms,
03/2015, Volume:
4, Issue:
1
Journal Article
Peer reviewed
Open access
The definition of Azumaya algebras over commutative rings \(R\) requires the tensor product of modules over \(R\) and the twist map for the tensor product of any two \(R\)-modules. Similar ...constructions are available in braided monoidal categories, and Azumaya algebras were defined in these settings. Here, we introduce Azumaya monads on any category \(\mathbb{A}\) by considering a monad \((F,m,e)\) on \(\mathbb{A}\) endowed with a distributive law \(\lambda: FF\to FF\) satisfying the Yang–Baxter equation (BD%please define -law). This allows to introduce an opposite monad \((F^\lambda,m\cdot \lambda,e)\) and a monad structure on \(FF^\lambda\). The quadruple \((F,m,e,\lambda)\) is called an Azumaya monad, provided that the canonical comparison functor induces an equivalence between the category \(\mathbb{A}\) and the category of \(FF^\lambda\)-modules. Properties and characterizations of these monads are studied, in particular for the case when \(F\) allows for a right adjoint functor. Dual to Azumaya monads, we define Azumaya comonads and investigate the interplay between these notions. In braided categories (V\(,\otimes,I,\tau)\), for any V-algebra \(A\), the braiding induces a BD-law \(\tau_{A,A}:A\otimes A\to A\otimes A\), and \(A\) is called left (right) Azumaya, provided the monad \(A\otimes-\) (resp. \(-\otimes A\)) is Azumaya. If \(\tau\) is a symmetry or if the category V admits equalizers and coequalizers, the notions of left and right Azumaya algebras coincide.
We consider -complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology (called generalized ...cohomology by M. Dubois-Violette) only vanishes on injective functors providing a well defined functor on the stable category. For left truncated -complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive -complexes is proved to be isomorphic to an Ext functor of an indecomposable -complex inside the abelian functor category. Finally we show that for the monoidal structure of -complexes a Clebsch-Gordan formula holds, in other words the fusion rules for -complexes can be determined.
As shown by S. Eilenberg and J.C. Moore (1965), for a monad \(F\) with right adjoint comonad \(G\) on any catgeory \(\mathbb{A}\), the category of unital \(F\)-modules \(\mathbb{A}_F\) is isomorphic ...to the category of counital \(G\)-comodules \(\mathbb{A}^G\). The monad \(F\) is Frobenius provided we have \(F=G\) and then \(\mathbb{A}_F\simeq \mathbb{A}^F\). Here we investigate which kind of equivalences can be obtained for non-unital monads (and non-counital comonads).
The theories of (Hopf) bialgebras and weak (Hopf) bialgebras have been introduced for vector space categories over fields and make heavily use of the tensor product. As first generalisations, these ...notions were formulated for monoidal categories, with braidings if needed. The present authors developed a theory of bimonads and Hopf monads H on arbitrary categories A, employing distributive laws, allowing for a general form of the Fundamental Theorem for Hopf algebras. For τ-bimonads H, properties of braided (Hopf) bialgebras were captured by requiring a Yang–Baxter operator τ:HH→HH. The purpose of this paper is to extend the features of weak (Hopf) bialgebras to this general setting including an appropriate form of the Fundamental Theorem. This subsumes the theory of braided Hopf algebras (based on weak Yang–Baxter operators) as considered by Alonso Álvarez and others.
Extending modules are generalizations of injective modules and, dually, lifting modules generalize projective supplemented modules. This duality exhibits a certain asymmetry. While the theory of ...extending modules is well documented in monographs and text books, the purpose of this monograph is to provide a thorough study of supplements and projectivity conditions needed to investigate classes of modules related to lifting modules.
QF functors and (co)monads Mesablishvili, Bachuki; Wisbauer, Robert
Journal of algebra,
02/2013, Volume:
376
Journal Article
Peer reviewed
Open access
One reason for the universal interest in Frobenius algebras is that their characterisation can be formulated in arbitrary categories: a functor K:A→B between categories is Frobenius if there exists a ...functor G:B→A which is at the same time a right and left adjoint of K; a monad F on A is a Frobenius monad provided the forgetful functor AF→A is a Frobenius functor, where AF denotes the category of F-modules. With these notions, an algebra A over a field k is a Frobenius algebra if and only if A⊗k− is a Frobenius monad on the category of k-vector spaces.
The purpose of this paper is to find characterisations of quasi-Frobenius algebras by just referring to constructions available in any categories. To achieve this we define QF functors between two categories by requiring conditions on pairings of functors which weaken the axioms for adjoint pairs of functors. QF monads on a category A are those monads F for which the forgetful functor UF:AF→A is a QF functor. Applied to module categories (or Grothendieck categories), our notions coincide with definitions first given K. Morita (and others). Further applications show the relations of QF functors and QF monads with Frobenius (exact) categories.
Algebras Versus Coalgebras Wisbauer, Robert
Applied categorical structures,
04/2008, Volume:
16, Issue:
1-2
Journal Article
Peer reviewed
Open access
Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible ...to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 1970s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a
Hopf monad
on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between the base category and the category of the associated bimodules. For a set
G
, the endofunctor
G
× – on the category of sets shares these properties if and only if
G
admits a group structure. Finally, we report about the possibility of subsuming algebras and coalgebras in the notion of
(
F
,
G
)-dimodules
associated to two functors
between different categories. This observation, due to Tatsuya Hagino, was an outcome from the theory of categorical data types and may also be of use in classical algebra.
Although coalgebras and coalgebraic structures are well-known for a long time it is only in recent years that they are getting new attention from people working in algebra and module theory. The ...purpose of this survey is to explain the basic notions of the coalgebraic world and to show their ubiquity in classical algebra. For this, we recall the basic categorical notions and then apply them to linear algebra and module theory. It turns out that a number of results proven there were already contained in categorical papers from decades ago.
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