Preadditive Category Research Articles

Flat model structures and Gorenstein objects in functor categories

We construct a flat model structure on the category ${_{\mathcal {Q},\,R}\mathsf {Mod}}$ of additive functors from a small preadditive category $\mathcal {Q}$ satisfying certain conditions to the module category ${_{R}\mathsf {Mod}}$ over an associative ring $R$ , whose homotopy category is the $\mathcal {Q}$ -shaped derived category introduced by Holm and Jørgensen. Moreover, we prove that for an arbitrary associative ring $R$ , an object in ${_{\mathcal {Q},\,R}\mathsf {Mod}}$ is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat, projective coresolving Gorenstein flat) if and only if so is its value on each object of $\mathcal {Q}$ , and hence improve a result by Dell'Ambrogio, Stevenson and Šťovíček.

Heavily separable functors of the second kind and applications

We introduce heavily separable functors of the second kind and study them in three different situations. The first of these is with restrictions and extensions of scalars for modules over small preadditive categories. The second is with free functors taking values in Eilenberg-Moore categories associated to a monad or a comonad. Finally, we consider entwined modules and give if and only if conditions for heavy separability of the second kind for functors forgetting either the comodule action or the module action.

Additive Grothendieck Pretopologies and Presentations of Tensor Categories

We study how tensor categories can be presented in terms of rigid monoidal categories and Grothendieck topologies and show that such presentations lead to strong universal properties. As the main tool in this study, we define a notion on preadditive categories which plays a role similar to (a generalisation of) the notion of a Grothendieck pretopology on an unenriched category. Each such additive pretopology defines an additive Grothendieck topology and suffices to define the sheaf category. This new notion also allows us to study the noetherian and subcanonical nature of additive topologies, to describe easily the join of a family of additive topologies and to identify useful universal properties of the sheaf category.

A monoidal analogue of the 2-category anti-equivalence between ABEX and DEF

We prove that the 2-category of skeletally small abelian categories with exact monoidal structures is anti-equivalent to the 2-category of fp-hom-closed definable additive categories satisfying an exactness criterion. For a fixed finitely accessible category C with products and a monoidal structure satisfying the appropriate assumptions, we provide bijections between the fp-hom-closed definable subcategories of C , the Serre tensor-ideals of C fp - mod and the closed subsets of a Ziegler-type topology. For a skeletally small preadditive category A with an additive, symmetric, rigid monoidal structure we show that elementary duality induces a bijection between the fp-hom-closed definable subcategories of Mod - A and the definable tensor-ideals of A - Mod .

Locally finitely presented and coherent hearts

Starting with a Grothendieck category \mathcal{G} and a torsion pair \mathbf{t}=(\mathcal{T},\mathcal{F}) in \mathcal{G} , we study the local finite presentability and local coherence of the heart \mathcal{H}_{\mathbf{t}} of the associated Happel–Reiten–Smalø t -structure in the derived category \mathbf{D}(\mathcal{G}) . We start by showing that, in this general setting, the torsion pair \mathbf{t} is of finite type, if and only if it is quasi-cotilting, if and only if it is cosilting. We then proceed to study those \mathbf{t} for which \mathcal{H}_{\mathbf{t}} is locally finitely presented, obtaining a complete answer under some additional assumptions on the ground category \mathcal{G} , which are general enough to include all locally coherent Grothendieck categories, all categories of modules and several categories of quasi-coherent sheaves over schemes. The third problem that we tackle is that of local coherence. In this direction, we characterize those torsion pairs \mathbf{t}=(\mathcal{T},\mathcal{F}) in a locally finitely presented \mathcal{G} for which \mathcal{H}_{\mathbf{t}} is locally coherent in two cases: when the tilted t -structure in \mathcal{H}_{\mathbf{t}} is assumed to restrict to finitely presented objects, and when \mathcal{F} is cogenerating. In the last part of the paper, we concentrate on the case when \mathcal{G} is a category of modules over a small preadditive category, giving several examples and obtaining very neat (new) characterizations in this more classical setting, underlying connections with the notion of an elementary cogenerator.

