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Journal of Homotopy and Related Structures

, Volume 14, Issue 4, pp 993–1042 | Cite as

Hearts and towers in stable \(\infty \)-categories

  • Domenico Fiorenza
  • Fosco LoregianEmail author
  • Giovanni Luca Marchetti
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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 Open image in new window , where J is a totally ordered set equipped with a monotone \(\mathbb {Z}\)-action.

Keywords

Stable \(\infty \)-categories Triangulated categories t-structures Tilting theory Semiorthogonal decompositions Stability conditions on triangulated categories 

1 Introduction

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

C. Palahniuk

An elementary and yet fundamental theorem in algebraic topology asserts that every sufficiently nice connected topological space X fits into a “tower”
$$\begin{aligned} \cdots \longrightarrow X_2\longrightarrow X_1\longrightarrow X_0=X \end{aligned}$$
where each \(X_n\) is n-connected and each map \(X_n\longrightarrow X_{n-1}\) is a fibration that induces isomorphisms in \(\pi _{>n}\), and has an Eilenberg–MacLane space \(K(\pi _{n}(X),n-1)\) as its fiber. This result admits an immediate generalization to an arbitrary ambient category which is “good enough” for homotopy theory. It is indeed a statement about the decomposition of an initial morphism \(* \longrightarrow X\) into a tower of fibrations whose fibers have homotopy concentrated in a single degree. It is nevertheless only in the setting of \((\infty ,1)\)-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 \(* \longrightarrow X\) with respect to the collection of factorization systems \((n\textsc {-conn}, n\textsc {-trunc})\) whose right classes are given by n-truncated morphisms [17, 5.2.8.16].
Of course, a similar construction can be exported to stable homotopy theory, where the analogue of the factorization system \((n\textsc {-conn}, n\textsc {-trunc})\) is given by the canonicalt-structure \(\mathfrak {t}\) on the category of spectra, determined by the objects whose homotopy groups vanish in negative an non-negative degree, respectively,
$$\begin{aligned} \mathcal {S}\text {p}_{\ge 0} = \{ A_* \in \mathcal {S}\text {p}\mid \pi _i(A_*)=0;\; i< 0 \}\\ \mathcal {S}\text {p}_{< 0} = \{ B_* \in \mathcal {S}\text {p}\mid \pi _i(B_*)=0;\; i\ge 0 \}, \end{aligned}$$
together with all its shifts\(\mathfrak {t}_n=\mathfrak {t}[n]\).1 In this context, it becomes natural to consider the whole \(\{\mathfrak {t}_n\mid n\in {\mathbb {Z}}\}\) as a single object, namely the orbit of \(\mathfrak {t}\) under the canonical action of the group of integers on the class \(\text {ts}(\mathcal {S}\text {p})\) of t-structures on the category of spectra. 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 \(\text {ts}(\mathcal {S}\text {p})\), and the natural total order of \({\mathbb {Z}}\): more formally, the group homomorphism \({\mathbb {Z}}\longrightarrow \text {Aut}(\text {ts}(\mathcal {S}\text {p}))\) defining the action is also a monotone mapping.

The aim of this paper is to investigate the consequences of taking this point of view further on the classical theory of t-structures. In particular, we describe all the terminology we need about partially ordered groups and their actions in Sect. 2, and then we specialize the discussion to \(\mathbb {Z}\)-actions on partially ordered sets. This is motivated by the fact that the class of t-structures on a given stable \(\infty \)-category carries a natural choice of such an action.

Even if we employ a rather systematic approach, we do not aim at reaching a complete generality, but instead at gathering a number of useful results and nomenclature we can refer to along the present article. Among various possible choices, we mention specialized references as [2, 8, 10] for an extended discussion of the theory of actions on ordered groups.

We introduce the definition of a slicing of a poset J (Definition 3.1) and of a J-slicing of a stable \(\infty \)-category Open image in new window (Definition 3.11) in Sect. 3: of course these are not original definitions, as the notion is classical in order theory under different names; our only purpose here is to collect the minimal amount of theory for the sake of clarity. Namely, in the same spirit of Dedekind’s construction of real numbers, we consider decompositions of a poset J (more often than not, a totally ordered one) into an upper and lower set, and associate with each such a decomposition a t-structure on an ambient \((\infty ,1)\)-category Open image in new window . The totally ordered set J will be assumed to be equipped with a monotone action of \(\mathbb {Z}\) and the correspondencewill be monotone and \({\mathbb {Z}}\)-equivariant.

Now, following [1], (bounded) t-structures on a triangulated category can be seen as the datum of a set of (co)homological functors indexed by integers; thus a J-slicings can be seen as a generalization to “fractal” or “ non-integer” cohomological dimensions now indexed by J and not by \(\mathbb {Z}\). Namely, each J-slicing induces local cohomology objects depending on an interval, as discussed in Sect. 4.

In this framework many homological features appear as a shadow of clear constructions with totally ordered sets with \(\mathbb {Z}\)-actions. For instance, when the totally ordered set J has a heart \(J^\heartsuit \), i.e., when there’s a \(\mathbb {Z}\)-equivariant monotone morphism \(J \longrightarrow {\mathbb {Z}}\), a J-slicing on a stable \(\infty \)-category Open image in new window is precisely the datum of a t-structure on Open image in new window together with a collection of torsion theories parametrized by \(J^\heartsuit \) on the heart of Open image in new window , which turns out to be an abelian \(\infty \)-category. This is shown Sects. 5 and 7.

In Sect. 6, we recover the theory of classical semi-orthogonal decompositions by considering the case when \({\mathbb {Z}}\) acts trivially on J. 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 \(\infty \)-category Open image in new window is equivalent to the datum of a finite type semi-orthogonal decomposition of Open image in new window , together with bounded t-structures on the slices and collections of torsion theories on the hearts of these t-structures (Theorem 8.2). It is worth mentioning that, under the finiteness conditions of Theorem 8.2, when \(J={\mathbb {R}}\) the notion of J-slicing as discussed in the present paper actually becomes a reinterpretation of Bridgeland’s definition of slicing of a triangulated category [5]. This can be better appreciated by switching to the general approach to ‘stability data’ introduced by [9].

Finally, in Sect. 7 we show how the functoriality of the association \(J\mapsto \{\text {J-slicings}\}\) gives rise to an elegant and synthetic reformulation of classical tilting theory [12].

1.1 Notation and conventions

We will work within the framework of stable \(\infty \)-categories in the sense of [18]. The reader who prefers to work in the more traditional framework of triangulated categories will find no difficulty in pretending that all higher categories are instead categories and that all fiber sequences are distinguished triangles; indeed, many of our statements and constructions are actually adjustments of classical arguments valid in triangulated categories. However a few proofs become more natural when stated in the language of stable \(\infty \)-categories (for example, theorems whose proof involves a certain universal property unavailable in the triangulated world).

To make the article more self-contained and enjoyable to a reader with no previous exposure to stable \(\infty \)-categories, we recall here the minimal amount of \(\infty \)-categorical notions we will make use of. As in [17], we will call ‘\(\infty \)-category’ an \((\infty ,1)\)-category, i.e. an higher category whose k-morphisms are invertible (up to homotopy) for any \(k>1\). In colloquial terms, this means that an \(\infty \)-category has objects, morphisms, homotopies between morphisms, homotopies between homotopies, etc., and such homotopies are all invertible. As shown in [13, 17], the entire theory of categories transports to \(\infty \)-categories. In particular, an \(\infty \)-category Open image in new window can have finite co/limits.

Definition

(Stable\(\infty \)-category) An \(\infty \)-category Open image in new window is said to be stable if it has all finite limits and all finite colimits, and if in addition it satisfies the so-called pullout axiom: a diagramin Open image in new window (or, more formally, an object of the functor category Open image in new window ) is a pushout if and only if it is a pullback.

We will henceforth call these diagrams pullout diagrams or simply pullouts. The pullout axiom is inherently \(\infty \)-categorical: the only ordinary category with finite limits and colimits satisfying it is the trivial category (i.e. the terminal additive category with a single zero object \(\mathbf{0}\)). It is also extremely powerful: if Open image in new window is a stable \(\infty \)-category, then its homotopy category Open image in new window is triangulated. In other words, this single and extremely simple axiom subsumes all of the axiomatic of triangulated categories (the notorious octahedral axiom included).

Unfortunately not every triangulated category can be realized as the homotopy category of a stable \(\infty \)-category, see [20]; stable \(\infty \)-categories therefore only give rise to ‘well-behaved’ triangulated categories. It should however be remarked that ill-behaved ones are often artificial: the reader can then safely assume that essentially every ‘reasonable’ triangulated category is the homotopy category Open image in new window of some stable \(\infty \)-category Open image in new window .

As already said, a particularly pleasant consequence of this good behaviour is the fact that in a stable \(\infty \)-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]:

Lemma

(Sator lemma) If \((\mathcal {E},\mathcal {M})\) is a factorization system on a stable \(\infty \)-category Open image in new window , with the property that both \(\mathcal {E}\) and \(\mathcal {M}\) saisfy the ‘two out of three’ property, then for every object Open image in new window an initial arrow \(0\longrightarrow A\) lies in \(\mathcal {E}\) (resp. in \(\mathcal {M}\)) if and only if the terminal arrow \(A\longrightarrow 0\) of the same object lies in \(\mathcal {E}\) (resp. in \(\mathcal {M}\)).

A final line to conclude this introductory subsection: not to spoil the reader’s fun, while avoiding to to hide them their meaning, translations of the quotes opening each section are provided immediately before the bibliography.

2 Posets with \(\mathbb {Z}\)-actions

This section introduces the terminology about partially ordered groups and their actions, and then specializes the discussion to \(\mathbb {Z}\)-actions. We do not aim at a complete generality, but instead at gathering a number of useful results and nomenclature which is useful to have at hand. Among various possible choices, we mention specialized references as [2, 8, 10] for an extended discussion of the theory of actions on ordered groups.

We begin by recalling the well known fact that the category Open image in new window of partially ordered sets and monotone maps is cartesian closed. Namely, the product order on the cartesian product \(P\times Q\) of two posets \((P,\le )\) and \((Q,\le )\), given by \((x,y)\le (x',y')\) if and only if \(x\le x'\) and \(y\le y'\) makes \((P\times Q,\le )\) together with the projections on the factors satisfy the universal property of the product in the category of posets. Moreover, the set Open image in new window of monotone maps from \((P,\le )\) to \((Q,\le )\) has a natural order on it given by \(f\le g\) if and only if \(f(x)\le g(x)\) for every x in P. With this order one has an adjunction

Remark 2.1

The product order is not the only standard order one puts on the product \(P\times Q\) of two posets P and Q. Another commonly used one is the lexicographic order defined by \((x,y)\le _{{\text {lex}}} (x',y')\) if and only if \(x< x'\) or \(x=x'\) and \(y\le y'\). The lexicographic order does not make \(P\times Q\) be the product of P and Q in the category of posets, but it still has a few peculiar properties that, as we are going to see, are relevant to the theory of slicings.

Remark 2.2

Let \((P,\le )\) be a poset, and let \(\sim \) be an equivalence relation on the set P. One says that \(\sim \) is compatible with the order relation if \(x\le y\), \(x\sim x'\) and \(y\sim y'\) imply \(x'\le y'\) or \(x'\sim y'\). When this happens the quotient set \(P/_{\!\sim }\) inherits a order relation from P by \([x]\le [y]\) if and only if \(x\le y\) or \(x\sim y\). Moreover the projection to the quotient \(P\longrightarrow P/_{\!\sim }\) is a morphism of posets.

Definition 2.3

A partially ordered group (“po-group” for short) is a pair \((G, \le )\) consisting of a group G and of a partial order relation \(\le \) on G such that the group multiplication \(\cdot :G\times G\longrightarrow G\) is a map of posets, where \(G\times G\) is endowed with the product order: for any two pairs (gh) and \((g',h')\) with \(g\le g'\) and \(h\le h'\) we have \(gg' \le hh'\).

Remark 2.4

In the literature on the subject it is customary draw a distinction between a left po-group and a right po-group. We choose to ignore this subtlety, since all the po-groups we will be dealing with will be ordered by two-sided congruences.

Remark 2.5

If \((G,\le )\) is a po-group, the inversion Open image in new window is an antitone antiautomorphism of groups, i.e. we have that \(g\le h\) if and only if \(h^{-1}\le g^{-1}\). Moreover the set \(G^+\) of positive elements, i.e. the set \(\{g\in G\mid 1\le g\}\) is closed under conjugation.

Example 2.6

Let \((P,\le )\) be a poset, and let Open image in new window be the automorphism group of P as a poset, i.e., the set of monotone bijections of p into itself. Then Open image in new window inherits an order relation by its inclusion in the poset Open image in new window , and this makes Open image in new window a partially ordered group. This is the standard po-group structure on Open image in new window .

Remark 2.7

Any group G can be seen as a po-group with the trivial order relation \(g\le h\) if and only if \(g=h\). It is worth noticing that on finite groups the trivial order is the only possible po-group structure. Namely, assume \(g\le h\) and let \(k=g^{-1}h\). Then we have \(1\le k\le k^2\le k^2\le \cdots \le k^{\text {ord}(k)}=1\) and so \(k=1\), i.e., \(g=h\).

Definition 2.8

A homomorphism of po-groups consists of a group morphism \(f:G\longrightarrow H\) which is also a monotone mapping. This defines a category \(\mathcal {PoGrp}\) of partially ordered groups and their homomorphisms.

Definition 2.9

Let \((G,\le )\) be a po-group. A G-poset is a partially ordered set \((P, \le )\) endowed with a po-group homomorphism Open image in new window to the group of order isomorphisms of P with its standard po-group structure.

Remark 2.10

Equivalently, a G-poset is a partially ordered set P together with a group action \(G\times P\longrightarrow P\) which is a morphism of posets, where on \(G\times P\) one has the product order.

Example 2.11

Every po-group G is a G-poset with the multiplication action of G on itself.

Remark 2.12

An equivalence relation \(\sim \) on a G-poset P is said to be compatible with the G-action if \(x\sim y\) implies \(g\cdot x\sim g\cdot y\) for any g in G. If \(\sim \) is compatible both with the order and with the G-action then the quotient set \(P/_{\!\sim }\) is naturally a G-poset with the G-action \(g\cdot [x]=[g\cdot x]\). Moreover the projection to the quotient is a morphism of G-posets.

We now specialize our discussion to the case \(G=\mathbb {Z}\)

Definition 2.13

A \(\mathbb {Z}\)-poset is a partially ordered set \((P,\le )\) together with a group action
$$\begin{aligned} +_P:P\times \mathbb {Z}\longrightarrow P :(x,n) \mapsto x+_Pn \end{aligned}$$
which is a morphism of partially ordered sets, when \(\mathbb {Z}\) is regarded with its usual total order.

Remark 2.14

It is immediate to see that a \(\mathbb {Z}\)-poset is equivalently the datum of a poset \((P,\le )\) together with a monotone bijection \(\rho :P\longrightarrow P\) such that \(x\le \rho (x)\) for any x in P. The function \(\rho \) and the action are related by the identity \(\rho (x)=x+_P1\).

Notation 2.15

To avoid a cumbersome accumulation of indices, the action \(+_P\) will be often denoted as a simple “\(+\)”. This is meant to evoke in the reader the most natural example of a \(\mathbb {Z}\)-poset, given by \({\mathbb {Z}}\) itself, and to simplify our notation for the axioms of an action:
$$\begin{aligned} {\left\{ \begin{array}{ll} (x +_P m) +_P n = x +_P (m+n);\\ x +_P 0 = x. \end{array}\right. } \end{aligned}$$
We will also write \(x-n\) for \(x+_P(-n)\).

Example 2.16

The poset \((\mathbb {Z},\le )\) of integers with their usual order is a \(\mathbb {Z}\)-poset with the action given by the usual sum of integers. The poset \((\mathbb {R},\le )\) of real numbers with their usual order is a \(\mathbb {Z}\)-poset for the action given by the sum of real numbers with integers (seen as a subring of real numbers).

Example 2.17

Given any poset \((P,\le )\), the poset \(\mathbb {Z}\times _{\text {lex}}P\) carries a natural \({\mathbb {Z}}\)-action given by \((n,x)+1=(n+1,x)\), i.e., by the standard \(\mathbb {Z}\)-action on the first factor and by the trivial \(\mathbb {Z}\)-action on the second factor.

Remark 2.18

If \((P,\le )\) is a finite poset, then the only \(\mathbb {Z}\)-action it carries is the trivial one. Indeed, if \(\rho :P\longrightarrow P\) is the monotone bijection associated with the \(\mathbb {Z}\)-action, one sees that \(\rho \) is a finite order element in Open image in new window , by the finiteness of P. Therefore there exists an \(n\ge 1\) such that \(\rho ^n=\text {id}_P\). It follows that, for any x in P,
$$\begin{aligned} x\le x+1\le \cdots \le x+n=x \end{aligned}$$
and so \(x=x+1\).

