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Levelwise Modules and Localization in Derivators

  • Author(s): Lagkas Nikolos, Ioannis
  • Advisor(s): Balmer, Paul
  • et al.
Abstract

We prove that the idempotent completion of a strong additive derivator~$\mathbb D$ is also a strong additive derivator which is moreover stable if~$\mathbb D$ is. This recovers the main results in~\cite{Balmer-idempotent} and~\cite{IdempotentCompletionpretriangulated} assuming that the (pre)triangulated categories we are working with are the base of a strong additive (stable) derivator (which happens in all examples one encounters in practice). We also show that given a separable cocontinuous monad on a triangulated derivator, the levelwise Eilenberg-Moore categories of modules glue together to a triangulated derivator. This allows us to give examples of derivators that are stable but not strong. Furthermore, we study localization of right derivators under classes that are closed under colimits and, as a special case, recover Franke's result~\cite{Franke} that we can form Verdier quotients of small triangulated derivators. Finally, we study compact objects in big triangulated derivators. We show that any compact object can be written as a retract of a finite direct colimit of a coherent diagram that is pointwise in a chosen generating set. We also show that levelwise and pointwise compact objects over small diagrams coincide. As an application, we extend the equivalence of~\cite{Neeman-Thomason} to an equivalence of~$\mathbf{Dir}_{\text f}$-derivators.

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