t-Structures are Normal Torsion Theories

Abstract

Abstract. We characterize t-structures in stable ∞-categories as suitable quasicategorical factorization systems. More precisely we show that a t-structure t on a stable ∞-category C is equivalent to a normal torsion theory F on C, i.e. to a factorization system F = (E, M) where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts.

Hearts and towers in stable $$\infty $$-categories

“…A closer look at this example makes it evident that this action is also monotone with respect to the natural structure of partially ordered class of ts(Sp), and the natural total 1 Here and in the rest of the paper we are implicitly using the equivalence between t-structures and normal torsion theories: if C is a stable ∞-category with a terminal object, there exists an antitone Galois connection between the poset Rex(C) of reflective subcategories of C and the poset pf(C) of prefactorization systems on C such that r (F) is a 3-for-2 class. This adjunction induces a bijective correspondence between the class of certain reflective and coreflective factorization systems called normal torsion theories and the class of t-structures on (the homotopy category of) C: this statement is the central result of [7] where it is called the Rosetta stone theorem, and motivates our choice to state our main results in the setting of stable (∞, 1)-categories. order of Z: more formally, the group homomorphism Z −→ Aut(ts(Sp)) defining the action is also a monotone mapping.…”

“…As already said, a particularly pleasant consequence of this good behaviour is the fact that in a stable ∞-category the notions of t-structure [1] and of normal factorization system (or normal torsion theory) [6] are naturally equivalent; this remains true in a triangulated category, although the equivalence is much less transparent (see [19], where this issue is framed in a fairly more general environment). This equivalence will be used several times along the discussion, as well as the following result from [7]:…”

“…In order to investigate properties of the (co-)reflections S and R, we recall the main result from [7]: there is an equivalence between t-structures on C and normal factorization systems on C, so that t can equivalently be seen as a Z-equivariant morphism O(J ) −→ fs(C), where fs(C) denotes the Z-poset of normal factorization systems of C. Explicitly, this equivalence is given as follows: given a t-structure (L, U) on C, the corresponding factorization system…”

“…For further information on normal factorization systems in stable ∞-categories we address the reader to [7,16].…”

Localization theory for derivators

“…Notice that, in the above notation, X 1 = Y 0 if and only if D(e)(X 1 , Y 0 ) = 0 and so, depending on the context, we sometimes also use the more common notation X 1 ⊥ Y 0 to mean the same as X 1 = Y 0 . Certain slight abuses of notation are now straightforward to understand: we can define orthogonality, as well as co/locality, with respect to a chosen class of {X α } α∈A ∈ D([1]) and this gives the usual Galois connection = ( ) ( ) = , which also allows to speak about the pairs (S, S = ) and ( ⊥ S, S) generated by a subprederivator S. The notion of coherent orthogonality is used in [LV17b] to lay the foundation of the theory of factorization systems on derivators, and then a theory of coherent t-structures as a consequence (see [FL16] for the characterization of tstructures as "normal torsion theories"). A general survey of the main features of factorization systems will be the subject of a subsequent work; for the moment we only record that [GLV17] already contains a rather general result, since it characterizes certain factorization systems as algebra structures for the "squaring" 2-monad of [KT93].…”