Remark 2.19

An obvious terminology: a G-fixed point for a G-poset P is an element \(p\in P\) kept fixed by all the elements of G under the G-action. Clearly, an element x of a \(\mathbb {Z}\)-poset P is a \(\mathbb {Z}\)-fixed point if and only if \(x+ 1 = x\), or equivalently \(x-1=x\). From this it immediately follows that if \(x\in P\) is a \(\le \)-maximal or \(\le \)-minimal element in the \(\mathbb {Z}\)-poset P, then it is a \(\mathbb {Z}\)-fixed point.

Remark 2.20

Given a poset P we can always define a partial order on the set \(P_{\! \mathchoice{\lhd \rhd }{\lhd \rhd }{\lhd \rhd }{\lhd \rhd } } = P\cup \{-\,\infty ,+\,\infty \}\) which extends the partial order on P by the rule \(-\infty \le x\le +\infty \) for any \(x\in P\).

Lemma 2.21

If \((P,\le )\) is a \(\mathbb {Z}\)-poset, then \((P_{\! \mathchoice{\lhd \rhd }{\lhd \rhd }{\lhd \rhd }{\lhd \rhd } },\le )\) carries a natural \(\mathbb {Z}\)-action extending the \(\mathbb {Z}\)-action on P, by declaring both \(-\infty \) and \(+\infty \) to be \(\mathbb {Z}\)-fixed points.

Proof

Adding a fixed point always gives an extension of an action, so we only need to check that the extended action is compatible with the partial order. This is equivalent to checking that also on \(P_{\! \mathchoice{\lhd \rhd }{\lhd \rhd }{\lhd \rhd }{\lhd \rhd } }\) the map \(x\mapsto x+1\) is a monotone bijection such that \(x\le x+1\), which is immediate. \(\square \)

Posets with \(\mathbb {Z}\)-actions naturally form a category Open image in new window , whose morphisms are \(\mathbb {Z}\)-equivariant morphisms of posets. More explicitly, if P and Q are \(\mathbb {Z}\)-posets with actions \(+_P\) and \(+_Q\), then a morphism of \(\mathbb {Z}\)-posets between them is a morphism of posets \(\varphi :P\longrightarrow Q\) such that
$$\begin{aligned} \varphi (x+_P n)=\varphi (x)+_Q n, \end{aligned}$$
for any \(x\in P\) and any \(n\in \mathbb {Z}\).

Remark 2.22

If P and Q are \(\mathbb {Z}\)-posets, then the hom-set Open image in new window is naturally a \(\mathbb {Z}\)-poset. Namely, as we have already remarked, Open image in new window is naturally a poset, and so Open image in new window inherits the poset structure as a subset. The \(\mathbb {Z}\)-action is given by \((\varphi +n)(x)=\varphi (x) +_Q n\). This makes Open image in new window a closed category.

Remark 2.23

Every poset can be seen as a \(\mathbb {Z}\)-poset with the trivial \(\mathbb {Z}\)-action. Since every monotone mapping is \(\mathbb {Z}\)-equivariant with respect to the trivial \(\mathbb {Z}\)-action, this gives a fully faithful embedding Open image in new window .

Lemma 2.24

The choice of an element x in a \(\mathbb {Z}\)-poset P is equivalent to the datum of a \(\mathbb {Z}\)-equivariant morphism \(\varphi :(\mathbb {Z},\le )\longrightarrow (P,\le )\). Moreover x is a \(\mathbb {Z}\)-fixed point if and only if the corresponding morphism \(\varphi \) factors \(\mathbb {Z}\)-equivariantly through \((*,\le )\), where \(*\) denotes the terminal object of Open image in new window .

Proof

To the element x one associates the \(\mathbb {Z}\)-equivariant morphism \(\varphi _x\) defined by \(\varphi _x(n)=x+n\). To the \(\mathbb {Z}\)-equivariant morphism \(\varphi \) one associates the element \(x_\varphi =\varphi (0)\). It is immediate to check that the two constructions are mutually inverse. The proof of the second part of the statement is straightforward. \(\square \)

Lemma 2.25

Let \(\varphi :(\mathbb {Z},\le )\longrightarrow (P,\le )\) be a \(\mathbb {Z}\)-equivariant morphism of \(\mathbb {Z}\)-posets. Then \(\varphi \) is either injective or constant.

Proof

Assume \(\varphi \) is not injective. then there exist two integers n and m with \(n>m\) such that \(\varphi (n)=\varphi (m)\). By \(\mathbb {Z}\)-equivariancy we therefore have
$$\begin{aligned} x_\varphi +(n-m)=x_\varphi , \end{aligned}$$
with \(n-m\ge 1\) and \(x_\varphi =\varphi (0)\). The conclusion then follows by the same argument used in Remark 2.18. \(\square \)

Lemma 2.26

Let \(\varphi :(P,\le )\longrightarrow (Q,\le )\) be a morphism of \(\mathbb {Z}\)-posets. Assume Q has a minimum and a maximum. Then \(\varphi \) extends to a morphism of \(\mathbb {Z}\)-posets \((P_{\! \mathchoice{\lhd \rhd }{\lhd \rhd }{\lhd \rhd }{\lhd \rhd } },\le )\longrightarrow (Q,\le )\) by setting \(\varphi (-\,\infty )=\min (Q)\) and \(\varphi (+\,\infty )=\max (Q)\).

Proof

Since \(\min (Q)\) and \(\max (Q)\) are \(\mathbb {Z}\)-fixed points by Remark 2.19, the extended \(\varphi \) is a morphism of \(\mathbb {Z}\)-posets. Moreover, since \(\min (Q)\) and \(\max (Q)\) are the minimum and the maximum of Q, respectively, the extended \(\varphi \) is indeed a morphism of posets, and so it is a morphism of \(\mathbb {Z}\)-posets. \(\square \)

All of the above applies in particular to totally ordered sets. We will denote by Open image in new window the full subcategory of totally ordered sets, and by Open image in new window the full subcategory of \(\mathbb {Z}\)-actions on totally ordered sets.

Lemma 2.27

Let \((P, \le )\) be a totally ordered \({\mathbb {Z}}\)-poset. The relation \(x\sim y\) if and only if there are integers \(a,b \in {\mathbb {Z}}\) such that \(x + a \le y \le x + b \) is an equivalence relation on P compatible with both the order and the \({\mathbb {Z}}\)-action. It therefore induces a morphism of \({\mathbb {Z}}\)-posets \(P\longrightarrow P/_{\!\sim }\) given by the projection to the quotient. Moreover, \(P/_{\!\sim }\) is totally ordered and the \({\mathbb {Z}}\)-action on the quotient is trivial.

Proof

Checking that \(\sim \) is an equivalence relation is immediate: reflexivity is manifest; symmetry reduces to noticing that \(x + a \le y \le x + b \) is equivalent to \(y-b\le x\le y-a\); transitivity follows by the fact that \(x + a \le y \le x + b \) and \(y + c \le z \le y + d\) together imply \(x+ (a+ c) \le z \le x+(b + d)\). To see that \(\sim \) is compatible with the order relation, let \(x\le y\) and let \(x\sim x'\) and \(y\sim y'\). Then there exist abc and d in \({\mathbb {Z}}\) such that \(x + a \le x' \le x + b \) and \(y + c \le y' \le y + d\). Since P is totally ordered, either \(x'\le y'\) or \(y'\le x'\). In the second case we have \(y'\le x' \le x + b\le y+b \le y'+b-c\), and so \(x'\sim y'\) by definition of the relation \(\sim \). The compatibility of \(\sim \) with the \({\mathbb {Z}}\)-action is straightforward. Therefore by Remarks 2.2 and 2.12 we see that the projection to the quotient \(P\longrightarrow P/_{\!\sim }\) is a morphism of \({\mathbb {Z}}\)-posets. Since the order on P is total, so is also the order induced by \(\sim \) on the quotient set. Finally, to see that the \({\mathbb {Z}}\) action on \(P/_{\!\sim }\) is trivial, just notice that for any x in P, we have \(x\le x+1\le x+1\) and so \([x]+1=[x]\). \(\square \)

Remark 2.28

If the \({\mathbb {Z}}\)-action on the totally ordered set P is trivial, then the equivalence relation from Lemma 2.27 is trivial as well: \(x\sim y\) if and only if \(x=y\).

Lemma 2.29

Let \((P, \le )\) be a totally ordered \({\mathbb {Z}}\)-poset, and let \(\sim \) be the equivalence relation from Lemma 2.27. Then either \([x]=\{x\}\) or \([x]={\mathbb {Z}}\times _{\text {lex}}[x,x+1)\), where \([x,x+1)=\{y\in P\mid x\le y<x+1\}\)

Proof

Let \(x\in P\); then either \(x=x+1\) or \(x<x+1\). In the first case x is a fixed point for the \(\mathbb {Z}\)-action on P and so the equivalence relation \(\sim \) is the trivial one: \(y\sim x\) if and only if \(y=x\). If \(x<x+1\) then the interval \([x,x+1)\) is nonempty, as \(x\in [x,x+1)\).

Let \(\varphi :{\mathbb {Z}}\times _{\text {lex}}[x,x+1)\longrightarrow P\) the map defined by \((n,y)\mapsto y+n\). The map \(\varphi \) is a morphism of \(\mathbb {Z}\)-posets.

Indeed, if \((n,y)\le _{\text {lex}} (n',y')\) either \(n<n'\) or \(n=n'\) and \(y\le y'\). In the first case we have \(n+1\le n'\) and so \(y+n< x+1+n\le x+n'\le y'+n'\), whereas in the second case we have \(y+n\le y'+n=y'+n'\). The map \(\varphi \) is also injective.

To conclude we only need to show that \(\varphi :{\mathbb {Z}}\times _{\text {lex}}[x,x+1)\longrightarrow [x]\) is surjective. Let \(z\in [x]\). By definition of the equivalence relation there exist ab in \({\mathbb {Z}}\) such that \(x+a\le z\le x+b\). Since x is not a \(\mathbb {Z}\)-fixed point, we have \(a\le b\) and \(z\in [x+a,x+b+1)\). Writing
$$\begin{aligned}{}[x+a,x+b+1)=\bigcup _{n=a}^{b}[x+n,x+n+1) \end{aligned}$$
we see that there exists \(n\in \mathbb {Z}\) such that \(z\in [x,x+1)+n\), i.e., \(z=\varphi (n,y)\) for some y in \([x,x+1)\). \(\square \)

Proposition 2.30

The fully faithful embedding Open image in new window given by trivial \(\mathbb {Z}\)-actions on totally ordered sets has a left adjoint.

Proof

For any totally ordered \(\mathbb {Z}\)-poset P, let \(\iota (P)\) be the \({\mathbb {Z}}\)-poset \(P/_{\!\sim }\), where \(\sim \) is the equivalence relation from Lemma 2.27. Then Open image in new window is a functor, since if \(f:P\longrightarrow Q\) is a morphism of \({\mathbb {Z}}\)-posets then \(x+a\le y\le x+b\) implies
$$\begin{aligned} f(x)+a=f(x+a)\le f(y)\le f(x+b)=f(x)+b \end{aligned}$$
and so f induces a well defined morphism of sets \({\tilde{f}}:\iota (P)\longrightarrow \iota (Q)\). It is immediate to see that \({\tilde{f}}\) is actually a morphism of \(\mathbb {Z}\)-posets and that \(f\rightsquigarrow {\tilde{f}}\) preserves identities and compositions of morphisms. Finally, to see that I is a right adjoint to the trivial action embedding Open image in new window , let P be a a totally ordered \({\mathbb {Z}}\)-poset and Q be a totally ordered set. Since I is a functor, a morphism of \({\mathbb {Z}}\)-posets \(f:P\longrightarrow Q^\flat \) induces a morphism of posets \({\tilde{f}}:\iota (P)\longrightarrow \iota (Q^\flat )\). Moreover, since the \({\mathbb {Z}}\)-action on \(Q^\flat \) is trivial, we have \(\iota (Q^\flat )\cong Q\), see Remark 2.28. Therefore, \(f\mapsto {\tilde{f}}\) defines a mapwhich we want to show is a bijection. Assume \({\tilde{f}}_1={\tilde{f}}_2\). Then, for any x in P we have \(f_1(x)\sim f_2(x)\) in \(Q^\flat \). Since the equivalence relation on \(Q^\flat \) is trivial, this means \(f_1=f_2\), so \(f\rightsquigarrow {\tilde{f}}\) is injective. Let now \(\varphi \) be a morphism of posets, \(\varphi :\iota (P)\longrightarrow Q\), and let \(f=\varphi \circ \pi \) where \(\pi :P\longrightarrow \iota (P)\) is the projection to the quotient. We have \({\tilde{f}}([x])=[f(x)]=f(x)=\varphi (\pi (x))=\varphi ([x])\), and so \(\varphi ={\tilde{f}}\), i.e., \(f\rightsquigarrow {\tilde{f}}\) is surjective. \(\square \)

3 Histoire d’\(\mathcal {O}(J)\)

Sa liberté était pire que n’importe quelle chaîne

D. Aury

Recall that a lower set in a poset J is a subset \(L\subseteq J\) such that if \(x\in L\) and \(y\le x\) then \(y\in L\); the set of lower sets of J is denoted \({\downharpoonleft } J\) and it naturally a partially ordered set. Dually, one defines upper sets and the set \({\upharpoonleft }J\) of upper sets with its natural partial order.

Definition 3.1

Let J be a poset. A slicing of J is a pair (LU), where L is a lower set in J, U is an upper set, \(L\cap U=\emptyset \) and \(L\cup U=J\). The collection of all slicings of J will be denoted by \(\mathcal {O}(J)\).

Remark 3.2

Since the complement of an upper set is a lower set and vice versa, the projection on the second factor is a bijection
$$\begin{aligned} \mathcal {O}(J)\xrightarrow {\sim } {\upharpoonleft }J \end{aligned}$$
This induces a natural partial order on \(\mathcal {O}(J)\): we set \((L_1,U_1)\le (L_2,U_2)\) if and only if \(U_2\subseteq U_1\). Notice that \(\mathcal {O}(J)\) has a minimum given by the slicing \((\emptyset ,J)\) and a maximum given by the slicing \((J,\emptyset )\).

Remark 3.3

If J is totally ordered, then so is \(\mathcal {O}(J)\). Namely, let \(U_1\) and \(U_2\) two upper sets in J and assume that \(U_1\) is not a subset of \(U_2\). Then there exists an element x in \(U_1\) which is not in \(U_2\). If \(y\in U_2\) then either \(y\le x\) or \(y\ge x\) since J is totally ordered. But since \(U_2\) is an upper set \(y\le x\) would imply \(x\in U_2\) against our assumption. This means that \(y\ge x\) and, since \(U_1\) is an upper set, this implies \(y\in U_1\). Therefore if \(U_1\) is not a subset of \(U_2\) we have that \(U_2\subseteq U_1\).

Remark 3.4

If J is a \({\mathbb {Z}}\)-poset, then so is \(\mathcal {O}(J)\). The natural \({\mathbb {Z}}\)-action on \(\mathcal {O}(J)\) is given by
$$\begin{aligned} (L,U)+n=(L+n,U+n), \end{aligned}$$
where \(L+n=\{x+n\,|\, x\in L\}\) and \(U+n=\{x+n\,|\, x\in U\}\).

Remark 3.5

Every element x in J determines two slicings of J: \(((-\infty ,x),[x,+\infty ))\) and \(((-\infty ,x],(x,+\infty ))\). Here \((-\infty ,x)\) is the lower set \(\{y\in J\,|\, y<x\}\), and similarly for \((-\infty ,x]\), \((x,+\infty )\) and \([x,+\infty )\). This gives two natural morphisms of posets \(J\longrightarrow \mathcal {O}(J)\). If J is a \({\mathbb {Z}}\)-poset, then these morphisms are \({\mathbb {Z}}\)-equivariant.

The construction of \(\mathcal {O}(J)\) is actually functorial in J so that we have the following

Lemma 3.6

The map \(J\rightsquigarrow \mathcal {O}(J)\) defines a functorwhere Open image in new window denotes the category of \(\mathbb {Z}\)-posets with minimum and maximum and with \(\mathbb {Z}\)-morphism of posets preserving them (these maps are called \(\{0,1\}\)-homomorphisms in lattice theory, see [11]).

