Related papers: Admissible weak factorization systems on extriangu…
The aim of this paper is to provide an expansion to Abe-Nakaoka's heart construction of the following two different realizations of the module category over the endomorphism ring of a rigid object in a triangulated category: Buan-Marsh's…
We prove that, assuming Vop\venka's principle, every small projectivity class in a locally presentable category is accessible.
Tannaka Duality describes the relationship between algebraic objects in a given category and their representations; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful…
Right triangulated categories can be thought of as triangulated categories whose shift functor is not an equivalence. We give intrinsic characterisations of when such categories have a natural extriangulated structure and are appearing as…
In this article, we initiate the study of hereditary extriangulated categories. Many important categories arising in representation theory in connection with various theories of mutation are hereditary extriangulated. Special cases include…
We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair $(\mathcal{A},\mathcal{B})$ in an exact category $\mathcal{C}$, $\mathcal{A}$ coincides…
Let $\mathcal{C}$ be a triangulated category. We first introduce the notion of balanced pairs in $\mathcal{C}$, and then establish the bijective correspondence between balanced pairs and proper classes $\xi$ with enough $\xi$-projectives…
We show that various derived categories of torsion modules and contramodules over the adic completion of a commutative ring by a weakly proregular ideal are full subcategories of the related derived categories of modules. By the work of…
We extend the framework of combinatorial model categories, so that the category of small presheaves over large indexing categories and ind-categories would be embraced by the new machinery called class-combinatorial model categories. The…
In this mostly expository paper, we present recent progress on infinite (weak) cluster categories that are related to triangulations of the disk, with and without a puncture. First we recall the notion of a cluster category. Then we move to…
Based on the novel notion of `weakly counital fusion morphism', regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields and multiplier bimonoids in braided…
In this article we prove that all the inclusions between the 'classical' and naturally defined full triangulated subcategories of a weakly approximable triangulated category are intrinsic (in one case under a technical condition). This…
This work extends the idea of matched pairs presented by Majid in \cite{Majid} and Takeuchi in \cite{Takeuchi} for the context of weak bialgebras and weak Hopf algebras. We introduce, also inspired by partial matched pairs…
Under appropriate conditions, if one picks a commutative algebra A with action of group G in braided monoidal category C, the category of A modules in C obtains a natural crossed G-braided structure. In the case of general commutative…
Bondal-Kapranov's notion of enhanced triangulated categories behaves well in the framework of localization theory, in the sense that the Verdier quotient of triangulated categories can be lifted to the Drinfeld dg quotient of…
In this paper, we study (complete) cotorsion pairs in extriangulated categories. First, we study a relationship between an interval of the poset of cotorsion pairs and the poset of cotorsion pairs in the heart associated to the interval.…
We call a triangulated category \emph{hereditary} provided that it is equivalent to the bounded derived category of a hereditary abelian category, where the equivalence is required to commute with the translation functors. If the…
We distinguish between faint, weak, strong and strict localizations of categories at morphism families and show that this framework captures the different types of derived functors that are considered in the literature. More precisely, we…
A convergent algorithm for nonnegative matrix factorization with orthogonality constraints imposed on both factors is proposed in this paper. This factorization concept was first introduced by Ding et al. with intent to further improve…
We establish a correspondence between consistent comprehension schemes and complete orthogonal factorisation systems. The comprehensive factorisation of a functor between small categories arises in this way. Similar factorisation systems…