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 , where J is a totally ordered set equipped with a monotone mathbb {Z}-action.

Highlights

  • If you’re going to read this, don’t bother

  • This result admits an immediate generalization to an arbitrary ambient category which is “good enough” for homotopy theory. It is a statement about the decomposition of an initial morphism ∗ −→ X into a tower of fibrations whose fibers have homotopy concentrated in a single degree. It is only in the setting of (∞, 1)-category theory that this result can be given its cleanest conceptualization: the tower of a pointed object X is the result of the factorization of ∗ −→ X with respect to the collection of factorization systems (n- conn, n- trunc) whose right classes are given by n-truncated morphisms [17, 5.2.8.16]

  • Semi-orthogonal decompositions and J -slicing with hearts are essentially the only two interesting classes, as shown by the structure theorem we prove in section Sect. 8: under suitable finiteness assumptions, the datum of a J -slicing on a stable ∞-category C is equivalent to the datum of a finite type semi-orthogonal decomposition of C, together with bounded t-structures on the slices and collections of torsion theories on the hearts of these t-structures (Theorem 8.2)

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Summary

Introduction

An elementary and yet fundamental theorem in algebraic topology asserts that every sufficiently nice connected topological space X fits into a “tower”. When the totally ordered set J has a heart J Œ, i.e., when there’s a Z-equivariant monotone morphism J −→ Z, a J -slicing on a stable ∞-category C is precisely the datum of a t-structure on C together with a collection of torsion theories parametrized by J Œ on the heart of C, which turns out to be an abelian ∞-category.

Notation and conventions
Posets with Z-actions
A tale of intervals
Interval decompositions and towers
Bridgeland slicings
Hearts of J-slicings
Abelianity of the heart
Semi-orthogonal decompositions
Abelian slicings and tiltings
Concluding remarks
Full Text
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