Proof

By the above remarks the only thing we have to prove is functoriality. For any morphism of \(\mathbb {Z}\)-posets \(f:J_1\longrightarrow J_2\), we set \(\mathcal {O}(f):U\longrightarrow f^{-1}(U)\). It is immediate to see that \(f^{-1}(U)\) is an upper set in \(J_2\) for any upper set U in \(J_1\) an that \(f^{-1}(U+1)=f^{-1}(U)+1\), so that \(\mathcal {O}(f)\) is indeed a morphism of \(\mathbb {Z}\)-posets from \(\mathcal {O}(J_2)\) to \(\mathcal {O}(J_1)\). Moreover we have \(\mathcal {O}(\text {id}_J)=\text {id}_{\mathcal {O}(J)}\) and \(\mathcal {O}(fg)=\mathcal {O}(g)\mathcal {O}(f)\), and \(\mathcal {O}(f)(\emptyset )=\emptyset \) and \(\mathcal {O}(f)(J_2)=J_1\). \(\square \)

Remark 3.7

Since the minimum and the maximum of a \(\mathbb {Z}\)-poset, when they exist, are necessarily fixed points of the \(\mathbb {Z}\)-action, we see that the inclusion Open image in new window induces a \(\mathbb {Z}\)-poset structure on Open image in new window making this inclusion a morphism of \(\mathbb {Z}\)-posets.

Example 3.8

The morphism \(n\mapsto [n,+\infty )\) induces an isomorphism of \({\mathbb {Z}}\)-posets \({\mathbb {Z}}_{ \mathchoice{\lhd \rhd }{\lhd \rhd }{\lhd \rhd }{\lhd \rhd } }\xrightarrow {\sim } \mathcal {O}({\mathbb {Z}})\). The morphisms \(x\mapsto (x,+\infty )\) and \(x\mapsto [x,+\infty )\) together induce an isomorphism of posets \({\mathbb {Z}}\)-posets
$$\begin{aligned} ({\mathbb {R}}\times _{\text {lex}} [\mathbf 1 ])_{ \mathchoice{\lhd \rhd }{\lhd \rhd }{\lhd \rhd }{\lhd \rhd } }\xrightarrow {\sim } \mathcal {O}({\mathbb {R}}), \end{aligned}$$
where \([\mathbf 1 ]\) is the totally ordered set \(\{0 < 1\}\).

Definition 3.9

Before introducing the main definition of this section, let us recall that a t-structure on a stable \(\infty \)-category Open image in new window consists of a pair Open image in new window of full sub-\(\infty \)-categories satisfying the following properties:

We introduce further terminology as a separate remark:

Remark 3.10

The categories Open image in new window and Open image in new window are called the lower sub-\(\infty \)-category and the upper sub-\(\infty \)-category of the t-structure \(\mathfrak {t}\), respectively.

The collection Open image in new window of all t-structures on a stable \(\infty \)-category Open image in new window is a poset with respect to following order relation: given two t-structures Open image in new window and Open image in new window , one has \(\mathfrak {t}_1 \le \mathfrak {t}_2\) if and only if Open image in new window .

The ordered group \(\mathbb {Z}\) acts on Open image in new window in a way that is fixed by the action of the generator \(+1\); this maps a t-structure Open image in new window to the shiftedt-structure Open image in new window . Since \(\mathfrak {t}\le \mathfrak {t}[1]\) one sees that Open image in new window is naturally a \(\mathbb {Z}\)-poset. Finally, the poset Open image in new window has a minimum and a maximum given by Open image in new window and Open image in new window , respectively. These are called the trivialt-structures.

Definition 3.11

Let \((J,\le )\) be a \(\mathbb {Z}\)-poset. A J-slicing of a stable \(\infty \)-category Open image in new window is a \(\mathbb {Z}\)-equivariant morphism of posets Open image in new window respecting minima and maxima on both sides. We denote as Open image in new window the class of all J-slicings of the category Open image in new window ;2

More explicitly, a J-slicing is a family \(\{\mathfrak {t}_{(L,U)}\}_{(L,U)\in \mathcal {O}(J)}\) of t-structures on Open image in new window such that
  1. (i)

    \(\mathfrak {t}_{(L_1,U_1)}\le \mathfrak {t}_{(L_2,U_2)}\) if \((L_1,U_1)\le (L_2,U_2)\) in \(\mathcal {O}(J)\);

     
  2. (ii)

    \(\mathfrak {t}_{(L,U)+1}=\mathfrak {t}_{(L,U)}[1]\) for any \((L,U)\in \mathcal {O}(J)\).

     
  3. (iii)
     

Remark 3.12

Of course, Open image in new window : this, together with 3.6 and 3.7, gives that Open image in new window is a functor.

Notation 3.13

We will denote the lower and the upper sub-\(\infty \)-categories of the t-structure \(\mathfrak {t}_{(L,U)}\) by Open image in new window and Open image in new window , respectively, i.e., we write Open image in new window . For \(i\in J\), we will write Open image in new window , Open image in new window , Open image in new window and Open image in new window for Open image in new window and Open image in new window , respectively. Note that, by \(\mathbb {Z}\)-equivariancy, we have Open image in new window , and similarly for the other cases.

Example 3.14

By Lemma 2.24 and Example 3.8, a \(\mathbb {Z}\)-slicing on Open image in new window is equivalent to the datum of a t-structure Open image in new window . One has Open image in new window for any \(n\in {\mathbb {Z}}\), consistently with the Notation 3.13, Open image in new window and Open image in new window . Notice that by our Remark 2.18, as soon as Open image in new window is a proper subcategory of Open image in new window , then the inclusion Open image in new window is proper for all \(n\in \mathbb {Z}\), i.e. the orbit \(\mathfrak {t}+ \mathbb {Z}\) is an infinite set. The equivalence between t-structures and \(\mathbb {Z}\)-slicings can also be seen in the light of Remark 3.12: for every \(\mathbb {Z}\)-poset P with minimum and maximum one has a distinguished isomorphism Open image in new window given by \(\varphi \mapsto \varphi ([0,+\infty ))\)

Example 3.15

By Example 3.8, an \(\mathbb {R}\)-slicing on Open image in new window is the datum of two t-structures Open image in new window and Open image in new window on Open image in new window for any \(\lambda \in \mathbb {R}\) in such a way that Open image in new window , etc., and with the inclusions Open image in new window for any \(\lambda \in {\mathbb {R}}\) andfor any \(\lambda _1<\lambda _2\) in \({\mathbb {R}}\). \({\mathbb {R}}\)-slicings have been introduced in [5], where they are called simply “slicings”. Actually [5] imposes more restrictive conditions to ensure “compactness” of the factorization, we will come back to this later. Compare also [9].

Remark 3.16

Since the subcategories Open image in new window and Open image in new window are the lower and the upper subcategories of a t-structure \(\mathfrak {t}_{(L,U)}\) they are reflexive and coreflective, respectively. In particular we have reflection and coreflection functorsFor X an object in Open image in new window we will occasionally write \(X_L\) for \(R_LX\) and \(X_U\) for \(S_UX\), and similarly for morphisms. Finally, by composing \(R_L\)and \(S_U\) with the inclusions of Open image in new window and Open image in new window in Open image in new window , we can look at \(R_L\) and \(S_U\) as endofunctors of Open image in new window .
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 Open image in new window and normal factorization systems on Open image in new window , so that \(\mathfrak {t}\) can equivalently be seen as a \(\mathbb {Z}\)-equivariant morphism Open image in new window , where Open image in new window denotes the \({\mathbb {Z}}\)-poset of normal factorization systems of Open image in new window . Explicitly, this equivalence is given as follows: given a t-structure Open image in new window on Open image in new window , the corresponding factorization system \((\mathcal {E},\mathcal {M})\) is characterized bySince we are going to use this fact several times, we recall that both the class \(\mathcal {E}\) and the class \(\mathcal {M}\) have the 3-for-2 property. In particular this implies the Sator lemma:
$$\begin{aligned} (0\longrightarrow X)&\text { is in }\mathcal {E}\text { if and only if }(X\longrightarrow 0) \text { is in }\mathcal {E}\\ (X\longrightarrow 0)&\text { is in }\mathcal {M}\text { if and only if }(0\longrightarrow X) \text { is in }\mathcal {M}\end{aligned}$$
For further information on normal factorization systems in stable \(\infty \)-categories we address the reader to [7, 16].

Remark 3.17

Notice that the left class \(\mathcal {E}\) of the normal factorization system \((\mathcal {E},\mathcal {M})\) corresponds to the right class Open image in new window of the corresponding t-structure Open image in new window . One could avoid this position switch by writing the pair of classes in a t-structure as Open image in new window , however we preferred to keep the upper class on the right to agree with the standard orientation on the line of real numbers.

Remark 3.18

Since \((\mathcal {E},\mathcal {M})\) are a factorization system, the class \(\mathcal {M}\) is closed under pullbacks and the class \(\mathcal {E}\) is closed under pushouts. Together with the Sator lemma this implies that Open image in new window and Open image in new window are extension closed.

Lemma 3.19

Let \(\mathfrak {t}\) be a J-slicing of Open image in new window and let \((L_0,U_0)\) and \((L_1,U_1)\) two slicings of J with \((L_0,U_0)\le (L_1,U_1)\). Then we have the natural isomorphisms
  1. (i)

    \(R_{L_0}R_{L_1}\cong R_{L_1}R_{L_0}\cong R_{L_0}\);

     
  2. (ii)

    \( S_{U_0}S_{U_1}\cong S_{U_1}S_{U_0}\cong S_{U_1}\);

     
  3. (iii)

    \(R_{L_0}S_{U_1}\cong S_{U_1}R_{L_0}\cong 0\).

     

Proof

It is enough to prove (i) and (iii), as the proof of (ii) is dual to (i).

We denote \(\mathfrak {t}_0\) and \(\mathfrak {t}_1\) the t-structures corresponding to \((L_0,U_0)\) and \((L_1,U_1)\), respectively, and by \((\mathcal {E}_0,\mathcal {M}_0)\) and \((\mathcal {E}_1,\mathcal {M}_1)\) the corresponding normal torsion theories. Since \((L_0,U_0)\le (L_1,U_1)\) we have \(\mathcal {M}_{0} \subseteq \mathcal {M}_{1}\) and \(\mathcal {E}_1\subseteq \mathcal {E}_0\).

The object \(S_1X\) is obtained by \((\mathcal {E}_1, \mathcal {M}_1)\)-factoring the arrow \(0\longrightarrow X\) as \(0\xrightarrow {\mathcal {E}_1}S_1X\xrightarrow {\mathcal {M}_1}X\). Since \(\mathcal {E}_1\subseteq \mathcal {E}_0\), this shows that \(0\longrightarrow S_1X\) is in \(\mathcal {E}_0\) and so, by the Sator lemma, also \(S_1X\longrightarrow 0\) is in \(\mathcal {E}_0\). Now the object \(R_0S_1X\) is obtained by the \((\mathcal {E}_0, \mathcal {M}_0)\)-factorization of the terminal morphism \(S_1X\longrightarrow 0\) as \(S_1X\xrightarrow {\mathcal {E}_0}R_0S_1X\xrightarrow {\mathcal {M}_0}0\).

By the 3-for-2 property for \(\mathcal {E}_0\) we see that \(R_0S_1X \longrightarrow 0\) lies in \(\mathcal {E}_{0}\cap \mathcal {M}_{0}\), hence it an isomorphism, so that \(R_0S_1 X\cong 0\). The proof that \(S_1R_0X\cong X\) is perfectly dual. This proves (iii).

The proof of (i) goes as follows. The reflection \(R_0R_1 X\) is defined by the \((\mathcal {E}_0,\mathcal {M}_0)\)-factorization \(R_1X\xrightarrow {\mathcal {E}_0}R_0R_1X\xrightarrow {\mathcal {M}_0}0\). Since \(\mathcal {E}_1\subseteq \mathcal {E}_0\) and \(X\longrightarrow R_1X\) is in \(\mathcal {E}_1\) we see that \(X\longrightarrow R_0R_1X\longrightarrow 0\) is already the \((\mathcal {E}_0,\mathcal {M}_0)\)-factorization of \(X\longrightarrow 0\) and so by uniqueness of the factorization we have \(R_0R_1X\cong R_0X\). Finally, the reflection \(R_1R_0 X\) is defined by the \((\mathcal {E}_1,\mathcal {M}_1)\)-factorization \(R_0X\xrightarrow {\mathcal {E}_1}R_1R_0X\xrightarrow {\mathcal {M}_1}0\). Since \(R_0X\longrightarrow 0\) is in \(\mathcal {M}_0\subseteq \mathcal {M}_1\), by the 3-for-2 property we have that \(R_0X\longrightarrow R_1R_0X\) is in \(\mathcal {E}_1\cap \mathcal {M}_1\) and so it is an isomorphism. \(\square \)

Remark 3.20

Notice how, in the proof of the above lemma, one sees that applying \(R_{L_0}\) to the natural morphism \(R_{L_1}X\longrightarrow 0\) we get a natural morphism \(R_{L_1}X\longrightarrow R_{L_0} X\), an so one has a natural transformation \(R_{L_1}\longrightarrow R_{L_0}\). Dually, we have a natural transformation \(S_{U_1}\longrightarrow S_{U_0}\).

Lemma 3.21

Let \(\mathfrak {t}\) be a J-slicing of Open image in new window and let \((L_0,U_0)\) and \((L_1,U_1)\) two slicings of J with \((L_0,U_0)\le (L_1,U_1)\). Then we have natural isomorphisms
$$\begin{aligned} S_{U_0}R_{L_1}\cong R_{L_1}S_{U_0} \end{aligned}$$
Moreover \(S_{U_0}R_{L_1}\) is the fiber of the natural transformation \(R_{L_1}\longrightarrow R_{L_0}\) and \(R_{L_1}S_{U_0}\) is the cofiber of the natural transformation \(S_{U_1}\longrightarrow S_{U_0}\)

Proof

In the same notation as in the proof of Lemma 3.19 we have that \(\mathcal {E}_1\subseteq \mathcal {E}_0\) and \(\mathcal {M}_0\subseteq \mathcal {M}_1\). Since both \((\mathcal {E}_0,\mathcal {M}_0)\) and \((\mathcal {E}_1,\mathcal {M}_1)\) are normal factorization systems, the isomorphisms of Lemma 3.19 give the diagramwhere every square is a pullout. The fact that each class \(\mathcal {E}_i\) is closed under pushout and each \(\mathcal {M}_i\) is closed under pullback now gives that the arrows \(S_0X \longrightarrow F \longrightarrow 0\) and \(0\longrightarrow F \longrightarrow R_1X\) are respectively the \((\mathcal {E}_1, \mathcal {M}_1)\)-factorization of \(S_0X\longrightarrow 0\) and the \((\mathcal {E}_0, \mathcal {M}_0)\)-factorization of \(0\longrightarrow R_1X\), so that \(R_1 S_0 X \cong F \cong S_0 R_1 X\).
To prove the second part of the statement, notice that by definition of normal factorization system associated to the slicing \((L_0,U_0)\) we have a fiber sequenceand the conclusion follows from the natural isomorphism \(R_0R_1X\cong R_0X\). Dually one proves the statement on the cofiber of \(S_{U_0}\longrightarrow S_{U_1}\). \(\square \)

3.1 A tale of intervals

Although a few of the statements we are going to prove hold more generally for arbitrary \({\mathbb {Z}}\)-posets, for the remainder of this section we will restrict our attention to \({\mathbb {Z}}\)-posets which are totally ordered sets.

Definition 3.22

Let J be a poset. An interval in J is a subset \(I\subseteq J\) such that if \(x,y\in I\) and \(x\le z\le y\) in J, then \(z\in I\).

Example 3.23

Let J be a totally ordered \({\mathbb {Z}}\)-poset, and let \(\sim \) be the equivalence relation from Lemma 2.27. For \(i\in J\), let \(I_i\) be the equivalence class of i. Then \(I_i\) is an interval. Namely, id \(x,y\in I_i\) then there exist integers ab with \(i+a\le x\) and \(y\le i+b\) so if \(x\le z\le y\) then \(i+a\le z\le i_b\) and so \(z\sim i\).

Clearly, the intersection of a lower set and an upper set is an interval. Remarkably, in totally ordered sets also the converse is true. Although this is a classical (and easy) result, we recall its proof for completeness.

Lemma 3.24

Let J be a totally ordered set. Then a subset \(I\subseteq J\) is an interval if and only if I can be written as the intersection of an upper set and a lower set.

Proof

Let
$$\begin{aligned} L_I=\bigcup _{x\in I} (-\infty ,x]; \qquad U_I=\bigcup _{x\in I} [x,+\infty ). \end{aligned}$$
Then clearly \(L_I\) is a lower set, \(U_I\) is an upper set and we have \(I\subseteq L_I\cap U_I\). Moreover, if \(y\in L_I\cap U_I\) then there exist \(x_0,x_1\) in I such that \(y\in (-\infty ,x_1]\cap [x_0,+\infty )=[x_0,x_1]\). Since \(x_0\le y\le x_1\) and I is an interval, we have \(y\in I\), and so \(L_I\cap U_I\subseteq I\). \(\square \)

Lemma 3.25

In a totally ordered set, the upper set and the lower set intersecting in a nonempty interval I are uniquely determined by I.

