Related papers: Higher extension closure and $d$-exact categories

Let B be an extriangulated category with enough projectives and enough injectives. Let C be a fully rigid subcategory of B which admits a twin cotorsion pair ((C,K),(K,D)). The quotient category B/K is abelian, we assume that it is…

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…

We show that the idempotent completion of an n-angulated category admits a unique n-angulated structure such that the inclusion is an n-angulated functor, which satisfies a universal property.

We introduce $n$-abelian and $n$-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that $n$-cluster-tilting subcategories of abelian (resp. exact) categories…

We construct a Caldero-Chapoton map on a triangulated category with a cluster tilting subcategory which may have infinitely many indecomposable objects. The map is not necessarily defined on all objects of the triangulated category, but we…

Let $\mathcal{E}=(\mathcal{A},\mathcal{S})$ be an exact category with enough projectives $\mathcal{P}$. We introduce the notion of support $\tau$-tilting subcategories of $\mathcal{E}$. It is compatible with existing definitions of support…

In this paper we study the dg-category of twisted perfect complexes on a ringed space with soft structure sheaf. We prove that this dg-category is quasi-equivalent to the dg-category of complexes of vector bundles on that space. This result…

In this paper, we associate a quiver with superpotential to each $d$-angulation of a (unpunctured) marked surface. We show that, under quasi-isomorphisms, the flip of a $d$-angulation is compatible with Oppermann's mutation of (the Ginzburg…

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…

We give a direct proof of the following known result: the Grothendieck group of a triangulated category with a silting subcategory is isomorphic to the split Grothendieck group of the silting subcategory. Moreover, we obtain its…

We show that an algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category if it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for…

We develop a homotopy theory for additive categories endowed with endofunctors, analogous to the concept of a model structure. We use it to construct the homotopy theory of a Hovey triple (which consists of two compatible complete cotorsion…

We construct an "almost involution" assigning a new DG-category to a given one, and use this construction to recover, say, the abelian category of graded modules over the graded ring $R^*$ from the DG-category of DG-modules over a DG-ring…

We initiate in this article the study of weakly exact structures, a generalization of Quillen exact structures. We introduce weak counterparts of one-sided exact structures and show that a left and a right weakly exact structure generate a…

In this paper, we study ideal approximation theory associated to almost $n$-exact structures in extension closed subcategories of $n$-angulated categories. For $n=3$, an $n$-angulated category is nothing but a classical triangulated…

We consider three categories arising from the higher Auslander algebras of type $A$ in relation to $d$-dimensional cluster combinatorics: $d$-exact subcategory of the module category of $A^d_{n+1}$ generated by the $d$-cluster-tilting…

For any CDG-ring $B^\bullet=(B^*,d,h)$, we show that the homotopy category of graded-projective (left) CDG-modules over $B^\bullet$ is equivalent to the quotient category of the homotopy category of graded-flat CDG-modules by its full…

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…

Let $n$ be a positive integer. We show that an $n$-extension closed subcategory of an $n$-exangulated category naturally inherits an $n$-exangulated structure through restriction of the ambient $n$-exangulated structure. Furthermore, we…

We prove that thick category $\mathcal{O}$ associated to a semi-simple complex finite dimensional Lie algebra is extension full in the category of all modules. We also prove the weak Alexandru conjecture both for regular blocks of thick…