A Note on the Abelianization Functor

It is well known that the abelianization of a group G can be computed as the cokernel of the diagonal morphism (1_G, 1_G): G → G × G in the category of groups. We generalize this to arbitrary regular subtractive categories, among which are the category of groups, the category of topological groups, and the categories of other group-like structures. We also establish that an abelian category is the same as a regular subtractive category in which every monomorphism is a kernel of some morphism.     

The Change-Base Issue for Ω-Categories

Let G ： Ω→Ω＇ be a closed unital map between commutative, unital quantales. G induces a functor G^- from the category of Ω-categories to that of Ω＇-categories. This paper is concerned with some basic properties of G^-. The main results are： （1） when Ω, Ω＇ are integral, G ： Ω→Ω＇ and F ： Ω＇→Ω are closed unital maps, F is a left adjoint of G^- if and only if F is a left adjoint of G; （2） G^- is an equivalence of categories if and only if G is an isomorphism in the category of commutative unital quantales and closed unital maps; and （3） a sufficient condition is obtained for G^- to preserve completeness in the sense that GA is a complete Ω＇-category whenever A is a complete Ω-category.     

Closedness Properties of Internal Relations III: Pointed Protomodular Categories

A pointed variety of universal algebras is protomodular in the sense of D. Bourn, if and only if it is classically ideal determined in the sense of A. Ursini (this result is due to D. Bourn and G. Janelidze). We prove a characterization theorem for pointed protomodular categories, which is a (pointed) categorical version of Ursini’s characterization theorem for classically ideal determined varieties, involving classically 0-regular algebras. A suitable simplification of the property of a pair of relations, which is used to define a classically 0-regular algebra, yields a new closedness property of a single binary relation – we show that a finitely complete pointed category is protomodular if and only if every binary internal relation R→A ² in it has this closedness property. Partially supported by South African National Research Foundation, and Georgian National Science Foundation (GNSF/ST06/3-004).     

Endomorphism category of an abelian category

Let 𝒞 be an additive category. Denote by End(𝒞) the endomorphism category of 𝒞, i.e., the objects in End(𝒞) are pairs (C,c) with C∈𝒞,c∈End_𝒞(C), and a morphism f:(C,c)→(D,d) is a morphism f∈Hom_𝒞(C,D) satisfying fc?=?df. This paper is devoted to an approach of the general theory of the endomorphism category of an arbitrary additive category. It is proved that the endomorphism category of an abelian category is again abelian with an induced structure without nontrivial projective or injective objects. Furthermore, the endomorphism category of any nontrivial abelian category is nonsemisimple and of infinite representation type. As an application, we show that two unital rings are Morita equivalent if and only if the endomorphism categories of their module categories are equivalent.     

Relative Semi-abelian Categories

We define a relative semi-abelian category as a pair (C,E), where C is a pointed category with finite limits, and E is a class of regular epimorphisms in C satisfying certain conditions, stronger than those defining a relative homological category. Some results on the equivalence of the so-called old-style and new-style axioms for the semi-abelian categories are extended to the relative case. Partially supported by the University of Cape Town Research Associateship and Georgia National Science Foundation (GNSF/ST06/3-004).     

Cover Relations on Categories

A cover relation on a category ℂ is a binary relation ⊏ on the class of morphisms of ℂ, which is defined only for those pairs of morphisms which have the same codomain, and which has the following two properties: (i) if f ⊏ g and h is composable with f, then hf ⊏ hg, (ii) if f ⊏ g and f is composable with e then fe ⊏ g. We study cover relations arising from a special type of factorization systems, and cover relations arising from a special type of monoidal structures. Supported by Claude Leon Foundation, INTAS (06-1000017-8609) and Georgian National Science Foundation (GNSF/ST06/3-004).     

Universal Jensen＇s Equations in Banach Modules over a C^＊-Algebra and Its Unitary Group

In this paper, we prove the generalized Hyers-Ulam-Rassias stability of universal Jensen‘s equations in Banach modules over a unital C^*-algebra. It is applied to show the stability of universal Jensen‘s equations in a Hilbert module over a unital C^*-algebra. Moreover, we prove the stability of linear operators in a Hilbert module over a unital C^*-algebra.     

Relative Regular Objects in Categories

We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely generated Grothendieck categories. Applications are given for categories of comodules over a coalgebra and for categories of graded modules, and a link to the theory of generalized inverses of matrices is presented. Some of the techniques we use are new, since dealing with arbitrary categories allows us to pass to the dual category.      

Axiomatizing subcategories of Abelian categories

We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find axioms which characterize generating cogenerating functorially finite subcategories, precluster tilting subcategories, and cluster tilting subcategories of abelian categories. As a consequence we prove that any d-abelian category is equivalent to a d-cluster tilting subcategory of an abelian category, without any assumption on the categories being projectively generated.     

On the structure of graded <Emphasis Type="Italic">λ</Emphasis>-Hopf algebras

Let G be an abelian group, B the G-graded λ-Hopf algebra with A being a bicharacter on G. By introducing some new twisted algebras （coalgebras）, we investigate the basic properties of the graded antipode and the structure for B. We also prove that a G-graded λ-Hopf algebra can be embedded in a usual Hopf algebra. As an application, it is given that if G is a finite abelian group then the graded antipode of a finite dimensional G-graded A-Hopf algebra is invertible.     

