We introduce and study (dual) strongly relative Rickart objects in abelian categories. We prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories. Our theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories.
We introduce (dual) relative CS-Rickart objects in abelian categories, as common generalizations of (dual) relative Rickart objects and extending (lifting) objects. We study direct summands and (co)products of (dual) relative CS-Rickart objects as well as classes all of whose objects are (dual) self-CS-Rickart. Applications are given to Grothendieck categories and, in particular, to module and comodule categories.
We study the transfer of (dual) relative CS-Rickart properties via functors between abelian categories. We consider fully faithful functors as well as adjoint pairs of functors. We give several applications to Grothendieck categories and, in particular, to (graded) module and comodule categories.
We introduce (dual) strongly relative CS-Rickart objects in abelian categories, as common generalizations of (dual) strongly relative Rickart objects and strongly extending (lifting) objects. We give general properties, and we study direct summands, (co)products of (dual) strongly relative CS-Rickart objects and classes all of whose objects are (dual) strongly self-CS-Rickart.
We show how the theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories. For each of them, we prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories.
We study the transfer of (dual) relative Rickart properties via functors between abelian categories, and we deduce the transfer of (dual) relative Baer property. We also give applications to Grothendieck categories, comodule categories and (graded) module categories, with emphasis on endomorphism rings.
We introduce and investigate (dual) relative split objects with respect to a fully invariant short exact sequence in abelian categories. We compare them with (dual) relative Rickart objects, and we study their behaviour with respect to direct sums and classes all of whose objects are (dual) relative split. We also introduce and study (dual) strongly relative split objects. Applications are given to Grothendieck categories, module and comodule categories.
We investigate relative CS-Baer objects in abelian categories in relationship with other relevant classes of objects such as relative Baer objects, extending objects, objects having certain summand intersection properties and relative CS-Rickart objects. Dual results are automatically obtained by applying the duality principle in abelian categories. We also study direct sums of relative CS-Baer objects, and we determine the complete structure of dual self-CS-Baer modules over Dedekind domains. Further applications are given to module categories.
We prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extending the well-established theory for module categories and Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples.
We extend to semi-abelian categories the notion of characteristic subobject, which is widely used in group theory and in the theory of Lie algebras. Moreover, we show that many of the classical properties of characteristic subgroups of a group hold in the general semi-abelian context, or in stronger ones.
Let $\mathcal{M}$ be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in $\mathcal{M}$. This is based on recent work of Holm and Jørgensen on cotorsion pairs in categories of quiver representations. An application on Ding projective and Ding injective representations of quivers over Ding-Chen rings is given.
We relativize the notion of a compact object in an abelian category with respect to a fixed subclass of objects. We show that the standard closure properties persist to hold in this case. Furthermore, we describe categorical and set-theoretical conditions under which all products of compact objects remain compact.
We show that an abelian category can be exactly, fully faithfully embedded into a module category as the right perpendicular subcategory to a set of modules or module morphisms if and only if it is a locally presentable abelian category with a projective generator, or in other words, the category of models of an additive algebraic theory of possibly infinite bounded arity. This includes the categories of contramodules over topological rings and other examples. Various versions of the definition of the right perpendicular subcategory are considered, all leading to the same class of abelian cate…
Basing ourselves on the concept of double central extension from categorical Galois theory, we study a notion of commutator which is defined relative to a Birkhoff subcategory B of a semi-abelian category A. This commutator characterises Janelidze and Kelly's B-central extensions; when the subcategory B is determined by the abelian objects in A, it coincides with Huq's commutator; and when the category A is a variety of omega-groups, it coincides with the relative commutator introduced by the first author.
Ladders of recollements of abelian categories are introduced, and used to address three general problems. Ladders of a certain height allow to construct recollements of triangulated categories, involving derived categories and singularity categories, from abelian ones. Ladders also allow to tilt abelian recollements, and ladders guarantee that properties like Gorenstein projective or injective are preserved by some functors in abelian recollements. Breaking symmetry is crucial in developing this theory.
Derived decompositions of abelian categories are introduced in internal terms of abelian subcategories to construct semi-orthogonal decompositions (or Bousfield localizations, or hereditary torsion pairs) in various derived categories of abelian categories. We give a sufficient condition for arbitrary abelian categories to have such derived decompositions and show that it is also necessary for abelian categories with enough projectives and injectives. For bounded derived categories, we describe which semi-orthogonal decompositions are determined by derived decompositions. The necessary and suf…
Recently Dupont proved that the categories of discrete and codiscrete (or connected) objects in an abelian 2-category are equivalent abelian categories. He posses also a question whether any abelian category comes in this way. We will give a rather trivial solution of this problem in the case when a given abelian category has enough projective or injective objects.
