Related papers: Exact dg categories I : Foundations
Given a negatively graded Calabi-Yau algebra, we regard it as a DG algebra with vanishing differentials and study its cluster category. We show that this DG algebra is sign-twisted Calabi-Yau, and realize its cluster category as a…
This note describes a strictly-unital $A_\infty$-category whose representations are exact triangles such that the three-fold symmetry on exact triangles is manifest on the $A_\infty$-category.
We extend results of Br\"ustle-Yang on ideal quotients of 2-term subcategories of perfect derived categories of non-positive dg algebras to a relative setting. We find a new interpretation of such quotients: they appear as prototypical…
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…
For any ring R we construct two triangulated categories, each admitting a functor from R-modules that sends projective and injective modules to 0. When R is a quasi-Frobenius or Gorenstein ring, these triangulated categories agree with each…
Extriangulated categories were introduced by Nakaoka and Palu to give a unification of properties in exact categories and extension-closed subcategories of triangulated categories. A notion of tilting pairs in an extriangulated category is…
Exact sequences are a well known notion in homological algebra. We investigate here the more vague properties of 'homotopical exactness', appearing for instance in the fibre or cofibre sequence of a map. Such notions of exactness can be…
In this paper, we first provide an explicit procedure to glue together hereditary exact model structures for the recollement of exact categories. To that end, we use the notion of cotorsion pairs and we investigate the gluing of complete…
For a higher Nakayama algebra $A$ in the sense of Jasso-K\"{u}lshammer, we show that the singularity category of $A$ is triangulated equivalent to the stable module category of a self-injective higher Nakayama algebra. This generalizes a…
We introduce a bicategory that refines the localization of the category of dg categories with respect to quasi-equivalences and investigate its properties via formal category theory. Concretely, we first introduce the bicategory of dg…
We introduce and study the new notion of an {\em exact factorization} $\mathcal{B}=\mathcal{A}\bullet \mathcal{C}$ of a fusion category $\mathcal{B}$ into a product of two fusion subcategories $\mathcal{A},\mathcal{C}\subseteq \mathcal{B}$…
Greenlees defined an abelian category A whose derived category is equivalent to the rational S^1-equivariant stable homotopy category whose objects represent rational S^1-equivariant cohomology theories. We show that in fact the model…
We introduce the notion of ST-pairs of triangulated subcategories, a prototypical example of which is the pair of the bound homotopy category and the bound derived category of a finite-dimensional algebra. For an ST-pair $(\C,\D)$, we…
We study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right…
We prove that a triangulated category which is the underlying category of a stable derivator has a filtered enhancement, providing an affirmative answer to a conjecture in [3].
For a triangulated category A with a 2-periodic dg-enhancement and a triangulated oriented marked surface S we introduce a dg-category F(S,A) parametrizing systems of exact triangles in A labelled by triangles of S. Our main result is that…
We define exact weights on a triangulated category to be nonnegative functions on objects satisfying a subadditivity condition with respect to exact triangles. Such weights induce a metric on objects in the triangulated category, which we…
$n$-exangulated categories were introduced by Herschend-Liu-Nakaoka which are a simultaneous generalization of $n$-exact categories and $(n+2)$-angulated categories. This paper consists of two results on $n$-exangulated categories: (1) we…
This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective `proper' alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from…
If ${\cal D}$ is a definable category then it may contain no nonzero finitely presented modules but, by a result of Makkai, there is a $\varinjlim$-generating set of strictly ${\cal D}$-atomic modules. These modules share some key…