Triangulated Categories

Abstract

In this survey we present the relatively new concept of 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.

Direct limits in the heart of a t-structure: The case of a torsion pair

“…products) of triangles are triangles (see [N,Proposition 1.2.1]), this is equivalent to saying that the left (resp. right) truncation functor τ U : D −→ U (resp.…”

Constructing cogenerators in triangulated categories and Brown representability

“…Together with the localization sequence whose existence is supposed in the hypothesis, this shows that both categories Loc(G) and U are equivalent to the Verdier quotient T /G ⊥ , hence they are equivalent to each other. Finally provided that objects in G are α-compact, we know by [12,Theorem 8.4.2] that Loc(G) satisfies Brown representability. Consequently the inclusion functor Loc(G) → T which preserves coproducts must have a right adjoint and a localization sequence Loc(G) → T → G ⊥ exists.…”

“…To be more specific, consider a triangulated category T . The definition and basic properties of triangulated categories are to be found in the standard reference [12]. A (co)homological functor on T is a (contravariant) functor F : T → A into an abelian category which sends triangles to long exact sequences.…”

“…We know when usual triangulated categories satisfy this property. For example, the derived category of a Grothendieck abelian category (including the derived category of modules or of quasicoherent sheaves) is α-compactly generated for some regular cardinal α, in the sense of [12,Definition 8.1.6] (note that Neeman calls such categories well generated). Consequently it satisfies Brown representability for contravariant functors, by [12,Theorem 8.4.2].…”

“…For example, the derived category of a Grothendieck abelian category (including the derived category of modules or of quasicoherent sheaves) is α-compactly generated for some regular cardinal α, in the sense of [12,Definition 8.1.6] (note that Neeman calls such categories well generated). Consequently it satisfies Brown representability for contravariant functors, by [12,Theorem 8.4.2]. In contrast, the homotopy category of an additive category does not satisfy Brown representability, unless the initial additive category is pure-semisimple (see [11]).…”