Factorization systems on (stable) derivators

Abstract

We define triangulated factorization systems on triangulated categories, and prove that a suitable subclass thereof (the normal triangulated torsion theories) corresponds bijectively to $t$-structures on the same category. This result is then placed in the framework of derivators regarding a triangulated category as the base of a stable derivator. More generally, we define derivator factorization systems in the 2-category $\mathrm{PDer}$, describing them as algebras for a suitable strict 2-monad (this result is of independent interest), and prove that a similar characterization still holds true: for a stable derivator $\mathbb D$, a suitable class of derivator factorization systems (the normal derivator torsion theories) correspond bijectively with $t$-structures on the base $\mathbb{D}(\mathbb{1})$ of the derivator. These two result can be regarded as the triangulated- and derivator- analogues, respectively, of the theorem that says that `$t$-structures are normal torsion theories' in the setting of stable $\infty$-categories, showing how the result remains true whatever the chosen model for stable homotopy theory is.