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Given a finite category T , we consider the functor category A T , where A can be any quasi-abelian category. Examples of quasi-abelian categories are given by any abelian category but also by non-exact additive categories as the categories of torsion(-free) abelian groups, topological abelian groups, locally compact abelian groups, Banach spaces and Fréchet spaces. In this situation, the categories of various internal categorical structures in A , such as the categories of internal n -fold groupoids, are equivalent to functor categories A T for a suitable category T. For a replete full subcategory S of T , we define F to be the full subcategory of A T whose objects are given by the functors F : T → A with F (T) = 0 for all T ∉ S. We prove that F is a torsion-free Birkhoff subcategory of A T. This allows us to study (higher) central extensions from categorical Galois theory in A T with respect to F and generalized Hopf formulae for homology. [ABSTRACT FROM AUTHOR]

After fixing a commutative ring with unit R, we present the definition of adequate category and consider the category of R-linear functors from an adequate category to the category of R-modules. We endow this category of functors with a monoidal structure and study monoids (generalized Green functors) over it. For one of these generalized Green functors, we define two new monoids, its commutant and its center, and study some of their properties and relations between them. This work generalizes the article [3]. [ABSTRACT FROM AUTHOR]

We study when the heart of a t-structure in a triangulated category D with coproducts is AB5 or a Grothendieck category. If D satisfies Brown representability, a t-structure has an AB5 heart with an injective cogenerator and coproduct-preserving associated homological functor if, and only if, the coaisle has a pure-injective t-cogenerating object. If D is standard well generated, such a heart is automatically a Grothendieck category. For compactly generated t-structures (in any ambient triangulated category with coproducts), we prove that the heart is a locally finitely presented Grothendieck category. We use functor categories and the proofs rely on two main ingredients. Firstly, we express the heart of any t-structure in any triangulated category as a Serre quotient of the category of finitely presented additive functors for suitable choices of subcategories of the aisle or the co-aisle that we, respectively, call t-generating or t-cogenerating subcategories. Secondly, we study coproduct-preserving homological functors from D to complete AB5 abelian categories with injective cogenerators and classify them, up to a so-called computational equivalence, in terms of pure-injective objects in D . This allows us to show that any standard well generated triangulated category D possesses a universal such coproduct-preserving homological functor, to develop a purity theory and to prove that pure-injective objects always cogenerate t-structures in such triangulated categories. [ABSTRACT FROM AUTHOR]

A correspondence functor is a functor from the category of finite sets and correspondences to the category of k -modules, where k is a commutative ring. By means of a suitably defined duality, new correspondence functors are constructed, having remarkable properties. In particular, their evaluation at any finite set is always a free k -module and an explicit formula is obtained for its rank. The results use some subtle new ingredients from the theory of finite lattices. [ABSTRACT FROM AUTHOR]

Let R be a ring, Q a small k -preadditive category, and let Q , R Mod be the category of k -linear functors from Q to the left R -module category R Mod. Given a cotorsion pair (, ℬ) in R Mod , we construct four cotorsion pairs (Γ () , Γ () ⊥) , (⊥ Γ (ℬ) , Γ (ℬ)) , (Θ () , Θ () ⊥) and (⊥ ξ (ℬ) , ξ (ℬ)) in Q , R Mod 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 N -periodic chain complexes of R -modules. [ABSTRACT FROM AUTHOR]

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Let Λ be an artin algebra and M be an n -cluster tilting subcategory of mod Λ. We show that M has an additive generator if and only if the n -almost split sequences form a basis for the relations for the Grothendieck group of M if and only if every effaceable functor M → A b has finite length. As a consequence we show that if mod Λ has an n -cluster tilting subcategory of finite type then the n -almost split sequences form a basis for the relations for the Grothendieck group of Λ. [ABSTRACT FROM AUTHOR]

For a category ℂ, a small category II, and a pre-cover relation ⊏ on ℂ we prove, under certain completeness assumptions on C, that a morphism g : B → C in the functor category ℂII has an ⊏II-image, where ⊏II is the pre-cover relation on ℂII induced by ⊏, as soon as each component of g has an ⊏-image. We then apply this to show that if a pointed category ℂ is: (i) algebraically cartesian closed; (ii) exact protomodular and action accessible; or (iii) admits normalizers, then the same is true of each functor category ℂII with I finite. In addition, our results give explicit constructions of ⊏II-images in functor categories using limits and ⊏-images in the underlying category. In particular, they can be used to give explicit constructions of both centralizers and normalizers in functor categories using limits and centralizers or normalizers (respectively) in the underlying category. [ABSTRACT FROM AUTHOR]

We prove that, for any fields k and F of characteristic 0 and any finite group T, the category of modules over the shifted Green biset functor (k R F) T is semisimple. [ABSTRACT FROM AUTHOR]

