2017/05/24 by Fosco Loregiàn, Loregian, Fosco, Simone Virili +1 · 1 citation
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.1705.08565
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 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 \mathbbD(\mathbb1) 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 ∞-categories, showing how the result remains true whatever the chosen model for stable homotopy theory is.