The homotopy category of pseudofunctors and translation cohomology

We develop the obstruction theory of the 2-category of abelian track categories, pseudofunctors and pseudonatural transformations by using the cohomology of categories. The obstructions are defined in Baues-Wirsching cohomology groups. We introduce translation cohomology to classify endomorphisms in the 2-category of abelian track categories. In a sequel to this paper we will show, under certain conditions which are satisfied by all homotopy categories, that a translation cohomology class determines the exact triangles of a triangulated category.     

The Popescu-Gabriel theorem for triangulated categories

The Popescu-Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differential graded categories. An analogous result for topological triangulated categories has recently been obtained by A. Heider.     

On the functoriality of cohomology of categories

In this paper we show that the Baues-Wirsching complex used to define cohomology of categories is a 2-functor from a certain 2-category of natural systems of abelian groups to the 2-category of chain complexes, chain homomorphisms and relative homotopy classes of chain homotopies. As a consequence we derive (co)localization theorems for this cohomology.     

When the heart of a faithful torsion pair is a module category

An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category (Mitchell (1964) [17]). A tilting object in an abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts (Colpi et al. (2007) [8]). It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By Colpi et al. (2007) [8], the problem simplifies in understanding when, given an associative ring R and a faithful torsion pair (X,Y) in the category of right R-modules, the heartH(X,Y)of the t-structure associated with (X,Y) is equivalent to a category of modules. In this paper, we give a complete answer to this question, proving necessary and sufficient conditions on (X,Y) for H(X,Y) to be equivalent to a module category. We analyze in detail the case when R is right artinian.     

Definable categories

We introduce the notion of a definable category–a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are precisely the finite-injectivity classes. We prove a 2-duality between the 2-category of small exact categories and the 2-category of definable categories, and provide a new proof of its additive version. We further introduce a third vertex of the 2-category of regular toposes and show that the diagram of 2-(anti-)equivalences between three 2-categories commutes; the corresponding additive triangle is well-known.     

Exact categories

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3$3\times 3$-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel–Quillen embedding theorem in an appendix.     

Fibrations and partial products in a 2-category

We introduce a new intrinsic definition of fibrations in a 2-category, and show how it may be used (in conjunction with a suitable limit-colimit commutation condition) to define a 2-categorical version of the notion of partial product. We use these notions to show that partial products exist for all fibrations in the 2-category of (small) categories, and to identify the fibrations in the 2-category of toposes and geometric morphisms.     

Cotorsion pairs, model category structures, and representation theory

We make a general study of Quillen model structures on abelian categories. We show that they are closely related to cotorsion pairs, which were introduced by Salce [Sal79] and have been much studied recently by Enochs and coauthors [EJ00]. This gives a method of constructing model structures on abelian categories, which we illustrate by building two model structures on the category of modules over a (possibly noncommutative) Gorenstein ring. The homotopy category of these model structures is a generalization of the stable module category much used in modular representation theory. This stable module category has also been studied by Benson [Ben97]. Received: 14 December 2000; in final form: 17 December 2001 / Published online: 5 September 2002     

Ideals,radicals, and structure of additive categories

Simple and semisimple additive categories are studied. We prove, for example, that an artinian additive category is (semi)simple iff it is Morita equivalent to a division ring(oid). Semiprimitive additive categories (that is, those with zero radical) are those which admit anoether full, faithful functor into a category of modules over a division ringoid.     

Generalized lax epimorphisms in the additive case

In this paper we call generalized lax epimorphism a functor defined on a ring with several objects, with values in an abelian AB5 category, for which the associated restriction functor is fully faithful. We characterize such a functor with the help of a conditioned right cancellation of another functor, constructed in a canonical way from the initial one. As consequences we deduce a characterization of functors inducing an abelian localization and also a necessary and sufficient condition for a morphism of rings with several objects to induce an equivalence at the level of two localizations of the respective module categories.     

Artin-Schelter regular algebras and categories

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin-Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin-Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin-Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the -category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.     

On t-structures and torsion theories induced by compact objects

First, we show that a compact object C in a triangulated category, which satisfies suitable conditions, induces a t-structure. Second, in an abelian category we show that a complex P· of small projective objects of term length two, which satisfies suitable conditions, induces a torsion theory. In the case of module categories, using a torsion theory, we give equivalent conditions for P· to be a tilting complex. Finally, in the case of artin algebras, we give a one-to-one correspondence between tilting complexes of term length two and torsion theories with certain conditions.     

On the cobordism and commutative monoid with cancellation approaches to conformal field theory

In the late 1980s, Graeme Segal axiomatized conformal field theory in terms of a cobordism category. In that same preprint he outlined a more symmetric trace approach, which was recently rigorized in terms of pseudo algebras over a 2-theory. In this paper, we treat the cobordism approach in the pseudo algebra context. We introduce a new algebraic structure on a bicategory, called a pseudo 2-algebra over a theory, as a means of comparison for the two approaches. The main result states that the 2-category of pseudo algebras over a fixed 2-theory is biequivalent to the 2-category of pseudo 2-algebras over a fixed theory in certain situations.     

Comparisons between singularity categories and relative stable categories of finite groups

We consider the relationship between the relative stable category of and the usual singularity category for group algebras with coefficients in a commutative noetherian ring. When the coefficient ring is self-injective we show that these categories share a common, relatively large, Verdier quotient. At the other extreme, when the coefficient ring has finite global dimension, there is a semi-orthogonal decomposition, due to Poulton, relating the two categories. We prove that this decomposition is partially compatible with the monoidal structure and study the morphism it induces on spectra.     

Adjoining adjoints

We construct the 2-category obtained from a category by freely adjoining a right adjoint for each morphism and isolate its universal property. Some others basic properties are also studied. Some examples in which the category is freely generated by a graph are discussed in detail. For these categories, the 2-cells are given a geometric interpretation and shown to be similar to certain diagrams which have appeared in the literature on C∗-algebras.     

Properties of abelian categories via recollements

A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an example where the converse does not hold and explore multiple sufficient conditions for it to hold.     

Orbifolds and groupoids

We define a 2-category structure (Pre-Orb) on the category of reduced complex orbifold atlases. We construct a 2-functor F from (Pre-Orb) to the 2-category (Grp) of proper étale effective groupoid objects over the complex manifolds. Both on (Pre-Orb) and (Grp) there are natural equivalence relations on objects: (a natural extension of) equivalence of orbifold atlases on (Pre-Orb) and Morita equivalence in (Grp). We prove that F induces a bijection between the equivalence classes of its source and target.     

Ext-projectives in suspended subcategories

Let H be a hereditary abelian k-category with tilting object and denote the bounded derived category of H. This paper is devoted to a study of suspended subcategories of by means of their Ext-projectives.     

Computations of atom spectra

This is a contribution to the theory of atoms in abelian categories recently developed in a series of papers by Kanda. We present a method that enables one to explicitly compute the atom spectrum of the module category over a wide range of non‐commutative rings. We illustrate our method and results by several examples.     

On the (non)vanishing of some “derived” categories of curved dg algebras

Since curved dg algebras, and modules over them, have differentials whose square is not zero, these objects have no cohomology, and there is no classical derived category. For different purposes, different notions of “derived” categories have been introduced in the literature. In this article, we show that for some concrete curved dg algebras, these derived categories vanish. This happens for example for the initial curved dg algebra whose module category is the category of precomplexes, and for certain deformations of dg algebras.