Related papers: Definable functors between triangulated categories

We give a construction of triangulated categories as quotients of exact categories where the subclass of objects sent to zero is defined by a triple of functors. This includes the cases of homotopy and stable module categories. These…

We provide various ways to characterise $\Sigma$-pure-injective objects in a compactly generated triangulated category. These characterisations mimic analogous well-known results from the model theory of modules. The proof involves two…

We establish a criterion for determining when a family of geometric functors is jointly conservative through the lens of purity in compactly generated triangulated categories. We introduce the notion of pure descendability and we apply it…

We define the notion of a $\lambda$-definable category, a generalisation of the notion of definable category from the model theory of modules. Let ${\cal C}$ be a $\lambda$-accessible additive category. We characterise the additive functors…

We provide a framework to triangulate subfactor categories of additive categories with additive endofunctors. It is proved that such a framework is sufficiently flexible to cover many instances in algebra and geometry where abelian, exact…

We develop some aspects of the theory of derivators, pointed derivators, and stable derivators. As a main result, we show that the values of a stable derivator can be canonically endowed with the structure of a triangulated category.…

Given a fixed tensor triangulated category S we consider triangulated categories T together with an S-enrichment which is compatible with the triangulated structure of T. It is shown that, in this setting, an enriched analogue of Brown…

In this paper we consider a construction in an arbitrary triangulated category T which resembles the notion of a Moore spectrum in algebraic topology. Namely, given a compact object C of T satisfying some finite tilting assumptions, we…

The paper contains general results on the uniqueness of a DG enhancement for triangulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent sheaves, for the triangulated categories of…

An (additive) functor F from an additive category A to an additive category B is said to be objective, provided any morphism f in A with F(f) = 0 factors through an object K with F(K) = 0. In this paper we concentrate on triangle functors…

For a triangulated category with products we develop a method for constructing a nice set of cogenerators, allowing us to prove a formal criterion in order to satisfy Brown representability for covariant functors. We apply this criterion…

We extend Deligne's notion of determinant functor to tensor triangulated categories. Specifically, to account for the multiexact structure of the tensor, we define a determinant functor on the 2-multicategory of triangulated categories and…

In a triangulated symmetric monoidal closed category, there are natural dualities induced by the internal Hom. Given a monoidal functor f^* between two such catgories and adjoint couples (f^*,f_*) and (f_*,f^!), we prove the necessary…

Two pertinent questions for any support theory of a monoidal triangulated category are whether it is functorial and if the tensor product property holds. To this end, we consider the complete prime spectrum of an essentially small monoidal…

This paper has been withdrawn and replaced by arXiv:1309.5035. In this paper we describe some examples of so called spherical functors between triangulated categories, which generalize the notion of a spherical object. We also give…

This paper is dedicated to the study of weight complexes (defined on triangulated categories endowed with weight structures) and their applications. We introduce pure (co)homological functors that "ignore all non-zero weights"; these have a…

Graduated locally finitely presentable categories are introduced, examples include categories of sets, vector spaces, posets, presheaves and Boolean algebras. A finitary functor between graduated locally finitely presentable categories is…

The question of when the derived category of a ring satisfies Brown--Adams representability is revisited via studying the transfer of pure homological dimension along definable functors: it is shown that, for any ring, the pure global…

We initiate the study of derived functors in the setting of extriangulated categories. By using coends, we adapt Yoneda's theory of higher extensions to this framework. We show that, when there are enough projectives or enough injectives,…

We prove that the homological and Balmer spectra in tensor-triangular geometry are functorial in certain definable functors, thereby providing an alternative perspective on functoriality in tensor-triangular geometry from the viewpoint of…