Related papers: Brown--Adams representability for triangulated cat…

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…

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…

Let T be a triangulated category with coproducts, C the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [Adams71]: All contravariant homological functors C --> Ab are the…

We define quasi--locally presentable categories as big unions of coreflective subcategories which are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a…

It is well known that cohomology with compact supports is not a homotopy invariant but only a proper homotopy one. However, as the proper category lacks of general categorical properties, a Brown representability theorem type does not seem…

Consider a complete abelian category which has an injective cogenerator. If its derived category is left--complete we show that the dual of this derived category satisfies Brown representability. In particular this is true for the derived…

We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated…

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…

In this paper, we establish a dual framework for Neeman's results concerning triangulated categories with compact silting objects by employing Brown--Comenetz duality. This framework introduces an intrinsic non-compact subcategory, provides…

In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak…

We prove that any digraph Brown functor -- i.e. a contravariant functor from the homotopy category of finite directed graphs to the category of abelian groups, satisfying the triviality axiom, the additivity axiom, and the Mayer-Vietoris…

We exhibit a triangulated category T having both products and coproducts, and a triangulated subcategory S of T which is both localizing and colocalizing, for which neither a Bousfield localization nor a colocalization exists. It follows…

We develop the general formalism of approximable triangulated categories, and prove two representability theorems.

In a well generated triangulated category T, given a regular cardinal a, we consider the following problems: given a functor from the category of a-compact objects to abelian groups that preserves products of <a objects and takes exact…

In this paper we give necessary and sufficient conditions for a functor to be representable in a strongly generated triangulated category which has a linear action by a graded ring, and we discuss some applications and examples.

We prove general adjoint functor theorems for weakly (co)complete $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of (co)complete $\infty$-categories, so these $n$-categories do not admit all small…

We prove the following result of V. Voevodsky. If $S$ is a finite dimensional noetherian scheme such that $S=\cup_\alpha\Spec(R_\alpha)$ for {\em countable} rings $R_\alpha$, then the stable motivic homotopy category over $S$ satisfies…

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…

We prove that any contravariant functor from the homotopy category of finite directed graphs to abelian groups satisfying the additivity axiom and the Mayer-Vietoris axiom is representable.

A theorem of Bondal and Kapranov lifts representations of cohomological functors from semiorthogonal decompositions of triangulated categories. We present a version of this result for triangulated functors. To this end, we introduce…