Proof

Let \(I\subseteq J\) be a interval and let
$$\begin{aligned} U_I=\bigcap _{U\supseteq I} U; \qquad L_I=\bigcap _{L\supseteq I} L, \end{aligned}$$
with U and L ranging over the upper sets and the lower sets in J containing I, respectively. Then it is clear that \(I\subseteq U_I\cap L_I\) and we want to show that actually \(I=U_I\cap L_I\) and that if \(I={\tilde{U}}\cap {\tilde{L}}\) then \({\tilde{U}}=U_I\) and \({\tilde{L}}=L_I\). By Lemma 3.24 there exist an upper set \({\tilde{U}}\) and a lower set \({\tilde{L}}\) such that \(I={\tilde{U}}\cap {\tilde{L}}\). By definition of \(U_I\) and \(L_I\) we have \(U_I\subseteq {\tilde{U}}\) and \(L_I\subseteq {\tilde{L}}\). Therefore \(I\subseteq L_I\cap U_I\subseteq {\tilde{L}}\cap {\tilde{U}}=I\) and so \(I=U_I\cap L_I\). Now we want to show that \(U_I={\tilde{U}}\). Since \(U_I\subseteq U_0\) we only need to show that \({\tilde{U}}\subseteq U_I\). Let \(x\in {\tilde{U}}\) and let \(y\in I\). Since J is totally ordered, either \(x\le y\) or \(x\ge y\). In the first case, since \(L_0\) is a lower set, we have \(x\in {\tilde{L}}\) and so \(x\in {\tilde{L}}\cup {\tilde{U}}=I\subseteq U_I\). In the second case, since \(U_I\) is an upper set, we have directly \(x\in U_I\). \(\square \)

By the above lemma, the following definition is well-posed.

Definition 3.26

Let J be a totally ordered \({\mathbb {Z}}\)-poset and let Open image in new window be a J-slicing on a stable \(\infty \)-category Open image in new window . For every nonempty interval \(I=L_I\cap U_I\) in J we setWe also set Open image in new window .

Remark 3.27

The whole of J is an interval, with \(L_J=U_J=J\). From Definition 3.26 we obtain Open image in new window , as expected. Also, every upper set U is an interval, with \(U_U=U\) and \(L_U=J\). So from Definition 3.26 we find that the subcategory of Open image in new window associated to U as an interval is precisely the subcategory Open image in new window associated to U as an upper set. The same happens for lower sets. This shows that the notation introduced in Definition 3.26 is consistent with the notation for J-slicings.

Example 3.28

For every ij in J with \(i\le j\) one has the four intervals (ij), (ij], [ij), [ij] and consequently the four subcategories Open image in new window and Open image in new window of Open image in new window . In particular for every \(i\in J\) we have the interval [ii] consisting of the single element i. To avoid cumbersome notation, we will always write Open image in new window for Open image in new window . The subcategories Open image in new window with i ranging in J are called the slices of the J-slicing \(\mathfrak {t}\).

Definition 3.29

Let \(\mathfrak {t}\) be a J-slicing on Open image in new window . We say that Open image in new window is J-bounded ifSimilarly, we say that Open image in new window is J-left-bounded if Open image in new window and J-right-bounded if Open image in new window .

Remark 3.30

This notion is well known in the classical as well as in the quasicategorical setting: see [1, 18]. In particular, when \(\mathfrak {t}\) is a \(\mathbb {Z}\)-family of t-structures on Open image in new window , then Open image in new window is \(\mathbb {Z}\)-bounded (resp., \(\mathbb {Z}\)-left-bounded, \(\mathbb {Z}\)-right-bounded) if and only if Open image in new window is bounded (resp., left-bounded, right-bounded) with respect to the t-structure \(\mathfrak {t}_0\), agreeing with the classical definition of boundedness as given, e.g., in [1].

Remark 3.31

Since Open image in new window one immediately sees that Open image in new window is J-bounded if and only if Open image in new window is both J-left- and J-right-bounded.

The following remark is the first step towards the definition of factorization of morphisms associated with interval decompositions of J.

Remark 3.32

A nonempty interval in a totally ordered set J is equivalent to the datum of a pair of upper sets \(U_0\) and \(U_1\) with \(U_1\subseteq U_0\), i.e., to the datum of a strictly monotone morphism of posets \([\mathbf 1 ]\longrightarrow \mathcal {O}(J)\). Namely, we have seen that I is equivalent to the datum of an upper set \(U_I\) and a lower set \(L_I\), which are uniquely determined by I. Let us set \(U_0=U_I\) and \(U_1=J\setminus L_I\). Then, since \(\mathcal {O}(J)\) is totally ordered by Remark 3.3, we have either \(U_0\subseteq U_1\) or \(U_1\subseteq U_0\). If \(U_0\subseteq U_1\) then we have \(I=U_0\cap L_I\subseteq U_I\cap L_I=\emptyset \) against the assumption on I. So \(U_1\subseteq U_0\) and \(i\mapsto U_i\) for \(i=0,1\) defines a monotone map from \([\mathbf 1 ]\) to \(\mathcal {O}(J)\). Moreover this map is strictly monotone since we have excluded the possibility \(U_0\subseteq U_1\) and so we can’t have \(U_0=U_1\). By removing the assumption that I is nonempty, we can say that an interval in J is given by a (non necessarily strictly monotone) morphism of posets \([\mathbf 1 ]\longrightarrow \mathcal {O}(J)\). Actually this is not completely accurate, since all constant maps from \([\mathbf 1 ]\) to \(\mathcal {O}(J)\) will correspond to the empty interval. Yet it will be extremely convenient to always think of intervals as monotone maps to \(\mathcal {O}(J)\), so we will systematically adopt this point of view in what follows. In other words we will identify a monotone map \(I:[\mathbf 1 ]\longrightarrow \mathcal {O}(J)\) with the interval \(I=U_0\cap L_1\), where \(I(0)=(L_0,U_0)\) and \(I(1)=(L_1,U_1)\).

Remark 3.33

If \(I:[\mathbf 1 ]\longrightarrow \mathcal {O}(J)\) is an interval in a totally ordered set J, then \([U_0,U_1]\) is an interval in the totally ordered set \({\mathcal {O}}(J)\). It is easy to see that intersecting with I defines a bijection of totally ordered sets
$$\begin{aligned}{}[U_0,U_1]&\longrightarrow {\mathcal {O}}(I)\\ U&\mapsto U\cap I. \end{aligned}$$

Lemma 3.34

Let \(I:[\mathbf 1 ]\longrightarrow \mathcal {O}(J)\) be an interval in J, and let Open image in new window be the corresponding subcategory of Open image in new window , for a given J-slicing. Then the restriction of \(S_{U_0}\) to Open image in new window and the restriction of \(R_{L_1}\) to Open image in new window both take values in Open image in new window .

Proof

We split the proof in two cases. If \(I=\emptyset \) then \(U_0=U_1\) and so for any X in Open image in new window we have \(S_0X\cong S_1X=\mathbf{0 }\). So Open image in new window does take its values in Open image in new window in this case. If \(I\ne \emptyset \), then \(U_1\subseteq U_0\). Since \(S_{0}\) takes values in Open image in new window , we only need to show that it maps Open image in new window into itself. In other words we want to show that if Open image in new window then \(S_0X\xrightarrow {\sim } R_1S_0X\). From the fiber sequencewe see we are reduced to showing that \(S_1S_0X\cong \mathbf{0 }\). Since \(U_1\subseteq U_0\), we have \(S_1S_0X\cong S_1X\). But, since Open image in new window we have \(S_1X\cong \mathbf{0 }\). This concludes the proof in the case \(I\ne \emptyset \). The proof for \(R_{L_1}\) is completely analogous. \(\square \)

By the above lemma and by Lemma 3.21 we can give the following

Definition 3.35

Let \(I:[\mathbf 1 ]\longrightarrow \mathcal {O}(J)\) be an interval in J, and let Open image in new window be a J-slicing on a stable \(\infty \)-category Open image in new window . The functoris defined as the composition \({\mathcal {H}}^I=R_{L_1}S_{U_0}=S_{U_0}R_{L_1}\).

As for the functors \(R_L\) and \(S_U\) we will often implicitly compose \({\mathcal {H}}^I\) with the inclusion Open image in new window and look at it as an endofunctor of Open image in new window . Notice that if I is the empty interval then \({\mathcal {H}}^I\) is the zero functor.

Remark 3.36

By looking at a lower set L and to an upper set U as intervals, the above definition gives \({\mathcal {H}}^L=R_L\) and \({\mathcal {H}}^U=S_U\). In particular we find
$$\begin{aligned} {\mathcal {H}}^I={\mathcal {H}}^{U_0}{\mathcal {H}}^{L_1}={\mathcal {H}}^{L_1}{\mathcal {H}}^{U_0} \end{aligned}$$
and, by Lemma 3.21, \({\mathcal {H}}^I\) is the cofiber of the natural transformation \({\mathcal {H}}^{U_1}\longrightarrow {\mathcal {H}}^{U_0}\).

Remark 3.37

Let \(I,{\tilde{I}}\subseteq J\) two intervals, with \(I\subseteq {\tilde{I}}\). Then
$$\begin{aligned} {\mathcal {H}}^I{\mathcal {H}}^{{\tilde{I}}}={\mathcal {H}}^{{\tilde{I}}}{\mathcal {H}}^I={\mathcal {H}}^I. \end{aligned}$$
Namely, if I is empty, then there is nothing to prove. If I is nonempty, as I is a sub-interval of \({\tilde{I}}\) we have \(U_0\subseteq {\tilde{U}}_0\) and \(L_1\subseteq {\tilde{L}}_1\). Therefore \(({\tilde{L}}_0,{\tilde{U}}_0)\le (L_0,U_0)\le (L_1,U_1)\le ({\tilde{L}}_1,{\tilde{U}}_1)\), and so \(S_{U_0}R_{{\tilde{L}}_1}=R_{{\tilde{L}}_1}S_{U_0}\) by Lemma 3.21 as well as \(R_{L_1}R_{{\tilde{L}}_1}=R_{L_1}\) and \(S_{U_0}S_{{\tilde{U}}_0}=S_{U_0}\) by Lemma 3.19. Therefore,
$$\begin{aligned} {\mathcal {H}}^I{\mathcal {H}}^{{\tilde{I}}}=R_{L_1}S_{U_0}R_{{\tilde{L}}_1}S_{{\tilde{U}}_0}=R_{L_1}R_{{\tilde{L}}_1}S_{U_0}S_{{\tilde{U}}_0}=R_{L_1}S_{U_0}={\mathcal {H}}^I. \end{aligned}$$
The proof that \({\mathcal {H}}^{{\tilde{I}}}{\mathcal {H}}^I={\mathcal {H}}^I\) is similar.

Remark 3.38

If I and \({\tilde{I}}\) are two disjoint intervals in the totally ordered set J then either every element of I is strictly smaller than every element of \({\tilde{I}}\) or vice versa. If we are in the first case, then Open image in new window is right-orthogonal to Open image in new window , i.e., Open image in new window is contractible whenever Open image in new window and Open image in new window . Namely, by the assumption on I and \({\tilde{I}}\) we have \({\tilde{U}}_0\subseteq U_1\) and so Open image in new window . On the other hand, Open image in new window and Open image in new window is right-orthogonal to Open image in new window by definition of t-structure.

Remark 3.39

If I and \({\tilde{I}}\) are two disjoint intervals in the totally ordered set J , then
$$\begin{aligned} {\mathcal {H}}^I{\mathcal {H}}^{{\tilde{I}}}={\mathcal {H}}^{{\tilde{I}}}{\mathcal {H}}^I=0. \end{aligned}$$
Indeed, the statement is trivial if either I or \({\tilde{I}}\) are empty. When they are nonempty, up to exchanging the role of I and \({\tilde{I}}\) we may assume that every element of I is strictly smaller than every element of \({\tilde{I}}\) . Then we have \((L_0,U_0)\le (L_1,U_1)\le ({\tilde{L}}_0,{\tilde{U}}_0)\le ({\tilde{L}}_1,{\tilde{U}}_1)\) and so
$$\begin{aligned} {\mathcal {H}}^I{\mathcal {H}}^{{\tilde{I}}}=R_{L_1}S_{U_0}R_{{\tilde{L}}_1}S_{{\tilde{U}}_0}=R_{L_1}R_{{\tilde{L}}_1}S_{U_0}S_{{\tilde{U}}_0}=R_{L_1}S_{{\tilde{U}}_0}=0, \end{aligned}$$
by lemma 3.19. Similarly one shows that \({\mathcal {H}}^{{\tilde{I}}}{\mathcal {H}}^I=0\).

The above Remarks 3.36 and 3.39 are actually two particular instances of the following general result. The proof is completely analogous to those in the remarks above, and so it is omitted.

Proposition 3.40

Let I and \({\tilde{I}}\) be two intervals in a \({\mathbb {Z}}\)-toset, and let Open image in new window be a J-slicing on a stable \(\infty \)-category Open image in new window . Then
$$\begin{aligned} {\mathcal {H}}^I{\mathcal {H}}^{{\tilde{I}}}={\mathcal {H}}^{{\tilde{I}}}{\mathcal {H}}^I={\mathcal {H}}^{I\cap {\tilde{I}}}. \end{aligned}$$

We conclude this section with a notational convention, which will be useful later.

Notation 3.41

Consistently with the notation from Example 3.28, for every i in J we write \({\mathcal {H}}^i\) for \({\mathcal {H}}^{[i,i]}\).

4 Interval decompositions and towers

Remark 4.1

For the whole section \((J,\le )\) will be a fixed totally ordered \(\mathbb {Z}\)-poset and Open image in new window will be a J-slicing.

Definition 4.2

A \((k+2)\)-fold interval decomposition of J is a morphism of posets \(I_{[\mathbf k ]}:[\mathbf k ]\longrightarrow \mathcal {O}(J)\).

Notation 4.3

When no ambiguity is possible, the image of \(j\in \{0,1,\ldots ,k\}\) via \(I_{[\mathbf k ]}\) will be denoted simply by \((L_j,U_j)\). For every \(j=0,\ldots ,k+1\) the interval \(I_j=U_{j-1}\cap L_j\) is called the j-th interval in the decomposition, with the convention that \(U_{-1}=J=L_{k+1}\).

The factorization system associated with \((L_j,U_j)\) will be denoted by \((\mathcal {E}_j,\mathcal {M}_j)\). Notice that, since \(I_{[\mathbf k ]}\) is a morphism of posets we have \(\mathcal {E}_{j+1}\subseteq \mathcal {E}_j\) and \(\mathcal {M}_{j+1}\supseteq \mathcal {M}_j\).

This implies that the composition Open image in new window is a k-fold factorization system; in other words

Lemma 4.4

Let \((\mathcal {E}_j, \mathcal {M}_j)\) as above. Then every arrow \(f:X\longrightarrow Y\) in Open image in new window can be uniquely factored into a composition
$$\begin{aligned} X \xrightarrow {\mathcal {E}_{k}} Z_{k} \xrightarrow {\mathcal {E}_{k-1}\cap \mathcal {M}_{k}} Z_{k-1}\longrightarrow \cdots \longrightarrow Z_{1} \xrightarrow {\mathcal {E}_{0}\cap \mathcal {M}_{1}} Z_{0} \xrightarrow {\mathcal {M}_{0}} Y. \end{aligned}$$

Proof

Since \((\mathcal {E}_k,\mathcal {M}_k)\) is a factorization system, we have a (unique) factorization \(X \xrightarrow {\mathcal {E}_{k}} Z_{k} \xrightarrow {\mathcal {M}_{k}} Y\). Since \((\mathcal {E}_{k-1},\mathcal {M}_{k-1})\) is a factorization system, we can (uniquely) factor \(Z_k\longrightarrow Y\) as \(Z_k \xrightarrow {\mathcal {E}_{k-1}} Z_{k-1} \xrightarrow {\mathcal {M}_{k-1}} Y\). Since \(\mathcal {M}_{k-1}\subseteq \mathcal {M}_k\), the morphism \(Z_{k-1} \longrightarrow Y\) is also in \(\mathcal {M}_k\) and so, by the 3-for-2 property, also \(Z_k \longrightarrow Z_{k-1}\) is in \(\mathcal {M}_k\). Therefore \(Z_k \xrightarrow {\mathcal {E}_{k-1}} Z_{k-1}\) is in \(\mathcal {E}_{k-1}\cap \mathcal {M}_k\). then one concludes iterating this argument. \(\square \)

Definition 4.5

The sequence of morphism in the factorization of \(f:X\longrightarrow Y\) in Lemma 4.4 is called the \(I_{[\mathbf k ]}\)-tower of f, and it is denoted Open image in new window , or simply by Open image in new window when the interval decomposition \(I_{[\mathbf k ]}\) is clear from the context.