A correspondence functor is a functor from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. We determine exactly which simple correspondence functors are projective. We also determine which simple modules are projective for the algebra of all relations on a finite set. Moreover, we analyze the occurrence of such simple projective functors inside the correspondence functor F associated with a finite lattice and we deduce a direct sum decomposition of F. [ABSTRACT FROM AUTHOR]

Let X be a contravariantly finite resolving subcategory of mod - Λ , the category of finitely generated right Λ -modules. We associate to X the subcategory S X (Λ) of the morphism category H (Λ) consisting of all monomorphisms (A → f B) with A, B and Cok f in X . Since S X (Λ) is closed under extensions it inherits naturally an exact structure from H (Λ) . We will define two other different exact structures other than the canonical one on S X (Λ) , and completely classify the indecomposable projective (resp. injective) objects in the corresponding exact categories. Enhancing S X (Λ) with the new exact structure provides a framework to construct a triangle functor. Let mod - X ̲ denote the category of finitely presented functors over the stable category X ̲ . We then use the triangle functor to show a triangle equivalence between the bounded derived category D b (mod - X ̲) and a Verdier quotient of the bounded derived category of the associated exact category on S X (Λ) . Similar consideration is also given for the singularity category of mod - X ̲ . [ABSTRACT FROM AUTHOR]

As part of the study of correspondence functors, the present paper investigates their tensor product and proves some of its main properties. In particular, the correspondence functor associated to a finite lattice has the structure of a commutative algebra in the tensor category of all correspondence functors and a converse also holds. [ABSTRACT FROM AUTHOR]

A sufficient condition for the existence of recollements of functor categories is provided. Using this criterion, we show that a recollement of rings induces a recollement of their path rings (resp. incidence rings, monomial rings) over a locally finite quiver. Also, we present a covering technique for recollement of derived categories of functor categories. [ABSTRACT FROM AUTHOR]

From the viewpoint of higher homological algebra, we introduce pure semisimple n -abelian categories, which are analogs of pure semisimple abelian categories. Let Λ be an Artin algebra and M be an n -cluster tilting subcategory of Mod-Λ. We show that M is pure semisimple if and only if each module in M is a direct sum of finitely generated modules. Let m be an n -cluster tilting subcategory of mod-Λ. We show that Add (m) is an n -cluster tilting subcategory of Mod-Λ if and only if m has an additive generator if and only if Mod (m) is locally finite. This generalizes Auslander's classical results on pure semisimplicity of Artin algebras. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct singular equivalences between functor categories. As a special case, we show that there exists a singular equivalence arising from a cotilting module T , namely, the singularity category of (T ⊥) / [ T ] and that of (mod Λ) / [ T ] are triangle equivalent. In particular, the canonical module ω over a commutative Noetherian ring R induces a singular equivalence between (CM R) / [ ω ] and (mod R) / [ ω ] , which generalizes Matsui-Takahashi's theorem. Our result is based on a sufficient condition for an additive category A and its subcategory X so that the canonical inclusion X ↪ A induces a singular equivalence D sg (A) ≃ D sg (X) , which is a functor category version of Xiao-Wu Chen's theorem. [ABSTRACT FROM AUTHOR]

We show that for the path algebra A of an acyclic quiver, the singularity category of the derived category D b (mod A) is triangle equivalent to the derived category of the functor category of mod _ A , that is, D sg (D b (mod A)) ≃ D b (mod ( mod _ A)). This extends a result in [14] for the path algebra A of a Dynkin quiver. An important step is to establish a functor category analog of Happel's triangle equivalence for repetitive algebras. [ABSTRACT FROM AUTHOR]

Based on Beligiannis's theory in [Beligiannis, A. (2000). Relative homological algebra and purity in triangulated categories. J. Algebra 227(1):268–361], we introduce and study E -tilting objects in a triangulated category, where E is a proper class of triangles. We show that each E -tilting object cogenerates an E -cotorsion pair. Meanwhile, we also achieve some nice characterizations with respect to the E -tilting object. As an application, we provide a necessary and sufficient condition for a triangulated category to be E -1-Gorenstein. Finally, we give a one to one correspondence between the class of E -tilting objects and the class of tilting subcategories in a suitable functor category. [ABSTRACT FROM AUTHOR]

Abstract A generalized correspondence functor is a functor from the category of finite sets and T -generalized correspondences to the category of all k -modules, where T is a finite distributive lattice and k a commutative ring. We parametrize simple generalized correspondence functors using the notions of T -module and presheaf of posets. As an application, we prove finiteness and stabilization results. In particular, when k is a field, any finitely generated correspondence functor has finite length, and when k is noetherian, any subfunctor of a finitely generated functor is finitely generated. [ABSTRACT FROM AUTHOR]