We show that, with some technical conditions, an abelian category can be embedded into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.
We consider three (2-)categories and their (anti-)equivalence. They are the category of small abelian categories and exact functors, the category of definable additive categories and interpretation functors, the category of locally coherent abelian categories and coherent morphisms. These categories link algebra, model theory and "geometry".
In this paper, we study the relationship of Gorenstein projective objects among three Abelian categories in a recollement. As an application, we introduce the relation of $n$-Gorenstein tilting modules (and Gorenstein syzygy modules) in three Abelian categories. For a recollement of Abelian categories, we show that a resolving subcategory induce two resolving subcategories. On the other hand, we also prove that two resolving subcategories can induce a resolving subcategory. Moreover, we give the size relationship between the relative global dimensions of three Abelian categories.
In this paper, we show all k-linear abelian 1-Calabi-Yau categories over an algebraically closed field k are derived equivalent to either the category of coherent sheaves on an elliptic curve, or to the finite dimensional representations of k[[t]]. Since all abelian categories derived equivalent with these two are known, we obtain a classification of all k-linear abelian 1-Calabi-Yau categories up to equivalence.
For an abelian category $\mathcal{A}$, we establish the relation between its derived and extension dimensions. Then for an artin algebra $Λ$, we give the upper bounds of the extension dimension of $Λ$ in terms of the radical layer length of $Λ$ and certain relative projective (or injective) dimension of some simple $Λ$-modules, from which some new upper bounds of the derived dimension of $Λ$ are induced.
In this article, we study the relationship among maximal green sequences, complete forward hom-orthogonal sequences and stability functions in abelian length categories. Mainly, we firstly give a one-to-one correspondence between maximal green sequences and complete forward hom-orthogonal sequences via mutual constructions, and then prove that a maximal green sequence can be induced by a central charge if and only if it satisfies crossing inequalities. As applications, we show that crossing inequalities can be computed by $c$-vectors for finite dimensional algebras; finally, we give the Rotati…
We give a classification of nullity classes (or torsion classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda's classification of Serre subcategories.
In the efforts to define a 2-categorical analog of an abelian category, two (or three) notions of "abelian 2-categories" are defined. One is the relatively exact 2-category, and the other(s) is the (2-)abelian Gpd-category. We compare these notions. Since they proceed independently in their own way, in different settings and terminologies, it will be worth while to collect and unify them. In this paper, by comparing their definitions and arguments, we show the relationship among these classes of 2-categories.
We describe a general framework for prime, completely prime, semiprime, and primitive ideals of an abelian 2-category. This provides a noncommutative version of Balmer's prime spectrum of a tensor triangulated category. These notions are based on containment conditions in terms of thick subcategories of an abelian category and thick ideals of an abelian 2-category. We prove categorical analogs of the main properties of noncommutative prime spectra. Similar notions, starting with Serre subcategories of an abelian category and Serre ideals of an abelian 2-category, are developed. They are linked…
Basing ourselves on Janelidze and Kelly's general notion of central extension, we study universal central extensions in the context of semi-abelian categories. We consider a new fundamental condition on composition of central extensions and give examples of categories which do, or do not, satisfy this condition.
We study abelian group objects in $ω$-categories and discuss the well-known Dold-Kan correspondence from the perspective of $ω$-categories as a model for strict $\infty$-categories. The first part of the paper is intended to compile results from the existing literature and to fill some gaps therein. We go on to consider a parameterized Dold-Kan correspondence, i.e. a Dold-Kan correspondence for presheaves of $ω$-categories. The main result is to describe the descent or sheaf condition in terms of a glueing condition that is familiar for 1 and 2-stacks.
We define torsion pairs for quasi-abelian categories and give several characterisations. We show that many of the torsion theoretic concepts translate from abelian categories to quasi-abelian categories. As an application, we generalise the recently defined algebraic Harder-Narasimhan filtrations to quasi-abelian categories.
We introduce the extension groups between atoms in an abelian category. For a locally noetherian Grothendieck category, the localizing subcategories closed under injective envelopes are characterized in terms of those extension groups. We also introduce the virtual duals of the extension groups between atoms to measure the global dimension of the category. A new topological property of atom spectra is revealed and it is used to relate the projective dimensions of atoms with the Krull-Gabriel dimensions. As a byproduct of the topological observation, we show that there exists a spectral space t…
Let $\mathcal{M}$ be a small $n$-abelian category. We show that the category of absolutely pure group valued functors over $\mathcal{M}$, denote by $\mathcal{L}_2(\mathcal{M},\mathcal{G})$, is an abelian category and $\mathcal{M}$ is equivalent to a full subcategory of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ in such a way that $n$-kernels and $n$-cokernels are precisely exact sequences of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ with terms in $\mathcal{M}$. This gives a higher-dimensional version of the Freyd-Mitchell embedding theorem for $n$-abelian categories.
