Derivators, pointed derivators and stable derivators

Abstract

Abstract. 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. Moreover, the functors belonging to the stable derivator can be turned into exact functors with respect to these triangulated structures. Along the way, we give a simplification of the axioms of a pointed derivator and a reformulation of the base change axiom in terms of Grothendieck (op)fibration. Furthermore, we have a new proof that a combinatorial model category has an underlying derivator.

Biset Functors as Module Mackey Functors and its Relation to Derivators

“…In fact, to any derivator on the 2-category of finite categories, we can associate a Mackey functor on S. This gives a way to construct biset functors from derivators (since the inclusion F k B ֒→ Mack k (S) is shown to have a left adjoint, as below). We remark also that the theory of derivators is of recent interest by several researchers, as seen in [7], [9].…”

How to glue derived categories

“…In [26] much more is proved, but extracting exactly the claim of Proposition 2.20 is a rather non-trivial exercise (one should also keep in mind the fact that in [26], triangulated categories are called "classical triangulated categories", and a triangulated category in the sense [26, Definition 7.1.1] has a form of enhancement already hardcoded into it). More recently, a proof in the ∞-categorical setting appears in [34], and a complete and detailed proof in the setting of derivators is worked out in [24]. However, just as [24,26] proves much more than what we claim here, so the paper is rather long, and on the other hand, it still does not cover the case of model pairs.…”

“…More recently, a proof in the ∞-categorical setting appears in [34], and a complete and detailed proof in the setting of derivators is worked out in [24]. However, just as [24,26] proves much more than what we claim here, so the paper is rather long, and on the other hand, it still does not cover the case of model pairs. So we think that giving a short and self-contained proof is useful.…”

“…Then for any morphism f : i → i , the map (4.5) induces by adjunction a natural map 24) and if these maps are invertible, the inverse maps turn the collection…”

The p -order of topological triangulated categories

“…The respective 'stable' versions come with theorems showing that the underlying category of such a stable collection of homotopy categories is naturally triangulated, compare [Gr,Thm. 4.18] or [Fr,Sec.…”