Related papers: Derived categories of (one-sided) exact categories…
In this paper, we introduce quotients of exact categories by percolating subcategories. This approach extends earlier localization theories by Cardenas and Schlichting for exact categories, allowing new examples. Let $\mathcal{A}$ be a…
In this article, we show that the localization of an extriangulated category by a multiplicative system satisfying mild assumptions can be equipped with a natural, universal structure of an extriangulated category. This construction unifies…
The aim of this paper is to develop a framework for localization theory of triangulated categories $\mathcal{C}$, that is, from a given extension-closed subcategory $\mathcal{N}$ of $\mathcal{C}$, we construct a natural extriangulated…
The Verdier quotient $\mathcal{T}/\mathcal{S}$ of a triangulated category $\mathcal{T}$ by a triangulated subcategory $\mathcal{S}$ is defined by a universal property with respect to triangulated functors out of $\mathcal{T}$. However,…
Exact categories are a natural generalisation of abelian categories and provide a fertile ground to develop relative homological algebra. In this paper, starting from a class of relative Gorenstein projective objects in an exact category…
The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's…
It is well known that a resolving subcategory $\mathcal{A}$ of an abelian subcategory $\mathcal{E}$ induces several derived equivalences: a triangle equivalence $\mathbf{D}^-(\mathcal{A})\to \mathbf{D}^-(\mathcal{E})$ exists in general and…
One-sided exact categories are obtained via a weakening of a Quillen exact category. Such one-sided exact categories are homologically similar to Quillen exact categories: a one-sided exact category $\mathcal{E}$ can be (essentially…
For an exact category we provide two constructions of an ambient category in which the initial category is resolving: In the derived category and in the Gabriel--Quillen embedding. For the first construction we describe a pre-aisle and its…
For an exact dg category $\mathcal A$, we introduce its bounded dg derived category $\mathcal{D}^b_{dg}(\mathcal A)$ and establish the universal exact morphism from $\mathcal A$ to $\mathcal{D}^b_{dg}(\mathcal A)$. We prove that the dg…
Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study…
Quasi-abelian categories are abundant in functional analysis and representation theory. It is known that a quasi-abelian category $\mathcal{E}$ is a cotilting torsionfree class of an abelian category. In fact, this property characterizes…
We study Verdier quotients of diverse homotopy categories of a full additive subcategory $\mathcal E$ of an abelian category. In particular, we consider the categories $K^{x,y}({\mathcal E})$ for $x\in\{\infty, +,-,b\}$, and…
We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite dimensional algebras.…
We mainly investigate abelian quotients of the categories of short exact sequences. The natural framework to consider the question is via identifying quotients of morphism categories as modules categories. These ideas not only can be used…
We study the following generalization of singularity categories. Let X be a quasi-projective Gorenstein scheme with isolated singularities and A a non-commutative resolution of singularities of X in the sense of Van den Bergh. We introduce…
One-sided exact categories appear naturally as instances of Grothendieck pretopologies. In an additive setting they are given by considering the one-sided part of Keller's axioms defining Quillen exact categories. We study one-sided exact…
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 x 3-lemma and the snake lemma. We briefly discuss exact functors,…
In this work, we provide a simple way to construct $d$-abelian categories via bounded derived categories for certain values of $d$. Namely, let ${\mathcal C}$ be an abelian category, and let ${\mathcal C}[0,m]$ denote the full subcategory…
We classify certain subcategories in quotients of exact categories. In particular, we classify the triangulated and thick subcategories of an algebraic triangulated category, i.e. the stable category of a Frobenius category.