Related papers: Derived equivalences induced by nonclassical tilti…

For a balanced pair $(\mathcal{X},\mathcal{Y})$ in an abelian category, we investigate when the chain homotopy categories ${\bf K}(\mathcal{X})$ and ${\bf K}(\mathcal{Y})$ are triangulated equivalent. To this end, we realize these chain…

We consider an arbitrary Abelian category $\mathcal{A}$ and a subcategory $\mathcal{T}$ closed under extensions and direct summands, and characterize those $\mathcal{T}$ that are (semi-)special preenveloping in $\mathcal{A}$; as a…

We give an overview of recent developments in silting theory. After an introduction on torsion pairs in triangulated categories, we discuss and compare different notions of silting and explain the interplay with t-structures and…

We use the maximal faces of the $m$-cluster complex of type A to describe the m-cluster tilted algebras of type A as quivers with relations. We then classify connected components of m-cluster tilted algebras of type A up to derived…

We consider recollements of derived categories of dg-algebras induced by self orthogonal compact objects obtaining a generalization of Rickard's Theorem. Specializing to the case of partial tilting modules over a ring, we extend the results…

We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define…

In the work of Hoshino, Kato and Miyachi, the authors look at t-structures induced by a compact object, C, of a triangulated category, T, which is rigid in the sense of Iyama and Yoshino. Hoshino, Kato and Miyachi show that such an object…

The notion of relative derived category with respect to a subcategory is introduced. A triangle-equivalence, which extends a theorem of Gao and Zhang [Gorenstein derived categories, \emph{J. Algebra} \textbf{323} (2010) 2041-2057] to the…

We prove that any derived equivalence between derived discrete algebras is standard, i.e.\ is isomorphic to the derived tensor product by a two-sided tilting complex.

This note extends Quillen's Theorem A to a large class of categories internal to topological spaces. This allows us to show that under a mild condition a fully faithful and essentially surjective functor between such topological categories…

The present paper is devoted to study the homotopy category associated with a simplicial descent category (D,s,E) (arXiv:0808.3684v2). We prove that the class E of equivalences has a calculus of left fractions over a quotient category of D…

In this paper, we show that the tilting modules over a cluster-tilted algebra $A$ lift to tilting objects in the associated cluster category $\mathcal{C}_H$. As a first application, we describe the induced exchange relation for tilting…

First, we show that a compact object $C$ in a triangulated category, which satisfies suitable conditions, induces a $t$-structure. Second, in an abelian category we show that a complex $P^{\centerdot}$ of small projective objects of term…

We study when the heart of a t-structure in a triangulated category $\mathcal{D}$ with coproducts is AB5 or a Grothendieck category. If $\mathcal{D}$ satisfies Brown representability, a t-structure has an AB5 heart with an injective…

This article introduces pre-Hilbert $*$-categories: an abstraction of categories exhibiting "algebraic" aspects of Hilbert-space theory. Notably, finite biproducts in pre-Hilbert $*$-categories can be orthogonalised using the Gram-Schmidt…

We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let $\mathscr{X}$ denote an additive category with finite direct sums and split…

Recent results by Keller and Nicol{\'a}s and by Koenig and Yang have shown bijective correspondences between suitable classes of t-structures and co-t-structures with certain objects of the derived category: silting objects. On the other…

Let T be a tilting object in a triangulated category equivalent to the bounded derived category of a hereditary abelian category with finite dimensional homomorphism spaces and split idempotents. This text investigates the strong global…

The (dual) Dold-Kan correspondence says that there is an equivalence of categories $K:\cha\to \Ab^\Delta$ between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show…

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…