We put cluster tilting in ageneral framework by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal one-orthogonal subcategory) carries an abelian structure. These abelian quotients turn out to be module categories of Goreisten algebras of dimension at most one.
In this paper we define Gabriel monads as the idempotent monads associated to exact reflective localizations in Abelian categories and characterize them by a simple set of properties. The coimage of a Gabriel monad is a Serre quotient category. The Gabriel monad induces an equivalence between its coimage and its image, the localizing subcategory of local objects.
The goal of this thesis is to define a 2-dimensional version of abelian categories, where symmetric 2-groups play the role that abelian groups played in 1-dimensional algebra. Abelian and 2-abelian groupoid enriched categories are defined and it is proved that homology can be developed in them, including the existence of a long exact sequence of homology corresponding to an extension of chain complexes. This generalises known results for symmetric 2-groups. The examples include, in addition to symmetric 2-groups, the 2-modules on a 2-ring, which form a 2-abelian groupoid enriched category. Mor…
We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find intrinsic 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.
Semi-abelian and finitely cocomplete homological categories are characterized in terms of four resp. three simple axioms, in terms of the basic categorical notions introduced in the first few chapters of MacLane's classical book. As an immediate application we show that categories of diagrams in semi-abelian and similar categories are of the same type; in particular, the category of simplicial or Γ-objects in a semi-abelian category is semi-abelian.
A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. In this article we first determine an abstract characterization of the categories which are semi-localizations of an exact Mal'tsev category, by specializing a result due to S. Mantovani. We then turn our attention to semi-abelian categories, where a special type of semi-localizations are known to coincide with torsion-free subcategories. A new characterisation of protomodular categories in terms of binary relations is obtained, inspired by the one discovered in the point…
We study the notion of fundamental group in the framework of descent-exact homological categories. This setting is sufficiently wide to include several categories of "algebraic" nature such as the almost abelian categories, the semi-abelian categories, and the categories of topological semi-abelian algebras. For many adjunctions in this context, the fundamental groups are described by generalised Brown-Ellis-Hopf formulae for the integral homology of groups.
We introduce $n$-abelian and $n$-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that $n$-cluster-tilting subcategories of abelian (resp. exact) categories are $n$-abelian (resp. $n$-exact). These results allow to construct several examples of $n$-abelian and $n$-exact categories. Conversely, we prove that $n$-abelian categories satisfying certain mild assumptions can be realized as $n$-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable categ…
The purpose of this article is to study the existence of Deligne's tensor product of abelian categories by comparing it with the well-known ten- sor product of finitely cocomplete categories. The main result states that the former exists precisely when the latter is an abelian category, and moreover in this case both tensor products coincide. An example of two abelian categories whose Deligne tensor product does not exist is given.
There are well known identities that involve the Ext bifunctor, coproducts, and products in Ab4 and Ab4* abelian categories with enough projectives and enough injectives. Namely, for every such category $\mathcal{A}$, the isomorphisms $\operatorname{Ext}^n (\bigoplus_{i\in I}A_{i},X) \cong \prod_{i\in I} \operatorname{Ext}^n(A_{i},X)$ and $\operatorname{Ext}^n (X,\prod_{i\in I}A_{i}) \cong \prod_{i\in I}\operatorname{Ext}^n (X,A_{i})$ always exist. The goal of this paper is to show similar isomorphisms for the Yoneda Ext in Ab4 and Ab4* abelian categories with not necessarily enough projective…
We define quasi--locally presentable categories as big unions of coreflective subcategories which are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi--locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi--locally presentable and we obtain a new proof of Brown representability theorem. Examples of functors which are not representable are also given.
We describe (braided-)commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over these algebras and classify commutative algebras with finite number of simple local modules.
This paper is a continuation of our study of non-abelian Baues-Wirsching cohomologies. In our previous paper, we defined second non-abelian cohomology H2(C;D) of a small category C with coefficients in a so-called centralised natural system D. We proved that H2(C;D) classifies linear extensions of C by D, generalising the corresponding result for abelian natural systems. For an abelian natural system D, the third cohomology classifies certain abelian track categories. A track category is a 2-category where all 2-morphisms are isomorphisms. A track category is called abelian if for every 1-morp…
