Hearts and towers in stable infinity-categories

TL;DR

This paper unifies various classical and modern concepts in the theory of triangulated and stable infinity-categories through the lens of towers associated with J-slicings, revealing deep structural connections.

Contribution

It demonstrates that classical topics like t-structures, cohomology, semiorthogonal decompositions, tiltings, and Bridgeland's slicings are all special cases of a single tower construction in stable infinity-categories.

Findings

Unified framework for classical and modern triangulated category concepts

Characterization of bounded t-structures via hearts and cohomology functors

Connection between slicings, towers, and J-slicings in stable infinity-categories

Abstract

We exploit the equivalence between $t$-structures and normal torsion theories on a stable $\infty$-category to show how a few classical topics in the theory of triangulated categories, i.e., the characterization of bounded $t$-structures in terms of their hearts, their associated cohomology functors, semiorthogonal decompositions, and the theory of tiltings, as well as the more recent notion of Bridgeland's slicings, are all particular instances of a single construction, namely, the tower of a morphism associated with a $J$-slicing of a stable $\infty$-category $\mathcal C$, where $J$ is a totally ordered set equipped with a monotone $\mathbb{Z}$-action.