Remark 4.6

When Open image in new window is the stable \(\infty \)-category of spectra and \(X\longrightarrow 0\) and \(0\longrightarrow X\) are the terminal and the initial morphism of X, respectively, the above notation and construction is in line with the classical Postnikov and Whitehead towers of X, i.e., with the sequences
$$\begin{aligned}&X\longrightarrow \cdots \longrightarrow R_2X\longrightarrow R_1X\longrightarrow R_0X \longrightarrow 0\\&0\longrightarrow \cdots \longrightarrow S_2X\longrightarrow S_1X\longrightarrow S_0X \longrightarrow X \end{aligned}$$
of factorizations obtained from the (stable image of) the n-connected factorization system of [14].

Remark 4.7

If both X and Y are in Open image in new window , then the morphism \(X\longrightarrow Z_{k}\) in Open image in new window is an isomorphism. Indeed, by construction the morphism \(Z_{k}\longrightarrow Y\) is in \(\mathcal {M}_{k}\). Since both \(X\longrightarrow 0\) and \(Y\longrightarrow 0\) are in \(\mathcal {M}_{k}\) then also \(X\longrightarrow Y\) is in \(\mathcal {M}_{k}\) by 3-for-2, and so also \(X\longrightarrow Z_{k}\) in in \(\mathcal {M}_{k}\) again by 3-for-2. But by construction \(X\longrightarrow Z_{k}\) is in \(\mathcal {E}_{k}\), so it is an isomorphism. By the same argument one sees that if both X and Y are in Open image in new window , then the morphism \(Z_{0}\longrightarrow Y\) is an isomorphism.

Corollary 4.8

Let \(I_{[\mathbf k ]}\) be a \((k+2)\)-fold interval decomposition of J. Then for any object Y in Open image in new window , the tower Open image in new window is
$$\begin{aligned} 0 \xrightarrow {\mathcal {E}_{k}} {\mathcal {H}}^{U_k}Y \xrightarrow {\mathcal {E}_{k-1}\cap \mathcal {M}_{k}} {\mathcal {H}}^{U_{k-1}}Y\longrightarrow \cdots \longrightarrow {\mathcal {H}}^{U_1}Y \xrightarrow {\mathcal {E}_{0}\cap \mathcal {M}_{1}} {\mathcal {H}}^{U_0}Y \xrightarrow {\mathcal {M}_{0}} Y. \end{aligned}$$
Moreover, the arrows \(f_j:{\mathcal {H}}^{U_j}Y\longrightarrow {\mathcal {H}}^{U_{j-1}}Y\) in Open image in new window are such that Open image in new window , for \(j=0,\ldots ,k+1\).

Proof

From the \((k+2)\)-fold factorization
$$\begin{aligned} 0 \xrightarrow {\mathcal {E}_{k}} Y_{k} \xrightarrow {\mathcal {E}_{k-1}\cap \mathcal {M}_{k}} Y_{k-1}\longrightarrow \cdots \longrightarrow Y_{1} \xrightarrow {\mathcal {E}_{0}\cap \mathcal {M}_{1}} Y_{0} \xrightarrow {\mathcal {M}_{0}} Y, \end{aligned}$$
and from the fact that \(\mathcal {E}_{0}\supseteq \mathcal {E}_{1} \supseteq \dots \supseteq \mathcal {E}_{k}\) and each class \(\mathcal {E}_{j}\) is closed for composition, we see that \(0\longrightarrow Y_{j}\longrightarrow Y\) is the \((\mathcal {E}_j,\mathcal {M}_j)\)-factorization of \(0\longrightarrow Y\) and so \(Y_{j}=S_{U_j}Y={\mathcal {H}}^{U_j}Y\). One concludes by Lemma 3.21. \(\square \)

The above corollary motivates the following

Definition 4.9

Let \(f:X\longrightarrow Y\) be a morphism in Open image in new window . A \(I_{[\mathbf k ]}\)-weaved factorization for f is a factorization of f of the form
$$\begin{aligned} X \xrightarrow {f_{k+1}} Z_{k} \xrightarrow {f_{k}} Z_{k-1}\longrightarrow \cdots \longrightarrow Z_{1} \xrightarrow {f_{1}} Z_{0} \xrightarrow {f_{0}} Y. \end{aligned}$$
with Open image in new window , for \(j=0,\ldots ,k+1\).

Remark 4.10

If we call Open image in new window the class of \(I_{[\mathbf k ]}\)-weaved factorizations for a morphism \(f :X\longrightarrow Y\), it is immediate to see that we have a canonical identificationMoreover Corollary 4.8 precisely states that Open image in new window is an \(I_{[\mathbf k ]}\)-weaved factorization, and so we have the existence of \(I_{[\mathbf k ]}\)-weaved factorizations for arbitrary morphisms. The remainder of this section is devoted to proving the uniqueness of \(I_{[\mathbf k ]}\)-weaved factorizations. This reduces to proving the uniqueness of \(I_{[\mathbf k ]}\)-weaved factorizations for initial morphism, i.e., to showing that the only possible \(I_{[\mathbf k ]}\)-weaved factorization of \(0\longrightarrow Y\) is its tower.

Lemma 4.11

In the above notation, let \(f:X\longrightarrow Y\) be a morphism in Open image in new window . If X is in Open image in new window and \({{\,\mathrm{cofib}\,}}(f)\) is in Open image in new window then \(0\longrightarrow X\xrightarrow {f}Y\) is the \((\mathcal {E}_j,\mathcal {M}_j)\)-factorization of the initial morphism \(0\longrightarrow Y\) and Y is in Open image in new window . In particular f is in \(\mathcal {E}_{j-1}\cap \mathcal {M}_j\).

Proof

Since X is in Open image in new window , the morphism \(0\longrightarrow X\) is in \(\mathcal {E}_j\), and so to show that \(0\longrightarrow X\xrightarrow {f} Y\) is the \((\mathcal {E}_j,\mathcal {M}_j)\)-factorization of \(0\longrightarrow Y\) we are reduced to showing that \(f:X\longrightarrow Y\) is in \(\mathcal {M}_j\). Since \({{\,\mathrm{cofib}\,}}(f)\) is in Open image in new window , we have in particular that \({{\,\mathrm{cofib}\,}}(f)\longrightarrow 0\) is in \(\mathcal {M}_j\) and so \(0\longrightarrow {{\,\mathrm{cofib}\,}}(f)\) is in \(\mathcal {M}_j\) by the Sator lemma. Then we have a homotopy pullback diagramand so f is in \(\mathcal {M}_j\) by the fact that \(\mathcal {M}_j\) is closed under pullbacks. To show also that \(f\in \mathcal {E}_{j-1}\) let \(X\longrightarrow T\longrightarrow Y\) be the \((\mathcal {E}_{j-1},\mathcal {M}_{j-1})\)-factorization of f. Then, since \(\mathcal {E}_j\subseteq \mathcal {E}_{j-1}\), \(0\longrightarrow T\longrightarrow Y\) is the \((\mathcal {E}_{j-1},\mathcal {M}_{j-1})\)-factorization of \(0\longrightarrow Y\). So, by the normality of \((\mathcal {E}_{j-1},\mathcal {M}_{j-1})\) we get the diagramwhere all the squares are pullouts, and where we have used the Sator lemma, the fact that \({{\,\mathrm{cofib}\,}}(f)\longrightarrow 0\) is in \(\mathcal {M}_j\), that the classes \(\mathcal {E}\) are closed for pushouts while the classes \(\mathcal {M}\) are closed for pullbacks, and the 3-for-2 property for both classes. Since by hypothesis \(0\longrightarrow {{\,\mathrm{cofib}\,}}(f)\) is in \(\mathcal {E}_{j-1}\), we see that \(V=0\) and so \(T=Y\). Therefore, Open image in new window and \(f\in \mathcal {E}_{j-1}\cap \mathcal {M}_j\). \(\square \)

Corollary 4.12

Let Y an object in Open image in new window and let
$$\begin{aligned} 0 \xrightarrow {f_{k+1}} Y_{k} \xrightarrow {f_k}Y_{{k-1}}\longrightarrow \cdots \longrightarrow Y_{1} \xrightarrow {f_1} Y_{0} \xrightarrow {f_0} Y, \end{aligned}$$
be an \(I_{[\mathbf k ]}\)-weaved factorization for \(0\longrightarrow Y\). Then Open image in new window

Proof

By uniqueness of the k-fold factorization we only need to prove that \(f_j\in \mathcal {E}_{i_{k-1}}\cap \mathcal {M}_{i_k}\), which is immediate by repeated application of Lemma 4.11. \(\square \)

Remark 4.13

It’s an unavoidable temptation to think of the \(I_{[\mathbf k ]}\)-weaved factorization of a morphism f as of its tower Open image in new window . As the following counterexample shows, when f is not an initial morphism this is in general not true. Let \(J=\mathbb {Z}\), let \(k=0\) and let \(I_{[\mathbf 0 ]}:[\mathbf 0 ]\longrightarrow \mathcal {O}(\mathbb {Z})\) be the slicing \(U_0=[0,+\infty )\). Now take a morphism \(f:X\longrightarrow Y\) between two elements in Open image in new window . The object \({{\,\mathrm{cofib}\,}}(f)\) will lie in Open image in new window , since \(\mathcal {E}_0[-1]\) is closed for pushouts, but in general it will not be an element in Open image in new window . In other words, we will have, in general, a nontrivial \((\mathcal {E}_0,\mathcal {M}_0)\)-factorization of the initial morphism \(0\longrightarrow {{\,\mathrm{cofib}\,}}(f)\), i.e., a nontrivial tower Open image in new window . Pulling this back along \(Y\longrightarrow {{\,\mathrm{cofib}\,}}(f)\) we obtain the \(I_{[\mathbf 0 ]}\)-weaved factorization \(X\xrightarrow {f_2} Z\xrightarrow {f_1} Y\) of f, and this factorization will be nontrivial since its pushout is nontrivial. It follows that \((f_2,f_1)\), cannot be Open image in new window , which is the \((\mathcal {E}_0,\mathcal {M}_0)\)-factorization of f. Indeed, by the 3-for-2 property of \(\mathcal {M}_0\), the morphism f is in \(\mathcal {M}_0\), so its \((\mathcal {E}_0,\mathcal {M}_0)\)-factorization is trivial.

Remark 4.14

Let \(I_{[\mathbf k' ]}:[\mathbf k' ]\longrightarrow {\mathcal {O}}(J)\) be a refinement of an interval decomposition \(I_{[\mathbf k ]}:[\mathbf k ]\longrightarrow {\mathcal {O}}(J)\). This means that for every \(i=0,\ldots ,k+1\) we have an interval decomposition \(I_{[\mathbf{k }_i]}:[\mathbf{k }_i]\longrightarrow {\mathcal {O}}(I_{[\mathbf k ];i})\), where \(I_{[\mathbf k ];i}\) denotes the i-th interval in the subdivision \(I_{[\mathbf k ]}\). By the pasting law for pullouts it is immediate to see that for any morphism \(f:X \longrightarrow Y\) we have a canonical identification

4.1 Bridgeland slicings

Definition 4.15

A J-slicing Open image in new window of a stable \(\infty \)-category Open image in new window is called discrete if for any object X in Open image in new window one has \({\mathcal {H}}^i(X)=\mathbf{0 }\) for every i in J if and only if \(X=\mathbf{0 }\). A discrete J-slicing is said to be of finite type if for any object X one has \({\mathcal {H}}^i(X)\ne \mathbf{0 }\) only for finitely many elements \(i\in J\).

Example 4.16

A finite type discrete \({\mathbb {Z}}\)-slicing on Open image in new window is precisely the datum of a bounded t-structure on Open image in new window .

Suppose now that \(\mathfrak {t}\) is of finite type, so that for each Open image in new window one has \({\mathcal {H}}^i(X)=\mathbf{0 }\) but for a finite set \(\{ i_1^X< \cdots < i_{k_X}^X \} \subseteq J\) of indices i, depending on X. We can then build up a \((k+2)\)-fold interval decomposition \(I_{[ \mathbf{k }_X ]}^X\), depending on the object X, by setting \(U_j^X=(i_j, +\infty )\). As we are assuming J to be totally ordered, we have \(L_j^X=(-\infty , i_j]\). The next proposition shows that the tower of the initial morphism \(\mathbf{0 } \longrightarrow X\) associated to this interval decomposition is indeed the “finest one”.

Proposition 4.17

Let Open image in new window be a J-slicing of finite type and let X an object of Open image in new window . Then for all j we have
$$\begin{aligned} {\mathcal {H}}^{I_j^X}(X)={\mathcal {H}}^{i_j^X}(X) \qquad \text { and }\qquad {\mathcal {H}}^{(i_{j-1}^X,i_{j}^X)}(X)=\mathbf{0 }. \end{aligned}$$

Proof

Let us write \(i_j\), \(I_j\) and \(S_j\) for \(i_j^X\), \(I_j^X\) and \(S_{U_j}^X\), respectively. Now, for each \(\varphi \in J\), using By, Proposition 3.40 we have
$$\begin{aligned} {\mathcal {H}}^{i}({\mathcal {H}}^{(i_{j-1}, i_{j})}X)= {\left\{ \begin{array}{ll} {\mathcal {H}}^{i}X &{}\quad \text {if } i\in (i_{j-1},i_{j})\\ \mathbf{0 } &{}\quad \text {otherwise} \end{array}\right. }. \end{aligned}$$
But \({\mathcal {H}}^iX=\mathbf{0 }\) for \(i \not = i_1 , \ldots , i_{k_X}\), and so \({\mathcal {H}}^{i}({\mathcal {H}}^{(i_{j-1}, i_{j})}X)=\mathbf{0 }\) for all \(i \in J\). Since \(\mathfrak {t}\) is discrete, this gives \({\mathcal {H}}^{(i_{j-1}, i_{j})}X= \mathbf{0 }\), proving the second part of the statement. To prove the first part, recall that by Lemma 3.21, \({\mathcal {H}}^{I_j}(X)\) is the cofiber of \(S_{j}X\longrightarrow S_{j-1}X\), while \({\mathcal {H}}^{i_j}(X)\) is the cofiber of \(S_jX\longrightarrow S_{[i_j,+\infty )}X\). So we are reduced to show that \(S_{j-1}X=S_{[i_j,+\infty )}X\). Since \(i_{j} > i_{j-1}\) we have \([i_{j},+\infty )\subseteq (i_{j-1},+\infty )=U_{j-1}^X\) and so, by Lemma 3.19, \(S_{[i_j,+\infty )}X=S_{[i_j,+\infty )}S_{j-1}X\). Thus, we are reduced to show that \(S_{j-1}X=S_{[i_j,+\infty )}S_{j-1}X\), i.e., equivalently, that \(R_{(-\infty ,i_j)}S_{j-1}X=\mathbf{0 }\). This is immediate as \( R_{(-\infty ,i_j)}S_{j-1}X={\mathcal {H}}^{(i_{j-1}, i_j)}X\). \(\square \)

In particular the above tells us that, writing \(\varphi _j\) for \(i_{k_X-j}\), the cofiber of the j-th morphism of Open image in new window is Open image in new window . In other words, these towers are weaved factorizations with cofibers in the subcategories Open image in new window and so they correspond to the Harder-Narasimhan filtrations from [5]. That is, Bridgeland’s slicings (in their generalized version from [9]) are precisely the slicings of finite type in our sense. We show this in detail below.

Definition 4.18

A BridgelandJ-slicing on Open image in new window is a collection Open image in new window of full extension closed sub-\(\infty \)-subcategories satisfying:
  1. (i)

    Open image in new window for each \(\varphi \in J\);

     
  2. (ii)

    orthogonality: Open image in new window is contractible for each Open image in new window , Open image in new window for \(\varphi > \psi \) in J;

     
  3. (iii)
    for each object Open image in new window there is a finite set \(\{ \varphi _1> \cdots > \varphi _n \}\) and a factorization of the initial morphism \(\mathbf{0 } \longrightarrow X\)
    $$\begin{aligned} \mathbf{0 }=X_0 \xrightarrow {\alpha _1} \cdots \xrightarrow {\alpha _n} X_n=X \end{aligned}$$
    with Open image in new window for all \(i = 1, \ldots , n\).
     

