We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor HomC(T, −), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equiv...

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 (Vopěnka’s principle) is assumed true. It follows that, under the same assumptions, orthogonality sets up a bijective correspondence between localizing subcategories and colocalizing subcategories. Th...

We show by an example that in a Verdier triangulated category, there may exist two mutually nonisomorphic Verdier octahedra containing the same commutative triangle.

We show by an example that in a Verdier triangulated category, there may exist two mutually nonisomorphic Verdier octahedra containing the same commutative triangle.

We show by an example that in a Verdier triangulated category, there may exist two mutually nonisomorphic Verdier octahedra containing the same commutative triangle.

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 triangulated subcategories of finitely generated modules over a principal ideal domain. As a corollary we show that in the category of finitely generated modules over a PID, thic...

We define the Grothendieck group of an n-exangulated category. For n odd, we show that this shares many properties with exact or a triangulated In particular, classify dense complete subcategories category n-(co)generator in terms subgroups group. This unifies and extends results Thomason, Bergh–Thaule, Matsui Zhu–Zhuang for triangulated, $$(n+2)$$ -angulated, extriangulated categories, respect...

This article is designed to provide an introduction to some examples of triangulated categories that arise in the study of G-equivariant cohomology theories for a compact Lie group G. We focus on cohomology theories whose values are rational vector spaces since one may often give explicit algebraic constructions of the triangulated category in that case. As general references for equivariant co...

In this paper we establish an equivalence between the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W and triangulated category of singularities of the fiber of W over zero. We also proved that the category of graded D-branes of type B in such LG-models is connected by a fully faithful functor with the derived category of coherent sheaves on the...

For each object in a tensor triangulated category, we construct a natural continuous map from the object’s support—a closed subset of the category’s triangular spectrum—to the Zariski spectrum of a certain commutative ring of endomorphisms. When applied to the unit object this recovers a construction of P. Balmer. These maps provide an iterative approach for understanding the spectrum of a tens...