Reflective and coreflective subcategories

Given any additive category C with split idempotents, pseudokernels and pseudocokernels, we show that a subcategory B is coreflective if, and only if, it is precovering, closed under direct summands and each morphism in B has a pseudocokernel in C that belongs to B. We apply this result and its dual to, among others, preabelian and pretriangulated categories. As a consequence, we show that a subcategory of a preabelian category is coreflective if, and only it, it is precovering and closed under taking cokernels. On the other hand, if C is pretriangulated with split idempotents, then a subcategory B is coreflective and invariant under the suspension functor if, and only if, it is precovering and closed under taking direct summands and cones. These are extensions of well-known results for AB3 abelian and triangulated categories, respectively.By-side applications of these results allow us: a) To characterize the coreflective subcategories of a given AB3 abelian category which have a set of generators and are themselves abelian, abelian exact or module categories; b) to extend to module categories over arbitrary small preadditive categories a result of Gabriel and De la Peña stating that all fully exact subcategories are bireflective; c) to show that, in any Grothendieck category, the direct limit closure of its subcategory of finitely presented objects is a coreflective subcategory.

Gradual and Fuzzy Modules: Functor Categories

The categorical treatment of fuzzy modules presents some problems, due to the well known fact that the category of fuzzy modules is not abelian, and even not normal. Our aim is to give a representation of the category of fuzzy modules inside a generalized category of modules, in fact, a functor category, Mod−P, which is a Grothendieck category. To do that, first we consider the preadditive category P, defined by the interval P=(0,1], to build a torsionfree class J in Mod−P, and a hereditary torsion theory in Mod−P, to finally identify equivalence classes of fuzzy submodules of a module M with F-pair, which are pair (G,F), of decreasing gradual submodules of M, where G belongs to J, satisfying G=Fd, and ∪αF(α) is a disjoint union of F(1) and F(α)\G(α), where α is running in (0,1].

Category of Submodules of a Uniserial Module

Let R be a ring, K,M be R-modules, L a uniserial R-module, and X a submodule of L. The triple (K,L,M) is said to be X-sub-exact at L if the sequence K→X→M is exact. Let σ(K,L,M) is a set of all submodules Y of L such that (K,L,M) is Y -sub-exact. The sub-exact sequence is a generalization of an exact sequence. We collect all triple (K,L,M) such that (K,L,M) is an X-sub exact sequence, where X is a maximal element of σ(K,L,M). In a uniserial module, all submodules can be compared under inclusion. So, we can find the maximal element of σ(K,L,M). In this paper, we prove that the set σ(K,L,M) form a category, and we denoted it by C_L. Furthermore, we prove that C_Y is a full subcategory of C_L, for every submodule Y of L. Next, we show that if L is a uniserial module, then C_L is a pre-additive category. Every morphism in C_L has kernel under some conditions. Since a module factor of L is not a submodule of L, every morphism in a category C_L does not have a cokernel. So, C_L is not an abelian category. Moreover, we investigate a monic X-sub-exact and an epic X-sub-exact sequence. We prove that the triple (K,L,M) is a monic X-sub-exact if and only if the triple Z-modules (<img src=image/13424409_01.gif>, <img src=image/13424409_02.gif>, <img src=image/13424409_03.gif>) is a monic <img src=image/13424409_04.gif>-sub-exact sequence, for all R-modules N. Furthermore, the triple (K,L,M) is an epic X-sub-exact if and only if the triple Z-modules (<img src=image/13424409_05.gif>, <img src=image/13424409_06.gif>, <img src=image/13424409_07.gif>) is a monic <img src=image/13424409_08.gif>-sub-exact, for all R-module N.

Gabriel localization in functor categories

Gabriel showed that for a unital ring R, there exists a bijective correspondence between the set of Gabriel filters of R and the set of Giraud subcategories of In this paper we prove a result analogous to the one given by Gabriel: for a small preadditive category there exists a bijective correspondence between the left Gabriel filters of and Giraud subcategories of

Models for functor categories over self-injective quivers

Let [Formula: see text] be a ring, [Formula: see text] a small [Formula: see text]-preadditive category, and let [Formula: see text] be the category of [Formula: see text]-linear functors from [Formula: see text] to the left [Formula: see text]-module category [Formula: see text]. Given a cotorsion pair [Formula: see text] in [Formula: see text], we construct four cotorsion pairs [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] in [Formula: see text] and investigate when these cotorsion pairs are hereditary and complete. Moreover, under few assumptions, we show that there exist a recollement of homotopy categories induced by these cotorsion pairs. Finally, some applications are given in the category of [Formula: see text]-periodic chain complexes of [Formula: see text]-modules.

FI-modules over preadditive categories and torsion theories

We work with FI -modules over a small preadditive category R , viewed as a ring with several objects. Our aim is to study torsion theories for FI -modules. We are especially interested in torsion theories on finitely generated FI -modules and the category of what we call “shift finitely generated” FI -modules. We also apply these methods to study inductive descriptions of FI -modules over R .