Notation 4.19

For \(\mathcal {S}\) a subcategory of Open image in new window , we write \(\langle \mathcal {S}\rangle \) for the smallest extension closed full subcategory of Open image in new window containing \(\mathcal {S}\). If \(M \subseteq J\) is a subset and Open image in new window is a Bridgeland J-slicing on Open image in new window , we denote Open image in new window the extension-closed subcategory generated by Open image in new window with \(\varphi \in M\), i.e., we set

Remark 4.20

Set \(\langle \mathcal {S}\rangle _0=\mathbf{0 }\), define \(\langle \mathcal {S}\rangle _1\) as the full subcategory of Open image in new window generated by \(\mathcal {S}\) and \(\mathbf{0 }\), and define inductively \(\langle \mathcal {S}\rangle _n\) as the full subcategory of Open image in new window on those objects X which fall into a homotopy fiber sequencewith \(h,k\ge 1\), \(X_h\) in \(\langle \mathcal {S}\rangle _h\), \(X_k\) in \(\langle \mathcal {S}\rangle _k\) and \(h+k=n\). One clearly has
$$\begin{aligned} \langle \mathcal {S}\rangle _0\subseteq \langle \mathcal {S}\rangle _1 \subseteq \langle \mathcal {S}\rangle _2\subseteq \cdots \subseteq \langle \mathcal {S}\rangle . \end{aligned}$$
Moreover \(\bigcup _n \langle \mathcal {S}\rangle _n\) is clearly extension closed, so that
$$\begin{aligned} \langle \mathcal {S}\rangle =\bigcup _n \langle \mathcal {S}\rangle _n. \end{aligned}$$

Lemma 4.21

Let \(\mathcal {S}_1,\mathcal {S}_2\) be two subcategories of Open image in new window with Open image in new window , i.e., such that Open image in new window is contractible for any \(X\in \mathcal {S}_1\) and any \(Y\in \mathcal {S}_2\). Then Open image in new window and Open image in new window , and so Open image in new window

Proof

By Remark 4.20, to prove the first statement we are reduced to show that, if \(X\in \mathcal {S}_1\) and \(Y\in \langle \mathcal {S}_2\rangle _n\) then Open image in new window is contractible. We prove this by induction on n. For \(n=0,1\) there is nothing to prove by the assumption Open image in new window . For \(n\ge 2\), consider a fiber sequence \(Y_h\longrightarrow Y\longrightarrow Y_k\) with \(1\le h,k\) and \(h+k=n\) as in Remark 4.20. Since Open image in new window preserves homotopy fiber sequences, we get a homotopy fiber sequence of \(\infty \)-groupoidsBy the inductive hypothesis both Open image in new window and Open image in new window are contractible, so Open image in new window also is. The proof of the second statement is perfectly dual, due to the fact that in Open image in new window every fiber sequence is also a cofiber sequence, and Open image in new window transforms a cofiber sequence into a fiber sequence. \(\square \)

Lemma 4.22

Let (LU) be a slicing of J, and let Open image in new window and Open image in new window be defined according to Notation 4.19. Then Open image in new window .

Proof

Since by definition Open image in new window for \(\varphi >\psi \), the statement immediately follows from Lemma 4.21. \(\square \)

Lemma 4.23

In the above hypothesis and notation, every object Y of Open image in new window sits into a homotopy fiber sequence \(Y_{U}\longrightarrow Y\longrightarrow Y_{L}\) with Open image in new window and Open image in new window .

Proof

Let
$$\begin{aligned} \mathbf{0 }=X_0 \xrightarrow {\alpha _1} X_1\cdots \xrightarrow {\alpha _{{\bar{\imath }}}}X_{{\bar{\imath }}}\xrightarrow {\alpha _{{\bar{\imath }}+1}}X_{{\bar{\imath }}+1}\xrightarrow {}\cdots \xrightarrow {\alpha _n} X_n=X \end{aligned}$$
a factorization of the initial morphism \(\mathbf{0 } \longrightarrow X\) with Open image in new window for all \(i = 1, \cdots , n\), with \(\varphi _i>\varphi _{i+1}\) and with \(\varphi _{{\bar{\imath }}}\in U\) and \(\varphi _{{\bar{\imath }}+1}\in L\) (with \({\bar{\imath }}=-1\) or n when all of the \(\varphi _i\) are in L or in U, respectively). Consider the pullout diagramtogether with the
$$\begin{aligned} \mathbf{0 }=X_0 \xrightarrow {\alpha _1} X_1\cdots \xrightarrow {\alpha _{{\bar{\imath }}}}X_{{\bar{\imath }}} \end{aligned}$$
and
$$\begin{aligned} X_{{\bar{\imath }}}\xrightarrow {\alpha _{{\bar{\imath }}+1}}X_{{\bar{\imath }}+1}\xrightarrow {}\cdots \xrightarrow {\alpha _n} X_n=X. \end{aligned}$$
The first factorization shows that Open image in new window while the second factorization shows that Open image in new window . \(\square \)

Lemma 4.24

In the above hypothesis and notation, one has Open image in new window and Open image in new window .

Proof

Since the shift functor commutes with pushouts, if an object X is obtained by iterated extensions by objects in Open image in new window with \(\varphi \in U\), then X[1] is obtained by iterated extensions by objects in Open image in new window with \(\varphi \in U\). In other words, X[1] is an object in Open image in new window . Since U is an upper set, \(U+1\subseteq U\) and so Open image in new window . This proves that Open image in new window . The proof for Open image in new window is perfectly analogous. \(\square \)

The above Lemmas together give the following

Proposition 4.25

In the above hypothesis and notation, the map Open image in new window defines a J-slicing of Open image in new window , i.e., \(\mathfrak {t}\) is a \({\mathbb {Z}}\)-equivariant map of posets Open image in new window .

Proof

Lemmas 4.224.24 together precisely say that Open image in new window is a t-structure on Open image in new window . Equivariancy of the map is the fact that, as remarked in the proof of Lemma 4.24, one has Open image in new window . Finally, if \((L_0,U_0)\le (L_1,U_1)\) then we have \(U_1\subseteq U_0\) and so Open image in new window , which shows that the map \(\mathfrak {t}\) is a morphism of posets. \(\square \)

Proposition 4.26

Let J be a totally ordered \({\mathbb {Z}}\)-poset and let Open image in new window be a stable \(\infty \)-category. Then we have a bijection

Proof

The only thing left to be proven is that the above construction actually produces a discrete slicing of finite type. This is actually immediate once one realizes that the factorization \(\mathbf{0 }=X_0 \xrightarrow {\alpha _1} \cdots \xrightarrow {\alpha _n} X_n=X\) of the initial morphism \(0\longrightarrow X\) provided by the definition of Bridgeland slicing is actually the weaved factorization corresponding to the interval decomposition of J associated with the decreasing sequence \(\varphi _1> \cdots > \varphi _n\). One then directly sees that the two constructions indicated in the statement of the proposition are inverse each other. \(\square \)

Remark 4.27

Using the orthogonality condition Open image in new window for \(\varphi >\psi \), is not hard to prove by induction on the length of the factorizations that Definition 4.18 actually implies uniqueness of Bridgeland factorizations
$$\begin{aligned} \mathbf{0 }=X_0 \xrightarrow {\alpha _1} \cdots \xrightarrow {\alpha _n} X_n=X \end{aligned}$$
of initial morphisms \(0\longrightarrow X\). As a consequence one has well defined functorsgiven by
$$\begin{aligned} {\mathcal {H}}^\varphi _B(X)={\left\{ \begin{array}{ll} {{\,\mathrm{cofib}\,}}(\alpha _i) &{}\text { if } \varphi =\varphi _i\\ \mathbf{0 }&{}\text { otherwise} \end{array}\right. } \end{aligned}$$
As it is natural to expect, in the correspondence given by Proposition 4.26 the functor \({\mathcal {H}}^\varphi _B\) is identified with the functor \({\mathcal {H}}^\varphi \) associated with the discrete J-slicing. Moreover, one easily sees thatso that the correspondence of Proposition 4.26 can be defined sending the pair Open image in new window into Open image in new window , with inverse

5 Hearts of J-slicings

I watched a snail crawl along the edge of a straight razor. That’s my dream. That’s my nightmare. Crawling, slithering, along the edge of a straight razor...and surviving.

Col. W. E. Kurtz

Recall the equivalence relation \(x\sim y\) if and only if there are integers \(a,b \in \mathbb {Z}\) with \(a\le b\) such that \(x + a \le y \le x + b\) on a \(\mathbb {Z}\)-toset J from Lemma 2.27.

Lemma 5.1

The following are equivalent:
  1. (i)

    the \(\mathbb {Z}\)-toset J consists of a single equivalence class with respect to the equivalence relation \(\sim \);

     
  2. (ii)

    there exists an interval I in J such that the map \(\varphi :(n,x)\mapsto x+n\) is an isomorphism of \(\mathbb {Z}\)-tosets \(\mathbb {Z}\times _{\text {lex}}I\xrightarrow {\sim } J\) (where the \(\mathbb {Z}\)-action on I is the trivial one and the \(\mathbb {Z}\)-action on \(\mathbb {Z}\) is the translation).

     

Proof

That (i) implies (ii) is an immediate consequence of 2.29. To prove the converse implication notice that since \(\varphi \) is surjective every element in J is equivalent to an element in I. So we are reduced to show that all elements in I are equivalent each other. Let \(x,y\in I\). We can assume \(x\le y\). Since J is totally ordered we have either \(y\le x+1\) or \(x+1\le y\). In the latter case we have \(x\le x+1\le y\) and so \(x+1\in I\), since I is an interval. But then \(\varphi (1,x)=\varphi (0,x+1)\) against the hypothesis on \(\varphi \). So we are left with \(x\le y\le x+1\) which implies \(x\sim y\). \(\square \)

Definition 5.2

Let J be a \(\mathbb {Z}\)-toset. A heart for J is an interval \(J^\heartsuit \subseteq J\) such that \(\varphi :(n,x)\mapsto x+n\) is an isomorphism of \(\mathbb {Z}\)-tosets \(\mathbb {Z}\times _{\text {lex}}J^\heartsuit \xrightarrow {\sim } J\).

Remark 5.3

Of course, not every \(\mathbb {Z}\)-toset has a heart. It is easy to see that J has an heart if and only if there is a morphism of \(\mathbb {Z}\)-tosets \(\pi _\heartsuit :J \longrightarrow \mathbb {Z}\). An heart of J is given by \(\pi _\heartsuit ^{-1}(0)\) in this case.

Remark 5.4

It is immediate from the definition that \(J^\heartsuit \) is a heart of J if and only if \(J^\heartsuit +n\) is a heart of J, for every \(n\in \mathbb {Z}\).

Example 5.5

If \(J=\mathbb {Z}\), with the standard \(\mathbb {Z}\)-toset structure, the hearts of J are the singletons \(\{n\}\) with \(n\in \mathbb {Z}\). In particular \(\{0\}\) is the standard heart of \(\mathbb {Z}\), and all the other hearts are shifts of this. If \(J={\mathbb {R}}\), with the standard \(\mathbb {Z}\)-toset structure, then the hearts of J are the intervals of the form \([x,x+1)\) and those of the form \((x,x+1]\), with \(x\in {\mathbb {R}}\).

Example 5.6

Let \((J, \le )\) be a totally ordered \({\mathbb {Z}}\)-poset, and let \(\sim \) be the equivalence relation from Lemma 2.27. For every \(i\in J\) let \(I_i\) be the equivalence class of i. This is an interval in J, see Example 3.23. Moreover, by Lemma 2.29, \(I_i\) has a heart precisely when i is not a fixed point of the \(\mathbb {Z}\)-action.

Lemma 5.7

Let \(I:[\mathbf 1 ]\longrightarrow \mathcal {O}(J)\) be a heart of J. Then \(I(1)=I(0)+1\), i.e., \(U_1=U_0+1\) (equivalently, \(L_1=L_0+1\)).

Proof

Assume \(U_1\nsubseteq U_0+1\). Then there exists an element x in \(U_1\cap (L_0+1)\). Since I is a heart, there exists an element y in I and an integer n such that \(x=y+n\). If \(n\ge 1\) we have \(y+n\in U_0+n\subseteq U_0+1\) and so \(x\in (U_0+1)\cap (L_0+1)\), which is impossible. If \(n\le 0\) we have \(y+n\in L_1+n\subseteq L_1\) and so \(x\in L_1\cap U_1\) which again is impossible. Therefore \(U_1\subseteq U_0+1\). Now assume \(U_0+1\nsubseteq U_1\). Then there exists an element \(x\in (U_0+1)\cap L_1\). Let \(y=x-1\). Then \(y\in U_0\cap (L_1-1)\subseteq U_0\cap L_1=I\). Since \(U_0+1\subseteq U_0\) we also have \(x\in I\), and so \(\varphi (-1,x)=\varphi (0,y)\), which is impossible. Therefore \(U_1= U_0+1\). \(\square \)

Definition 5.8

Let \(J^\heartsuit \subseteq J\) be a heart of J and let Open image in new window be a J-slicing on a stable \(\infty \)-category Open image in new window . The subcategory Open image in new window of Open image in new window will be called a heart of the J-slicing \(\mathfrak {t}\) and will be denoted Open image in new window .

Notation 5.9

We denote the canonical projection to the heart as Open image in new window ; see Definition 3.35.

Example 5.10

We have seen in Example3.14 that a a \(\mathbb {Z}\)-slicing on a stable \(\infty \)-category Open image in new window is the same thing as the datum of a t-structure Open image in new window on Open image in new window . The standard heart Open image in new window is called the heart of the t-structure \(\mathfrak {t}\). The projection to the heart is the functor \({\mathcal {H}}^0\); see Notation 3.41.

From 4.8 we immediately get the following

Proposition 5.11

Let Open image in new window be a bounded t-structure on a stable \(\infty \)-category Open image in new window , and let Open image in new window be its standard heart. Then \(\mathfrak {t}\) is completely determined by the functors Open image in new window (and so by the functor \({\mathcal {H}}^0\) alone). More precisely, Open image in new window is the full subcategory of Open image in new window on the objects Y such that \({\mathcal {H}}^jY=0\) for any \(j< 0\), while Open image in new window is the full subcategory of Open image in new window on those objects Y for which \({\mathcal {H}}^jY=0\) for any \(j\ge 0\).

Remark 5.12

There is a rather evocative pictorial representation of the heart Open image in new window of an \({\mathbb {R}}\)-slicing, manifestly inspired by [5]: if we depict Open image in new window and Open image in new window as contiguous half-planes (refer to Fig. 1) then the action of the shift functor can be represented as an horizontal shift, and the closure properties of the two classes Open image in new window under positive and negative shifts are a direct consequence of the shape of these areas. With these notations, an object Z is in the heart Open image in new window if it lies in a “boundary region”, i.e. if it lies in Open image in new window , but \(Z[-1]\) lies in Open image in new window .

Fig. 1

Heart of an \({\mathbb {R}}\)-slicing

Let now \(J^\heartsuit \subseteq J\) be a heart, and let Open image in new window be the corresponding subcategory of C, relative to a given J-slicing \(\mathfrak {t}\). Writing I as \(I:[\mathbf 1 ]\longrightarrow \mathcal {O}(J)\), for any \(n\in \mathbb {Z}\) and any \(k\ge 0\) we can consider the interval decomposition \(I_{[\mathbf k ]}:[\mathbf k ]\longrightarrow \mathcal {O}(J)\) defined by setting \(I_{[\mathbf k ]}(j)=I(0)+j+n\) for \(0\le j\le k\). By Lemma 5.7, this corresponds to the collection of n contiguous intervals \(I+n,I+n+1,\ldots ,I+n+k-1\subseteq J\). The corresponding subcategories of Open image in new window will be Open image in new window , Open image in new window ,..., Open image in new window .

The existence and uniqueness of \(I_{[\mathbf k ]}\)-weaved factorizations then specializes to the following statement

Proposition 5.13

Let \(f:X\longrightarrow Y\) be a morphism in Open image in new window . Then for any integer n and any positive integer k there exists a unique factorization
$$\begin{aligned} X \xrightarrow {f_{n+k}} Z_{n+k-1} \xrightarrow {f_{n+k-1}}Z_{n+k-2}\longrightarrow \cdots \longrightarrow Z_{n+1} \xrightarrow {f_{n}} Z_{n} \xrightarrow {f_{n-1}} Y, \end{aligned}$$
of f such that \({{\,\mathrm{cofib}\,}}(f_j)\in \mathcal {C}^\heartsuit [j]\) for any \(j=n,\ldots ,n+k-1\), Open image in new window and Open image in new window .

