Monomorphisms, Epimorphisms, and Pull-Backs

The study of radicals in general categories has followed several lines of development. The problem of defining radical properties in general categories has been considered by Kurosh and Shul'geifer, see (7). Under mild conditions on their categories they obtain sufficient conditions for the existence of radical functors which are closely related to radical properties. Another approach is by Maranda (5) and Dickson (3) who studied idempotent radical functors and torsion theories in abelian categories. Our aim has been to study radical functors in as general a category as possible. To this end we introduce the concept of an R-category. The categories of rings, modules, near-rings, groups and Jordan algebras are all examples of R-categories.

We introduce a relative tilting theory in abelian categories and show that this work offers a unified framework of different previous notions of tilting, ranging from Auslander–Solberg relative tilting modules on Artin algebras to infinitely generated tilting modules on arbitrary rings. Furthermore, we see that it presents a tool for developing new tilting theories in categories that can be embedded nicely in an abelian category. In particular, we will show how the tilting theory in exact categories built this way, coincides with tilting objects in extriangulated categories introduced recently. We will review Bazzoni’s tilting characterization, the relative homological dimensions on the induced tilting classes and parametrise certain cotorsion-like pairs by using n-X-tilting classes. As an application, we show how to construct relative tilting classes and cotorsion pairs in Rep(Q,C) (the category of representations of a quiver Q in an abelian category C) from tilting classes in C, where Q is finite-cone-shape.

We give an introduction to constructive category theory by answering two guiding computational questions. The first question is: how do we compute the set of all natural transformations between two finitely presented functors like Ext \operatorname {Ext} and Tor \operatorname {Tor} over a commutative coherent ring R R ? We give an answer by introducing category constructors that enable us to build up a category which is both suited for performing explicit calculations and equivalent to the category of all finitely presented functors. The second question is: how do we determine the differentials on the pages of a spectral sequence associated to a filtered cochain complex only in terms of operations directly provided by the axioms of an abelian category? Its answer relies on a constructive method for performing diagram chases based on a calculus of relations within an arbitrary abelian category.

Cartan–Eilenberg categories were recently introduced by Guillen Santos, Navarro Aznar, Pascual and Roig. In the present paper, we give a method of constructing Cartan–Eilenberg categories for abelian categories, based on cotorsion pairs. In particular, we recover the left Cartan–Eilenberg structure for (un)bounded below chain complexes of modules (Pascual, Collect Math 63(2):203–216, 2012) and extend it to more general categories, including the category of (quasi-coherent)sheaves over a projective scheme.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Homotopy theory in general categories

The formal content of near set theory can be summarised in terms of three concepts: a perceptual system, a nearness relation and a near set. Perceptual systems and different forms of nearness relations have been already successfully related to important mathematical structures (e.g., approach spaces) and described in the frameworks of general topology and category theory. However, since near sets actually do not form any regular structure, there is lack of similar results about the concept of a near set. The main goal of this paper is to fill this gap and provide a mathematical basis for near sets on a similar abstract level as it has been already done for nearness relations. However, we take in the paper a down-to-earth approach; that is, instead of seeking the richest mathematical structures which express our intuitions (call it up-to-sky approach), we present in the paper the simplest category theoretic structures which, however, are rich enough to convey our ideas. Thus, although many times we actually deal with sheaves, we speak about them as pre-sheaves. The main reason is that a sheaf (in contrast to a pre-sheaf) embodies the idea of gluing (in the very similar way like manifolds, which are obtainable by gluing open subsets of Euclidean space), which is irrelevant to the study of near sets and rough sets. Furthermore, the concept a pre-sheaf is much simpler than a sheaf and should be easily “digested” by readers who are not very familiar with category theory.

Monads of sets

Covering theory for linear categories with application to derived categories

This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad. In Part 4 we define general monoidal categories equipped with an action operad equivariant structure, and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras. Many illustrations and examples are included. Assuming only basic category theory, this monograph is intended for graduate students and researchers. In addition to being a coherent reference for the topics covered, this book is also suitable for a graduate student seminar and a reading course.

Equivalences between cluster categories