  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.
