Related papers: Gluing Approximable Triangulated Categories
In this survey we present the relatively new concept of \emph{approximable triangulated categories.} We will show that the definition is natural, that it leads to powerful new results, and that it throws new light on old, familiar objects.…
We use the theory of approximable triangulated categories to give a condition for a proper DG-category to be reflexive in the sense of Kuznetsov and Shinder. To do this we provide another description of the completion of an approximable…
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…
Small, finite entities are easier and simpler to manipulate than gigantic, infinite ones. Consequently huge chunks of mathematics are devoted to methods reducing the study of big, cumbersome objects to an analysis of their finite building…
We provide an explicit procedure to glue (not necessarily compact) silting objects along recollements of triangulated categories with coproducts having a 'nice' set of generators, namely, well generated triangulated categories. This…
Recently, Chach\'olski, Neeman, Pitsch, and Scherer studied, in a series of three papers, model approximations for the unbounded category of cochain complexes over a commutative ring. These approximations allow to construct relative…
Let $R$ be a commutative ring. A full additive subcategory $\C$ of $R$-modules is triangulated if whenever two terms of a short exact sequence belong to $\C$, then so does the third term. In this note we give a classification of…
A recollement of triangulated categories describes one such category as being "glued together" from two others. This paper gives a precise criterion for the existence of a recollement of the derived category of a Differential Graded Algebra…
This paper surveys some recent results, concerning the intrinsicness of natural subcategories of weakly approximable triangulated categories. We also review the results about uniqueness of enhancements of triangulated categories, with the…
In this article, a new construction of derived equivalences is given. It relates different endomorphism rings and more generally cohomological endomorphism rings - including higher extensions - of objects in triangulated categories. These…
We develop the general formalism of approximable triangulated categories, and prove two representability theorems.
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…
Weakly approximable triangulated categories, introduced by Neeman, provide a powerful framework for studying localization phenomena in triangulated categories. In this paper, we establish new localization theorems showing that, under mild…
We provide a technique to glue simple-minded collections along a recollement of Hom-finite Krull-Schmidt triangulated categories over a field. This gluing technique for simple-minded collections is shown to be compatible with those for…
Recollements of triangulated categories may be seen as exact sequences of such categories. Iterated recollements of triangulated categories are analogues of geometric or topological stratifications and of composition series of algebraic…
This work is concerned with approximability (\`{a} la Neeman) and Rouquier dimension for triangulated categories associated to noncommutative algebras over schemes. Amongst other things, we establish that the category of perfect complexes…
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…
We shall study the existence of almost split sequences in tri-exact categories, that is, extension-closed subcategories of triangulated categories. Our results unify and extend the existence theorems for almost split sequences in abelian…
We prove new Brown representability theorems for triangulated categories using metric techniques as introduced in the work of Neeman. In the setting of algebraic geometry, this gives us new representability theorems for homological and…
This paper is a sequel to arXiv:1503.05523 and arXiv:1605.03934. We extend the classical Harrison-Matlis module category equivalences to a triangulated equivalence between the derived categories of the abelian categories of torsion modules…