The Popescu–Gabriel theorem for triangulated categories

Abstract

The Popescu-Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differential graded categories. An analogous result for topological triangulated categories has recently been obtained by A. Heider.

Parametrizing recollement data for triangulated categories

“…that X is an aisle in D and Y is closed under small coproducts. The fact that X is an aisle follows from [43,Corollary 3.12]. The fact that Y is closed under small coproducts follows from the first step, since the morphisms of I are morphisms between compact objects.…”

“…Indeed, this result is implicitly contained in Neeman's book [37] (see Proposition 8.4.2 and Corollary 4.4.3 there). Perhaps, a shorter way of proving it would be to use Corollary 3.12 of [43] and the adjoint functor argument (see Lemma 2.3). Therefore, well-generated triangulated categories form a class of triangulated categories which are aisled by 'global' reasons.…”

Localization theory for triangulated categories

“…For an alternative approach which simplifies the definition, see [23]. More recently, well generated categories with specific models have been studied; see [37,47] for work involving algebraic models via differential graded categories, and [20] for topological models. In [43], Rosický used combinatorial models and showed that there exist universal cohomological functors into locally presentable categories which are full.…”

Derived equivalences of algebras

“…We refer the reader to for further details on differential graded categories and derived categories over them. By a theorem of Porta [, Theorem 1.2], every algebraic triangulated category which is well generated in the sense of [, Definition 1.15, p. 15] is equivalent to a localization of the derived category of a small differential graded category. An analogous result of Porta's theorem for topological triangulated categories has recently been proved by Heider in .…”