The content of Proposition 5.13 becomes more interesting when Open image in new window is bounded with respect to the J-slicing \(\mathfrak {t}\) (see Definition 3.29), as in this case \({{\,\mathrm{cofib}\,}}(f)\) lies in Open image in new window for \(n\ll 0\) and in Open image in new window for \(n\gg 0\) . Namely, if Open image in new window is J-bounded, then \({{\,\mathrm{cofib}\,}}(f)\) lies in Open image in new window for some \(i\in J\). Since \(J^\heartsuit \) is a heart, there exists an element \(x\in I\) and an integer \(n_0\) such that \(i=x+n_0\), so that Open image in new window . As \(x\in I\) we have \(x\in U_0\) and so \([x,+\infty )\subseteq U_0\) therefore Open image in new window and so Open image in new window for any \(n\le n_0\). Dually one proves the statement for \(n\gg 0\). As an immediate consequence, by Remark 4.7 we see that in the J-bounded case if \(f:X\longrightarrow Y\) is any morphism in Open image in new window , then the morphisms \(0 \longrightarrow {{\,\mathrm{cofib}\,}}(f)_{n+k-1}\) and \({{\,\mathrm{cofib}\,}}(f)_{n} \longrightarrow {{\,\mathrm{cofib}\,}}(f)\) in Open image in new window are isomorphisms for \(n\ll 0\) and \(k \gg 0\). Since Open image in new window is the \(I_{[\mathbf k ]}\)-weaved factorization of \(0\longrightarrow {{\,\mathrm{cofib}\,}}(f)\) by Corollary 4.8, and since isomorphisms are preserved by pullouts we see that both \(X \xrightarrow {f_{n+k}} Z_{n+k-1} \) and \(Z_{n} \xrightarrow {f_{n-1}} Y\) are isomorphisms.

This leads to the following

Proposition 5.14

Let Open image in new window be a stable \(\infty \)-category which is bounded with respect to a given J-slicing \(\mathfrak {t}\). Let \(J^\heartsuit \subseteq J\) be a heart for J and let Open image in new window be the corresponding heart in Open image in new window . Then for any morphism \(f:X\longrightarrow Y\) in Open image in new window there exists an integer \(n_0\) and a positive integer \(k_0\) such that for any integer \(n\le n_0\) and any positive integer k with \(k\ge n_0-n+k_0\) there exists a unique factorization of f
$$\begin{aligned} X \xrightarrow {\sim } Z_{n+k-1} \xrightarrow {f_{n+k-1}}Z_{n+k-2}\longrightarrow \cdots \longrightarrow Z_{n+1} \xrightarrow {f_{n}} Z_{n} \xrightarrow {\sim } Y \end{aligned}$$
such that Open image in new window for any \(j=n,\ldots ,n+k-1\).

Remark 5.15

By uniqueness in Proposition 5.14, one has a well defined \(\mathbb {Z}\)-factorization
$$\begin{aligned} X=\lim (Z_j) \longrightarrow \cdots \longrightarrow Z_{j+1} \xrightarrow {f_{j}} Z_{j} \xrightarrow {f_{j-1}}Z_{j-1}\longrightarrow \cdots \longrightarrow \text {colim}(Z_j)=Y \end{aligned}$$
with j ranging over the integers, Open image in new window for any \(j\in \mathbb {Z}\) and with \(f_m\) being an isomorphism for \(|j|\gg 0\). We will refer to this factorization as the Open image in new window -weaved factorization of f. Notice how the boundedness of Open image in new window has played an essential role: when Open image in new window is not bounded, one still has towers for interval decomposition \(I_{[\mathbf k ]}:j\mapsto I(0)+j+n\) for arbitrary k and n, but in general they do not stabilize.

Remark 5.16

Let \(J^\heartsuit \subseteq J\) be a heart, and let (LU) be the slicing \(J^\heartsuit (0)\) of J and Open image in new window be the corresponding t-structure on Open image in new window . By Lemma 5.7 the standard heart of \(\mathfrak {t}\) is precisely Open image in new window . Moreover, by Corollaries 4.8 and 4.12, the Open image in new window -weaved factorization
$$\begin{aligned} 0=\lim (Y_j) \longrightarrow \cdots \longrightarrow Y_{j+1} \xrightarrow {f_{j}} Y_{j} \xrightarrow {f_{j-1}}Y_{j-1}\longrightarrow \cdots \longrightarrow \text {colim}(Y_j)=Y \end{aligned}$$
of an initial morphism \(0\longrightarrow Y\) is such that \({{\,\mathrm{cofib}\,}}(f_j)={\mathcal {H}}^jY\) for any \(j\in \mathbb {Z}\). Therefore we see that the t-structure Open image in new window can be completely read from Open image in new window -weaved factorizations: an object Y is in Open image in new window if and only if the Open image in new window -weaved factorization of \(0\longrightarrow Y\) satisfies \({{\,\mathrm{cofib}\,}}(f_j)=0\) for any \(j< 0\), while Y is in Open image in new window if and only if \({{\,\mathrm{cofib}\,}}(f_j)=0\) for any \(j\ge 0\). We are going to use this fact later to characterize hearts of t-structures among full subcategories of Open image in new window .

Remark 5.17

The content of Remark 5.16 can be elegantly expressed in terms of the functoriality of slicings (Remark 3.12). Namely, by Remark 5.3, to give a heart \(J^\heartsuit \) of J is equivalent to giving a \(\mathbb {Z}\)-equivariant morphism of \(\pi _\heartsuit :\mathbb {Z}\)-tosets \(J\longrightarrow \mathbb {Z}\), and by functoriality this induces a map

The heart Open image in new window of a J-slicing \(\mathfrak {t}\) on Open image in new window is then the standard heart of the corresponding t-structure \(\pi _\heartsuit (\mathfrak {t})\). Moreover, one easily sees that \(\mathfrak {t}\) is a bounded J-slicing if and only if \(\pi _\heartsuit (\mathfrak {t})\) is a bounded t-structure. Indeed, since \(\pi _\heartsuit \) is a morphism of tosets, for any \(i_1\le i_2\) in J we have \([i_1,i_2]\subseteq \pi _\heartsuit ^{-1}[\pi _\heartsuit (i_1),\pi _\heartsuit (i_2)]\). Vice versa, since \(\pi _\heartsuit \) is \(\mathbb {Z}\)-equivariant, for every \(j\in J\) we have \(\pi _\heartsuit (j+n)=\pi _\heartsuit (j)+n\) and so for every \(n_1\le n_2\) in \(\mathbb {Z}\) there exist \(j_1\le j_2\) in J such that \(\pi _\heartsuit (j_1)< n_1\) and \(\pi _\heartsuit (j_2)> n_2\). Let \(j\in J\) be such \(n_1\le \pi _\heartsuit (j)\le n_2\). If \(j>j_2\) then we would have \(\pi _\heartsuit (j)\ge \pi _\heartsuit (j_2)>n_2\), and so \(j\le j_2\). Similarly \(j\ge j_1\) and so \(\pi _{\heartsuit }^{-1} [n_1,n_2]\subseteq [j_1,j_2]\).

5.1 Abelianity of the heart

In the following section we present a proof of the fact that a heart Open image in new window of a J-slicing on a stable \(\infty \)-category Open image in new window is an abelian \(\infty \)-category.

In other words, Open image in new window is homotopy equivalent to its homotopy category Open image in new window , which is an abelian category; this is the higher-categorical counterpart of a classical result, first proved in [1, Thm. 1.3.6], which only relies on properties stated in terms of normal torsion theories in a stable \(\infty \)-category.

We begin with the following

Definition 5.18

(Additive \(\infty \)-category) An additive\(\infty \)-category is an additive category regarded as an \(\infty \)-category.

Remark 5.19

Equivalently, an additive \(\infty \)-category is a quasicategory \(\mathcal {A}\) such that
  1. (i)

    the hom space \(\mathcal {A}(X,Y)\) is a homotopically discrete infinite loop space for any XY, i.e., there exists an infinite sequence of \(\infty \)-groupoids \(\Xi _0, \Xi _1,\Xi _2,\ldots \), with Open image in new window and homotopy equivalences \(\Xi _i\cong \Omega \Xi _{i+1}\) for any \(i\ge 0\), such that \(\pi _n \Xi _0=0\) for any \(n\ge 1\);

     
  2. (ii)

    \(\mathcal {A}\) has a zero object, and (homotopy) biproducts.

     

Remark 5.20

If Open image in new window is a full additive \(\infty \)-subcategory of a stable \(\infty \)-category Open image in new window , then Open image in new window for any \(n_1>n_2\). Namely, for an additive Open image in new window , let Open image in new window and Open image in new window . Then \(X=Z_1[n_1]\) and \(Y=Z_2[n_2]\) for suitable Open image in new window and sowhere in the last equality we used the fact that Open image in new window is full. Since \(n_1-n_2>0\), the space Open image in new window is contractible by definition of additive \(\infty \)-category.

Remark 5.21

In the triangulated setting one does not have the compatibility between the looping operation on objects and on spaces of morphisms Open image in new window and so the othogonality condition Open image in new window for \(n_1>n_2\), when needed, has to be imposed by hands. This is nothing but the usual vanishing of negative Ext’s one frequently meets as a property of ‘good’ additive subcategories of a triangulated category.

Definition 5.22

(Abelian\(\infty \)-category) An abelian\(\infty \)-category is an abelian category regarded as an \(\infty \)-category.

Remark 5.23

Equivalently, an abelian \(\infty \)-category is an additive \(\infty \)-category \(\mathcal {A}\) such that
  1. (i)

    \(\mathcal {A}\) has (homotopy) kernels and cokernels;

     
  2. (ii)

    for any morphism f in \(\mathcal {A}\), the natural morphism from the coimage of f to the image (see Definition 5.31) of f is an equivalence.

     

The rest of the section is devoted to the proof of the following result:

Theorem 5.24

Let Open image in new window be a heart of a J-slicing \(\mathfrak {t}\) on a stable \(\infty \)-category Open image in new window . Then Open image in new window is an abelian \(\infty \)-category.

Remark 5.25

If \(J=\mathbb {Z}\), so that \(\mathfrak {t}\) is the datum of a t-structure on Open image in new window , and Open image in new window is the standard heart of Open image in new window , the the homotopy category Open image in new window is the abelian category arising as the standard heart of the t-structure \(h(\mathfrak {t})\) on the triangulated category Open image in new window .

In what follows, let \(I:[\mathbf 1 ]\longrightarrow \mathcal {O}(J)\) be an interval such that Open image in new window . The two factorization systems associated with I will be denoted by \((\mathcal {E}_0,\mathcal {M}_0)\) and \((\mathcal {E}_1,\mathcal {M}_1)\), respectively. By Lemma 5.7 we have \(\mathcal {M}_1=\mathcal {M}_0[1]\) and \(\mathcal {E}_1=\mathcal {E}_0[1]\).

Lemma 5.26

For any X and Y in Open image in new window , the hom space Open image in new window is a homotopically discrete infinite loop space.

Proof

Since Open image in new window is a full subcategory of Open image in new window , we have Open image in new window , which is an infinite loop space since Open image in new window is a stable \(\infty \)-category.

So we are left to prove that Open image in new window for \(n\ge 1\). Since Open image in new window , this is equivalent to showing that Open image in new window is contractible. Since X and Y are objects in Open image in new window , we have Open image in new window and Open image in new window . But Open image in new window is right object-orthogonal to Open image in new window , therefore Open image in new window is contractible. \(\square \)

The subcategory Open image in new window inherits the 0 object and biproducts (in fact, all finite limits) from Open image in new window , so in order to prove it is is abelian we are left to prove that it has kernels and cokernels, and that the canonical morphism from the coimage to the image is an equivalence.

Lemma 5.27

Let \(f:X\longrightarrow Y\) be a morphism in Open image in new window . Then \(\text {fib}(f)\) is in Open image in new window and \({{\,\mathrm{cofib}\,}}(f)\) is in Open image in new window .

Proof

Since both \(X\longrightarrow 0\) and \(Y\longrightarrow 0\) are in \(\mathcal {M}_1\), by the 3-for-2 property also f is in \(\mathcal {M}_1\). Since \(\mathcal {M}_1\) is closed for pullbacks, \(\text {fib}(f)\longrightarrow 0\) is in \(\mathcal {M}_1\) and so \(\text {fib}(f)\) is in Open image in new window . The proof for \({{\,\mathrm{cofib}\,}}(f)\) is completely dual. \(\square \)

Definition 5.28

Denote bythe \((\mathcal {E}_0,\mathcal {M}_0)\)-factorization of the morphism \(0\longrightarrow {{\,\mathrm{fib}\,}}(f)\) and bythe \((\mathcal {E}_1,\mathcal {M}_1)\)-factorization of the morphism \({{\,\mathrm{cofib}\,}}(f)\longrightarrow 0\). We call \(\ker (f)\) and \({{\,\mathrm{coker}\,}}(f)\) respectively the kernel and the cokernel of f in Open image in new window .

Remark 5.29

Since \({{\,\mathrm{cofib}\,}}(f)[-1]\cong {{\,\mathrm{fib}\,}}{f}\) and \((\mathcal {M}_1,\mathcal {E}_1)[-1]=(\mathcal {M}_0,\mathcal {E}_0)\), one can equivalently define \({{\,\mathrm{coker}\,}}(f)\) by declaring the \((\mathcal {E}_0,\mathcal {M}_0)\)-factorization of \({{\,\mathrm{fib}\,}}(f)\longrightarrow 0\) to be \({{\,\mathrm{fib}\,}}(f)\xrightarrow {\mathcal {E}_0}{{\,\mathrm{coker}\,}}(f)[-1]\xrightarrow {\mathcal {M}_0} 0\). Similarly, one can define \(\ker (f)\) by declaring the \((\mathcal {E}_1,\mathcal {M}_1)\)-factorization of \(0\longrightarrow {{\,\mathrm{cofib}\,}}(f)\) to be \(0\xrightarrow {\mathcal {E}_1}\ker (f)[1]\xrightarrow {\mathcal {M}_1} {{\,\mathrm{cofib}\,}}(f)\). By normality of the factorization system we therefore have the homotopy commutative diagramwhose square sub-diagram is a homotopy pullout.

Lemma 5.30

Both \(\ker (f)\) and \({{\,\mathrm{coker}\,}}(f)\) are in Open image in new window .

Proof

By construction \(\ker (f)\) is in Open image in new window , so we only need to show that \(\ker (f)\) is in Open image in new window . By definition of \(\ker (f)\), we have that \(\ker (f)\longrightarrow {{\,\mathrm{fib}\,}}(f)\) is in \(\mathcal {M}_0\). Since \(\mathcal {M}_0[-1]\subseteq \mathcal {M}_0\), we have that also \(\ker (f)[-1]\longrightarrow {{\,\mathrm{fib}\,}}(f)[-1]\) is in \(\mathcal {M}_0\). By Lemma 5.27, \({{\,\mathrm{fib}\,}}(f)[-1]\longrightarrow 0\) is in \(\mathcal {M}_1[-1]=\mathcal {M}_0\) and so we find that also \(\ker (f)[-1]\longrightarrow 0\) is in \(\mathcal {M}_0\). The proof for \({{\,\mathrm{coker}\,}}(f)\) is perfectly dual. \(\square \)

By definition of \(\ker (f)\) and \({{\,\mathrm{coker}\,}}(f)\), the defining diagram of \({{\,\mathrm{fib}\,}}(f)\) and \({{\,\mathrm{cofib}\,}}(f)\) can be enlarged as

where \(k_f\) and \(c_f\) are morphisms in Open image in new window .

Definition 5.31

Let \(f:X\longrightarrow Y\) be a morphism in Open image in new window . The image\({{\,\mathrm{\textsf {im}}\,}}(f)\) and the coimage\({{\,\mathrm{\textsf {coim}}\,}}(f)\) of f are defined as \({{\,\mathrm{\textsf {im}}\,}}(f)=\ker (c_f)\) and \({{\,\mathrm{\textsf {coim}}\,}}(f)={{\,\mathrm{coker}\,}}(k_f)\).

The following lemma shows that \(\ker (f)\) does indeed have the defining property of a kernel:

Lemma 5.32

The homotopy commutative diagramis a pullback diagram in Open image in new window .

Proof

A homotopy commutative diagrambetween objects in the heart is in particular a homotopy commutative diagram in Open image in new window so it is equivalent to the datum of a morphism \(k':K\longrightarrow {{\,\mathrm{fib}\,}}(f)\) in Open image in new window , with K an object in Open image in new window . By the orthogonality of \((\mathcal {E}_0,\mathcal {M}_0)\), this is equivalent to a morphism \({\tilde{k}}:K\longrightarrow \ker (f)\):\(\square \)

There is, obviously, a dual result showing that \({{\,\mathrm{coker}\,}}(f)\) is indeed a cokernel.

Lemma 5.33

The homotopy commutative diagramis a pushout diagram in Open image in new window .

Proposition 5.34

A homotopy commutative diagramwith vertices in Open image in new window is a pullout diagram in Open image in new window if and only if it is a pullout diagram in Open image in new window .