Uniqueness of uniform decompositions in exact categories

Two uniqueness theorems on uniform decompositions due to Krause, Diracca and Facchini are extended from abelian categories to weakly idempotent complete exact categories. We give applications to (quasi-)abelian categories, finitely accessible additive categories and exactly definable additive categories.     

The Left and Right Triangulated Structures of Stable Categories

Beligiannis and Marmaridis in 1994, constructed the one-sided triangulated structures on the stable categories of additive categories induced from some homologically finite subcategories. We extend their results to slightly more general settings. As an application of our results, we give some new examples of one-sided triangulated categories arising from abelian model categories. An interesting outcome is that we can describe the pretriangulated structures of the homotopy categories of abelian model categories via those of stable categories.     

Negative K-theory of derived categories

We define negative K-groups for exact categories and for ``derived categories' in the framework of Frobenius pairs, generalizing definitions of Bass, Karoubi, Carter, Pedersen-Weibel and Thomason. We prove localization and vanishing theorems for these groups. Dévissage (for noetherian abelian categories), additivity, and resolution hold. We show that the first negative K-group of an abelian category vanishes, and that, in general, negative K-groups of a noetherian abelian category vanish. Our methods yield an explicit non-connective delooping of the K-theory of exact categories and chain complexes, generalizing constructions of Wagoner and Pedersen-Weibel. Extending a theorem of Auslander and Sherman, we discuss the K-theory homotopy fiber of ε^⊕→ ε and its implications for negative K-groups. In the appendix, we replace Waldhausen's cylinder functor by a slightly weaker form of non-functorial factorization which is still sufficient to prove his approximation and fibration theorems.     

Covers of the Abelian Variety of Generalized MV-Algebras

Using the categorical equivalence of the class of generalized MV-algebras with the class of unital ?-groups, we describe all varieties of symmetric top abelian unital ?-groups that cover the variety  _u? of abelian unital ?-groups. Equivalently, we describe all cover varieties of the variety of MV-algebras, ?, within the variety of generalized MV-algebras admitting only one negation and each of whose maximal ideals is normal. In particular, there are continuum many representable varieties of generalized MV-algebras that cover ?.     

Subregular Spreads of Hermitian Unitals

In this paper, we consider the problem of constructing partitions of the points of a Hermitian unital into pairwise disjoint blocks, commonly known as spreads. We generalize a construction of Baker et al. (In Finite Geometry and Combinatorics, Vol. 191 of London Math. Soc. Lecture Not Ser., pages 17–30. Cambridge University Press, Cambridge, 1993.) to provide a new infinite family of spreads. Morover, we develop a structural connection between these new spreads of the Hermitian unital in PG(2, q²) and the subregular spreads of PG(3, q), allowing us to christen a new “subregular” family of spreads in the Hermitian unital in PG(2, q²).     

Free Monoid in Monoidal Abelian Categories

We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. Then we apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This gives a conceptual explanation of the form of the free operad, free dioperad and free properad.      

Skew Monoidal Monoids

Skew monoidal categories are monoidal categories with non-invertible “coherence” morphisms. As shown in a previous article, bialgebroids over a ring R can be characterized as the closed skew monoidal structures on the category Mod-R in which the unit object is R_R. This offers a new approach to bialgebroids and Hopf algebroids. Little is known about skew monoidal structures on general categories. In the present article, we study the one-object case: skew monoidal monoids (SMMs). We show that they possess a dual pair of bialgebroids describing the symmetries of the (co)module categories of the SMM. These bialgebroids are submonoids of their own base and are rank 1 free over the base on the source side. We give various equivalent definitions of SMM, study the structure of their (co)module categories, and discuss the possible closed and Hopf structures on a SMM.     

Gorenstein global dimension of recollements of abelian categories

Abstract

In this article, we investigate the Gorenstein global dimension with respect to the recollements of abelian categories. With the invariants spli and silp of the categories, we give some upper bounds of Gorenstein global dimensions of the categories involved in a recollement of abelian categories. We apply our results to some rings and artin algebras, especially to the triangular matrix artin algebras.     

Ideals in Morita Rings and Morita Semigroups

We characterize the lattice of all ideals of a Morita ring （semigroup） when the corresponding pair of rings （semigroups） in the Morita context are Morita equivalent s-unital （like-unitv） rings （semigroups）.     

Extensions of covariantly finite subcategories

Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We give an example to show that Gentle–Todorov’s theorem may fail in an arbitrary abelian category; however we prove a triangulated version of Gentle–Todorov’s theorem which holds for arbitrary triangulated categories; we apply Gentle–Todorov’s theorem to obtain short proofs of a classical result by Ringel and a recent result by Krause and Solberg. This project is partially supported by China Postdoctoral Science Foundation (No.s 20070420125 and 200801230). The author also gratefully acknowledges the support of K. C. Wong Education Foundation, Hong Kong.