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Bachuki Mesablishvili

Publications and source records attributed to Bachuki Mesablishvili.

14 recordsLinked to original sources
arXiv2025

Morita theory for quantales

Morita theory for quantales is developed. The main result of the paper is a characterization of those quantaloids (categories enriched in the symmetric monoidal closed category of sup-lattices) that are equivalent to modular categories over quantales. Based on this characterization, necessary and sufficient conditions are derived for two quantales to be Morita-equivalent, i. e. have equivalent module categories. As an application, it is shown that the category of internal sup-lattices in a Grothendieck topos is equivalent to the module category over a suitable chosen ordinary quantale.

math.CT↗
arXiv2022

Descent cohomology and factorizations of groups

The aim of the paper is to give a full classification of factorizations of groups in terms of descent cohomology (pointed) sets introduced in [5]. We show that descent cohomology includes Serre's non-abelian group cohomology as a special case. This enables us to generalize Serre's theory further to include monoids.

math.GR↗
arXiv2016

The Fundamental Theorem for weak braided bimonads

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 $\mathbb{A}$, 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 $τ:HH\to HH$. 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.

math.CT↗
arXiv2014

Generalised bialgebras and entwined monads and comonads

Jean-Louis Loday has defined generalised bialgebras and proved structure theorems in this setting which can be seen as general forms of the Poincaré-Birkhoff-Witt and the Cartier-Milnor-Moore theorems. It was observed by the present authors that parts of the theory of generalised bialgebras are special cases of results on entwined monads and comonads and the corresponding mixed bimodules. In this article the Rigidity Theorem of Loday is extended to this more general categorical framework.

math.CT↗
arXiv2013

Galois functors and generalised Hopf modules

As shown in a previous paper by the same authors, the theory of Galois functors provides a categorical framework for the characterisation of bimonads on any category as Hopf monads and also for the characterisation of opmonoidal monads on monoidal categories as right Hopf monads in the sense of Bruguieres and Virelizier. Hereby the central part is to describe conditions under which a comparison functor between the base category and the category of Hopf modules becomes an equivalence (Fundamental Theorem). For monoidal categories, Aguiar and Chase extended the setting by replacing the base category by a comodule category for some comonoid and considering a comparison functor to generalised Hopf modules. For duoidal categories, Bohm, Chen and Zhang investigated a comparison functor to the Hopf modules over a bimonoid induced by the two monoidal structures given in such categories. In both approaches fundamental theorems are proved and the purpose of this paper is to show that these can be derived from the theory of Galois functors.

math.CT↗
arXiv2012

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.

math.CT↗
arXiv2011

Notes on bimonads and Hopf monads

For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguières and A. Virelizier study opmonoidal monads on monoidal categories (which they called {\em bimonads}). In a recent joint paper with S. Lack the same authors define the notion of a {\em pre-Hopf monad} by requiring only a special form of the fusion operator to be invertible. In previous papers it was observed by the present authors that bimonads yield a special case %Hopf monads may be considered as a special case of an entwining of a pair of functors (on arbitrary categories). The purpose of this note is to show that in this setting the pre-Hopf monads are a special case of Galois entwinings. As a byproduct some new properties are detected which make a (general) bimonad on a Cauchy complete category to a Hopf monad. In the final section applications to cartesian monoidal categories are considered.

math.CT↗
arXiv2010

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 $a,b\in \A$, $$β_{a,b}:\A(a, G(b)) \to \A(T(a),b),$$ and we call it {\em rational} if these all are injective. In case $\bT=(T,m_T,e_T)$ is a monad and $\bG=(G,δ_G,\ve_G)$ 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.

math.CT↗
arXiv2009

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. Bruguières 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.

math.CT↗
arXiv2008

Bimonads and Hopf monads on categories

The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to {\em monoidal} categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on any category $\A$ and by this we retain some of the combinatorial complexity which makes the theory so interesting. As a basic tool we use distributive laws between monads and comonads (entwinings) on $\A$: we define a {\em bimonad} on $\A$ as an endofunctor $B$ which is a monad and a comonad with an entwining $λ:BB\to BB$ satisfying certain conditions. This $λ$ is also employed to define the category $\A^B_B$ of (mixed) $B$-bimodules. In the classical situation, an entwining $λ$ is derived from the twist map for vector spaces. Here this need not be the case but there may exist special distributive laws $τ:BB\to BB$ satisfying the Yang-Baxter equation ({\em local prebraidings}) which induce an entwining $λ$ and lead to an extension of the theory of {\em braided Hopf algebras}. An antipode is defined as a natural transformation $S:B\to B$ with special properties and for categories $\A$ with limits or colimits and bimonads $B$ preserving them, the existence of an antipode is equivalent to $B$ inducing an equivalence between $\A$ and the category $\A^B_B$ of $B$-bimodules. This is a general form of the {\em Fundamental Theorem} of Hopf algebras.

math.QA↗
arXiv2007

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.

math.QA↗
arXiv2006

On a generalization of Grothendieck's theorem

A wide generalization of the classical theorem of A. Grothendieck asserting that for any faithfully flat extension of commutative rings, the corresponding relative Picard group and the Amitsur 1-cohomology group with values in the units-functor are isomorphic, is obtained. This implies some known results that are concerned with extending to non-commutative rings of Grothendieck's theorem.

math.RA↗
arXiv2005

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.

math.QA↗