Abstract A correspondence functor is a functor from the category of finite sets and correspondences to the category of k -modules, where k is a commutative ring. A main tool for this study is the construction of a correspondence functor associated to any finite lattice T. We prove for instance that this functor is projective if and only if the lattice T is distributive. Moreover, it has quotients which play a crucial role in the analysis of simple functors. The special case of total orders yields some more specific and complete results. [ABSTRACT FROM AUTHOR]

For a (right and left) coherent ring A, we show that there exists a duality between homotopy categories and 핂b(mod-A). If A = Λ is an artin algebra of finite global dimension, this duality induces a duality between their subcategories of acyclic complexes, and As a result, it will be shown that, in this case, admits a Serre functor and hence has Auslander-Reiten triangles. [ABSTRACT FROM AUTHOR]

We investigate the representation theory of finite sets. The correspondence functors are the functors from the category of finite sets and correspondences to the category of k -modules, where k is a commutative ring. They have various specific properties which do not hold for other types of functors. In particular, if k is a field and if F is a correspondence functor, then F is finitely generated if and only if the dimension of F ( X ) grows exponentially in terms of the cardinality of the finite set X . Moreover, in such a case, F has actually finite length. Also, if k is noetherian, then any subfunctor of a finitely generated functor is finitely generated. [ABSTRACT FROM AUTHOR]

We introduce weakly FP-injective and weakly flat functors, and we show that a right R-module K is FP-injective if and only if (−, K) is a flat weakly FP-injective functor in $${(({{\rm mod}-R})^{\rm op},{\rm Ab})}$$ , whereas a left R-module N is flat if and only if $${-\otimes N}$$ is an FP-injective weakly flat functor in $${({{\rm mod}-R},{\rm Ab})}$$ . We give closure properties, we characterise classes of rings, and we study approximations by these classes of functors. [ABSTRACT FROM AUTHOR]

Let $$\mathcal {X}$$ be a resolving subcategory of an abelian category. In this paper we investigate the singularity category $$\mathsf {D_{sg}}(\underline{\mathcal {X}})=\mathsf {D^b}({\mathsf {mod}}\,\underline{\mathcal {X}})/\mathsf {K^b}({\mathsf {proj}}({\mathsf {mod}}\,\underline{\mathcal {X}}))$$ of the stable category $$\underline{\mathcal {X}}$$ of $$\mathcal {X}$$ . We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type $$(\mathsf {A}_1)$$ . We also generalize several results of Yoshino on totally reflexive modules. [ABSTRACT FROM AUTHOR]

The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalize two results of Happel by proving the existence of AR-triangles in Gorenstein-derived categories, provide situations for which relative derived categories with respect to Gorenstein projective and Gorenstein injective modules are equivalent, and finally study relations between the Gorenstein-derived category of a quiver and its image under a reflection functor. Some interesting applications are provided. [ABSTRACT FROM PUBLISHER]

We show that direct summands of certain additive functors arising as bifunctors with a fixed argument in an abelian category are again of that form whenever the fixed argument has finite length or, more generally, satisfies the descending chain condition on images of nested endomorphisms. In particular, this provides a positive answer to a conjecture of M. Auslander in the case of categories of finite modules over artin algebras. This implies that the covariant Ext functors are the only injectives in the category of defect-zero finitely presented functors on such categories. [ABSTRACT FROM AUTHOR]

A quasi-schemoid is a small category whose morphisms are colored with appropriate combinatorial data. In this paper, Mitchell's embedding theorem for a tame schemoid is established. The result allows us to give a cofibrantly generated model category structure to the category of chain complexes over a functor category with a schemoid as the domain. Moreover, a notion of Morita equivalence for schemoids is introduced and discussed. In particular, we show that every Hamming scheme of binary codes is Morita equivalent to the association scheme arising from the cyclic group of order two. In an appendix, we construct a new schemoid from an abstract simplicial complex, whose Bose–Mesner algebra is closely related to the Stanley–Reisner ring of the given complex. [ABSTRACT FROM AUTHOR]

In the first part of this article we formalize the concepts of terminal and initial object, categorical product [4] and natural transformation within a free-object category [1]. In particular, we show that this definition of natural transformation is equivalent to the standard definition [13]. Then we introduce the exponential object using its universal property and we show the isomorphism between the exponential object of categories and the functor category [12]. [ABSTRACT FROM AUTHOR]

Abstract: This paper introduces a concept of dimension of a triangulated category with respect to a fixed full subcategory. For the bounded derived category of an abelian category, upper bounds of the dimension with respect to a contravariantly finite subcategory and a resolving subcategory are given. Our methods not only recover some known results on the dimensions of derived categories in the sense of Rouquier, but also apply to various commutative and non-commutative noetherian rings. [Copyright &y& Elsevier]