Related papers: Triangulated categories without models

Let $n$ be an integer greater or equal than $3$. We give a simultaneous generalization of $(n-2)$-exact categories and $n$-angulated categories, and we call it one-sided $n$-suspended categories. One-sided $n$-angulated categories are also…

We prove that, in a triangulated category with combinatorial models, every localizing subcategory is coreflective and every colocalizing subcategory is reflective if a certain large-cardinal axiom (Vopenka's principle) is assumed true. It…

We show that in an essentially small rigid tensor triangulated category with connected Balmer spectrum there are no proper non-zero thick tensor ideals admitting strong generators. This proves, for instance, that the category of perfect…

In this article we construct various models for singularity categories of modules over differential graded rings. The main technique is the connection between abelian model structures, cotorsion pairs and deconstructible classes, and our…

In this paper we study triangular matrix categories using the theory of recollements of abelian categories. Given a triangular matrix category we construct two canonical recollements. We show that if certain funtors of these recollements…

We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the…

In this article we make the concept of a continuous family of triangles precise and prove the moduli functor classifying oriented triangles admits a fine moduli space but the functor classifying non-oriented triangles only admits a coarse…

In the terms of an `$n$-periodic derived category', we describe explicitly how the orbit category of the bounded derived category of an algebra with respect to powers of the shift functor embeds in its triangulated hull. We obtain a large…

In this paper, we explore when a locally finite triangulated category has dimension zero or finite representation type. We also study generation of derived categories by orthogonal subcategories.

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,…

We introduce and study several homological notions which generalise the discrete derived categories of D. Vossieck. As an application, we show that Vossieck discrete algebras have this property with respect to all bounded t-structures. We…

We give criteria for subcategories of a compactly generated algebraic triangulated category to be precovering or preenveloping. These criteria are formulated in terms of closure conditions involving products, coproducts, directed homotopy…

Let $\Ascr,\Bscr$ be exact categories with $\Ascr$ karoubian and $M$ be an exact functor. Under suitable adjonction hypotheses for $M$, we are able to show that the direct factors of the objects of $\Ascr$ of the form $MY$ with $Y \in…

We investigate the triangulated hull of the orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull will correspond…

A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a…

We classify all triangulated orbit categories of path-algebras of Dynkin diagrams that are triangle equivalent to a stable module category of a representation-finite self-injective standard algebra. For each triangulated orbit category T we…

For an abelian category $\mathcal{A}$ we investigate when the stable categories $\underline{\mathrm{GPro}}\mathrm{j}(\mathcal{A})$ and $\underline{\mathrm{GIn}}\mathrm{j}(\mathcal{A})$ are triangulated equivalent. To this end, we realize…

We prove that some subquotient categories of exact categories are abelian. This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated categories. As a particular case, if an exact category B with enough projectives and…

In the theory of triangulated categories, we propose to replace hearts of $t$-structures by proper abelian subcategories, which may be plentiful even when hearts are not. For instance, this happens in negative cluster categories. In support…

This paper consists of three results on Frobenius categories: (1) we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that any Frobenius category is equivalent to an…