Torsion Pairs in Categories of Modules over a Preadditive Category

It is a result of Gabriel that hereditary torsion pairs in categories of modules are in bijection with certain filters of ideals of the base ring, called Gabriel filters or Gabriel topologies. A result of Jans shows that this bijection restricts to a correspondence between (Gabriel filters that are uniquely determined by) idempotent ideals and TTF triples. Over the years, these classical results have been extended in several different directions. In this paper, we present a detailed and self-contained exposition of an extension of the above bijective correspondences to additive functor categories over small preadditive categories. In this context, we also show how to deduce parametrizations of hereditary torsion theories of finite type, Abelian recollements by functor categories, and centrally splitting TTFs.

Baer-Kaplansky classes in categories: transfer via functors

We study the transfer of Baer-Kaplansky classes via additive functors between preadditive categories. We show that the Baer-Kaplansky property is well behaved with respect to fully faithful functors, but not with respect to their usual generalizations such as faithful, full, separable, naturally full, Maschke or dual Maschke functors. We also present the transfer of the Baer-Kaplansky property between subclasses of static and adstatic objects induced by adjoint pairs of functors. We mainly give applications to (graded) module categories and comodule categories.

On Differential Torsion Theories and Rings with Several Objects

AbstractLet${\mathcal{R}}$be a small preadditive category, viewed as a “ring with several objects.” A right${\mathcal{R}}$-module is an additive functor from${\mathcal{R}}^{\text{op}}$to the category$Ab$of abelian groups. We show that every hereditary torsion theory on the category$({\mathcal{R}}^{\text{op}},Ab)$of right${\mathcal{R}}$-modules must be differential.

Definable categories and $\mathbb T$-motives

Making use of Freyd's free abelian category on a preadditive category we show that if T\colon D\rightarrow \mathcal A is a representation of a quiver D in an abelian category \mathcal A then there is an abelian category \mathcal A (T) , a faithful exact functor F_T\colon \mathcal A (T) \to \mathcal A and an induced representation \widetilde T\colon D \to \mathcal A (T) such that F_T\widetilde T= T universally. We then can show that \mathbb T -motives as well as Nori's motives are given by a certain category of functors on definable categories.

Derived equivalences of functor categories

Let Mod-S denote the category of S-modules, where S is a small pre-additive category. Using the notion of relative derived categories of functor categories, we generalize Rickard's theorem on derived equivalences of module categories over rings to Mod-S. Several interesting applications will be provided. In particular, it will be shown that derived equivalence of two coherent rings not only implies the equivalence of their homotopy categories of projective modules, but also implies that they are Gorenstein derived equivalent. As another application, it is shown that a good tilting module produces an equivalence between the unbounded derived category of the module category of the ring and the relative derived category of the module category of the endomorphism ring of it.

Maximal Ideals in Module Categories and Applications

We study the existence of maximal ideals in preadditive categories defining an order $$\preceq $$ between objects, in such a way that if there do not exist maximal objects with respect to $$\preceq $$ , then there is no maximal ideal in the category. In our study, it is sometimes sufficient to restrict our attention to suitable subcategories. We give an example of a category $$\mathbf {C}_F$$ of modules over a right noetherian ring R in which there is a unique maximal ideal. The category $$\mathbf {C}_F$$ is related to an indecomposable injective module F, and the objects of $$\mathbf {C}_F$$ are the R-modules of finite F-rank.

Cartesian modules over representations of small categories

We introduce the new concept of cartesian module over a pseudofunctor R from a small category to the category of small preadditive categories. Already the case when R is a (strict) functor taking values in the category of commutative rings is sufficient to cover the classical construction of quasi-coherent sheaves of modules over a scheme. On the other hand, our general setting allows for a good theory of contravariant additive locally flat functors, providing a geometrically meaningful extension of a classical Representation Theorem of Makkai and Paré. As an application, we relate and extend some previous constructions of the pure derived category of a scheme.

Category Theoretic Interpretation of Rings

We enhance the category of rings and the category of idempotented rings to 2-categories. After doing this, we prove an equivalence of 1-categories and 2-categories between the category of rings and the category of small preadditive categories with one object and between the category of idempotented rings and the category of small preadditive categories with finitely many objects. Under these equivalences, we demonstrate some analogues between notions in category theory and ring theory. KEYWORDS: Ring, Idempotent, Preadditive Category

Linear filters and hereditary torsion theories in functor categories

We introduce the notion of Gabriel filter for a preadditive category C and we show that there is a bijective correspondence between Gabriel filters of C and hereditary torsion theories in the category of additive functors (C,Ab), obtaining a generelization of the theorem given by Gabriel [Ga] and Maranda [Ma] which establishes a bijective correspondence between Gabriel filters for a ring and hereditary torsion theories in the corresponding category of modules.