Proof

Since Open image in new window is a full subcategory of Open image in new window it is clear that if the given diagram is a pullout in Open image in new window then it is automatically also a pullout in Open image in new window . Conversely, assume the given diagram is a pullout in Open image in new window . This means that \(X=\ker (g)\) and \(Z={{\,\mathrm{coker}\,}}(f)\). By definition of \(\ker \) and \({{\,\mathrm{coker}\,}}\) this implies that we have the following homotopy commutative diagram in Open image in new window where each square is a pullout (in Open image in new window ):Here \(X\longrightarrow \mathbf 0 \) is in \(\mathcal {E}_0\) by definition of \(\ker (g)\) and by the Sator lemma, and so also \({{\,\mathrm{fib}\,}}(g)\longrightarrow W\) is in \(\mathcal {E}_0\) since \(\mathcal {E}_0\) is closed under pushouts. The morphism \({{\,\mathrm{cofib}\,}}(f)\longrightarrow Z\) is in \(\mathcal {E}_1\) by definition of \({{\,\mathrm{coker}\,}}(f)\). From the pullout diagram in Open image in new window and the fact \(\mathcal {E}_1\) is closed under pushouts that follows that \(\mathbf 0 \longrightarrow W[1]\) is in \(\mathcal {E}_1\). Therefore \(\mathbf 0 \longrightarrow W\) is in \(\mathcal {E}_1[-1]=\mathcal {E}_0\) and so \(W\longrightarrow \mathbf 0 \) is in \(\mathcal {E}_0\). This shows that \({{\,\mathrm{fib}\,}}(g)\longrightarrow \mathbf 0 \) is in \(\mathcal {E}_0\) and so \(X\longrightarrow {{\,\mathrm{fib}\,}}(g)\) is an isomorphism. Therefore the given diagram is a pullback in Open image in new window and so a pullout in Open image in new window . \(\square \)

Lemma 5.35

For \(f:X\longrightarrow Y\) a morphism in Open image in new window , there is a homotopy commutative diagram where all squares are homotopy pullouts:uniquely determining an object Open image in new window .

Proof

Define \(Z_f\) as the homotopy pulloutHere the vertical arrow on the right is in \(\mathcal {E}_0\) since the vertical arrow on the left is in \(\mathcal {E}_0\) by definition of \({{\,\mathrm{coker}\,}}(f)\) (see Remark 5.29) and \(\mathcal {E}_0\) is preserved by pushouts. Next, paste on the left of this diagram the pullout given by Remark 5.29 and build the rest of the diagram by taking pullbacks or pushouts, recalling that \((\mathcal {E}_0,\mathcal {M}_0)[1]=(\mathcal {E}_1,\mathcal {M}_1)\). Use again Remark 5.29 and the fact that \(\mathcal {M}_1\) is closed under pullbacks to see that \(Z_f\longrightarrow Y\) is in \(\mathcal {M}_1\). Finally, we have
$$\begin{aligned} 0\xrightarrow {\mathcal {E}_0} X\xrightarrow {\mathcal {E}_0} Z_f\xrightarrow {\mathcal {M}_1} Y\xrightarrow {\mathcal {M}_1} 0, \end{aligned}$$
and so \(Z_f\) is in Open image in new window . \(\square \)

Proposition 5.36

There is an isomorphism \({{\,\mathrm{\textsf {im}}\,}}(f)\cong {{\,\mathrm{\textsf {coim}}\,}}(f)\).

Proof

By definition, \({{\,\mathrm{\textsf {im}}\,}}(f)\) and \({{\,\mathrm{\textsf {coim}}\,}}(f)\) are defined by the factorizationsandThe diagram in Lemma 5.35 shows that we have \({{\,\mathrm{fib}\,}}(c_f)=Z_f={{\,\mathrm{cofib}\,}}(k_f)\). Therefore, what we need to exhibit are the \((\mathcal {E}_0,\mathcal {M}_0)\) factorizations of \(0\longrightarrow Z_f\) and the \((\mathcal {E}_1,\mathcal {M}_1)\) factorization of \(Z_f\longrightarrow 0\). Since \(Z_f\) is an object in Open image in new window , these are
$$\begin{aligned} 0\xrightarrow {\mathcal {E}_0}Z_f\xrightarrow {\text {id}_{Z_f}}Z_f \end{aligned}$$
and
$$\begin{aligned} Z_f\xrightarrow {\text {id}_{Z_f}}Z_f\xrightarrow {\mathcal {M}_1}0, \end{aligned}$$
respectively, thus giving \({{\,\mathrm{\textsf {im}}\,}}(f)\cong Z_f\cong {{\,\mathrm{\textsf {coim}}\,}}(f)\). \(\square \)

5.2 Abelian subcategories as hearts.

By the results in the previous section and by Corollary 5.11 we see that hearts of bounded t-structures are very peculiar subcategories of a stable \(\infty \)-category Open image in new window : they are abelian and every morphism in Open image in new window admits a unique Open image in new window -weaved factorization. As we are going to show, these two properties precisely characterize hearts among full subcategories of Open image in new window .

Definition 5.37

Let \(f:X\longrightarrow Y\) be a morphism in Open image in new window , and let Open image in new window be an additive \(\infty \)-subcategory of Open image in new window . An Open image in new window -weaved factorization of f is a factorization
$$\begin{aligned} X=\lim (Z_j) \longrightarrow \cdots \longrightarrow Z_{j+1} \xrightarrow {f_{j}} Z_{j} \xrightarrow {f_{j-1}}Z_{j-1}\longrightarrow \cdots \longrightarrow \text {colim}(Z_j)=Y \end{aligned}$$
with j ranging over the integers, Open image in new window for any \(j\in \mathbb {Z}\) and with \(f_m\) being an isomorphism for \(|j|\gg 0\).

Proposition 5.38

Let Open image in new window be an additive full \(\infty \)-subcategory of a stable \(\infty \)-category Open image in new window such that any morphism \(f:X\longrightarrow Y\) in Open image in new window has an Open image in new window -weaved factorization. Then the collection of subcategories Open image in new window is a Bridgeland \(\mathbb {Z}\)-slicing of Open image in new window .

Proof

Looking at Definition 4.18, the only thing we need to prove is that Open image in new window for \(n_1>n_2\). This is Remark 5.20. \(\square \)

From Proposition 4.26 and Remark 4.27 we then immediately have the following converse of Corollary 5.11, corresponding to [5, Lemma 3.2].

Proposition 5.39

Let Open image in new window be a full additive \(\infty \)-subcategory of a stable \(\infty \)-category Open image in new window , such that any morphism in Open image in new window has a Open image in new window -weaved factorization, and let Open image in new window be the functors given by taking the cofibers of the n-th morphism in the Open image in new window -weaved factorization of the initial morphisms. Let Open image in new window be the full subcategory of Open image in new window on those objects X such that \({\mathcal {H}}^j_B(X)=0\) for any \(j< 0\), and let Open image in new window be the full subcategory of Open image in new window on those objects X such that \({\mathcal {H}}^j_B(X)=0\) for any \(j\ge 0\). Then Open image in new window is a t-structure on Open image in new window , the stable \(\infty \)-category Open image in new window is bounded with respect to Open image in new window , and the standard heart of Open image in new window is (equivalent to) Open image in new window . In particular Open image in new window is abelian.

Proof

The only thing left to prove is that the standard heart of Open image in new window is Open image in new window . To see this notice that an object Y lies in Open image in new window if and only if the Open image in new window -weaved factorization
$$\begin{aligned} 0 =\lim (Y_j)\longrightarrow \cdots \longrightarrow Y_{j+1} \xrightarrow {f_{j}} Y_{j} \xrightarrow {f_{j-1}}Y_{j-1}\longrightarrow \cdots \longrightarrow \text {colim}(Y_j)=Y \end{aligned}$$
of its initial morphism has \(\text {cofib}(f_j)=0\) for every \(j\ne 0\), and so it is of the form
$$\begin{aligned} \cdots 0 \longrightarrow 0\longrightarrow \cdots \longrightarrow 0\xrightarrow {f_{0}} Y \xrightarrow {\text {id}_Y}Y\xrightarrow {\text {id}_Y} \cdots \xrightarrow {\text {id}_Y}Y\xrightarrow {\text {id}_Y}\cdots , \end{aligned}$$
with Open image in new window . \(\square \)

6 Semi-orthogonal decompositions

La vie c’est ce qui se décompose à tout moment; c’est une perte monotone de lumière, une dissolution insipide dans la nuit, sans sceptres, sans auréoles, sans nimbes.

E. Cioran

In the previous section we have investigated the case when the equivalence relation \(\sim \) from Lemma 2.27 had a single equivalence class. At the opposite end is the case when each equivalence class consists of a single element. As \(x\sim x+1\) for any \(x\in J\), this is equivalent to requiring that the \(\mathbb {Z}\)-action is trivial. As noticed in Remark 2.18 this in particular happens when J is a finite finite totally ordered set. As we are going to show, this is another well investigated case in the literature: J-families of t-structures with a finite J capture the notion of semi-orthogonal decompositions for the stable \(\infty \)-category Open image in new window (see [3, 15] for the notion of semi-orthogonal decomposition in the classical triangulated context).
To fix notations for this section, let \(J=[\mathbf k ]\) be the totally ordered set on \((k+1)\) elements, i.e., \(J=\{0,1,\ldots ,k\}\), and let Open image in new window be a \([\mathbf k ]\)-slicing on Open image in new window . We have the maximal interval decomposition on \([\mathbf k ]\) given by intervals \(I_i=\{i\}\) for \(i=0,\ldots ,k\). This corresponds to the morphism of posets \([\mathbf k-1 ]\longrightarrow {\mathcal {O}}([\mathbf k ])\) given by \(i\mapsto \{x\ge i+1\}\). Using this decomposition we have that any morphism \(f:X\longrightarrow Y\) in Open image in new window has a unique factorization
$$\begin{aligned} X \xrightarrow {f_{k}} Z_{k-1} \xrightarrow {f_{k-1}}Z_{{k-2}}\longrightarrow \cdots \longrightarrow Z_{{1}} \xrightarrow {f_1} Z_{0} \xrightarrow {f_0} Y, \end{aligned}$$
with Open image in new window , and Open image in new window , for any \(0\le h <i\le k\).

What we are left to investigate are therefore the special features of the t-structures Open image in new window coming from the triviality of the \(\mathbb {Z}\)-action on \([\mathbf k ]\), and so on \({\mathcal {O}}([\mathbf k ])\). By \(\mathbb {Z}\)-equivariancy of the map Open image in new window , this implies that all the t-structures \(\mathfrak {t}_{i}\) are \(\mathbb {Z}\)-fixed points for the natural \(\mathbb {Z}\)-action on Open image in new window . Now, a rather pleasant fact is that fixed points of the \(\mathbb {Z}\)-action on Open image in new window are precisely those t-structures Open image in new window for which Open image in new window is a stable sub-\(\infty \)-category of Open image in new window . We will make use of the following

Lemma 6.1

Let \(\mathcal {B}\) be a full sub-\(\infty \)-category of the stable \(\infty \)-category Open image in new window ; then, \(\mathcal {B}\) is a stable sub-\(\infty \)-category of Open image in new window if and only if \(\mathcal {B}\) is closed under shifts in both directions and under pushouts in Open image in new window .

Proof

The “only if” part is trivial, so let us prove the “if” part. First of all let us see that under these assumptions \(\mathcal {B}\) is closed under taking fibers of morphisms. This is immediate: if \(f:X\longrightarrow Y\) is an arrow in \(\mathcal {B}\) (i.e. an arrow of Open image in new window between objects of \(\mathcal {B}\), by fullness), then \(f[-1]\) is again in \(\mathcal {B}\) since \(\mathcal {B}\) is closed with respect to the left shift. Since \(\mathcal {B}\) is closed under pushouts in Open image in new window , also \({{\,\mathrm{fib}\,}}(f)={{\,\mathrm{cofib}\,}}(f[-1])\) is in \(\mathcal {B}\). It remains to show how this implies that \(\mathcal {B}\) is actually stable, i.e. it is closed under all finite limits and satisfies the pullout axiom. Unwinding the assumptions on \(\mathcal {B}\), this boils down to showing that in the squarethe pullback B of \(f,g \in \hom (\mathcal {B})\) done in Open image in new window is actually an object of \(\mathcal {B}\); indeed, once this is shown, the square above will satisfy the pullout axiom in Open image in new window , so a fortiori it will have the universal property of a pushout in \(\mathcal {B}\). To this aim, let us consider the enlarged diagram of pullout squares in Open image in new window

The objects \(Z[-1], {{\,\mathrm{fib}\,}}(f)\) and \({{\,\mathrm{fib}\,}}(g)\) lie in \(\mathcal {B}\) by the first part of the proof, so the square \((\star )\) is in particular a pushout of morphism in \(\mathcal {B}\); by assumption, this entails that \(B\in \mathcal {B}\). \(\square \)

Remark 6.2

Obviously, a completely dual statement can be proved in a completely dual fashion: a full sub-\(\infty \)-category \(\mathcal {B}\) of a stable \(\infty \)-category Open image in new window is a stable sub-\(\infty \)-category if and only if it is closed under shifts in both directions and under pullbacks in Open image in new window .

Proposition 6.3

Let Open image in new window be a t-structure on a stable \(\infty \)-category Open image in new window , and let \((\mathcal {E},\mathcal {M})\) be the normal torsion theory associated to \(\mathfrak {t}\); then the following conditions are equivalent:
  1. (1)

    \(\mathfrak {t}\) is a fixed point for the \(\mathbb {Z}\)-action on \(\text {ts}(\mathcal {C})\), i.e., \(\mathfrak {t}[1]=\mathfrak {t}\) (or equivalently, Open image in new window , or equivalently Open image in new window );

     
  2. (2)

    Open image in new window is a stable sub-\(\infty \)-category of Open image in new window .

     
  3. (3)

    Open image in new window is a stable sub-\(\infty \)-category of Open image in new window .

     
  4. (4)

    \(\mathcal {E}\) is closed under pullback;

     
  5. (5)

    \(\mathcal {M}\) is closed under pushout.

     

Proof

‘(2) implies (1)’ is obvious. Namely, if Open image in new window is a stable sub-\(\infty \)-category of Open image in new window , then it is closed under shifts in both directions. Therefore Open image in new window , and so Open image in new window . Since, by definition of t-structure, Open image in new window , we have Open image in new window . To prove that ‘(1) implies (2)’, assume Open image in new window . This implies Open image in new window , so Open image in new window is closed under shifts in both directions. By Lemma 6.1, we then have only to show that Open image in new window is closed under pushouts in Open image in new window to conclude that Open image in new window is a stable \(\infty \)-subcategory of Open image in new window . Consider a pushout diagramin Open image in new window with A, B and C in Open image in new window . Since A and C are in Open image in new window we have that both \(0\longrightarrow A\) and \(0\longrightarrow C\) are in \(\mathcal {E}\). But \(\mathcal {E}\) has the 3-for-2 property, so also \(A\longrightarrow C\) is \(\mathcal {E}\). Since \(\mathcal {E}\) is closed for pushouts, this implies that also \(B\longrightarrow P\) is in \(\mathcal {E}\). But \(0\longrightarrow B\) in in \(\mathcal {E}\) since B is in Open image in new window , and therefore also \(0\longrightarrow P\) is in \(\mathcal {E}\), i.e., P is in Open image in new window .
We now prove that ‘(1) is equivalent to (4)’. Assume \(\mathcal {E}\) is closed under pullbacks. Then for any X in Open image in new window we have that \(0\longrightarrow X\) is in \(\mathcal {E}\), and so \(X[-1]\longrightarrow 0\) is in \(\mathcal {E}\). By the Sator lemma this implies that \(0\longrightarrow X[-1]\) is in \(\mathcal {E}\), i.e., that \(X[-1]\) is in Open image in new window . This shows that Open image in new window and therefore that \(\mathfrak {t}[1]= \mathfrak {t}\). Conversely, assume \(\mathfrak {t}[1]=\mathfrak {t}\), and consider a morphism \(f:X\longrightarrow Y\) in \(\mathcal {E}\). For any morphism \(B\longrightarrow Y\) in Open image in new window consider the diagramwhere all the squares are pullouts in Open image in new window . Since f is in \(\mathcal {E}\) and \(\mathcal {E}\) is closed for pushouts, also \(0\longrightarrow {{\,\mathrm{cofib}\,}}(f)\) is in \(\mathcal {E}\). This means that \({{\,\mathrm{cofib}\,}}(f)\) is in Open image in new window and so, since we are assuming that Open image in new window , also \(\text {fib}(f)={{\,\mathrm{cofib}\,}}(f)[-1]\) is in Open image in new window , i.e., \(0\longrightarrow \text {fib}(f)\) is in \(\mathcal {E}\). By the Sator lemma, \(\text {fib}(f)\longrightarrow 0\) is in \(\mathcal {E}\), which is closed for pushouts, and so \(A\longrightarrow B\) is in \(\mathcal {E}\). The proofs that ‘(1) if and only if (3)’ and ‘(1) if and only if (5)’ are perfectly dual. \(\square \)

Remark 6.4

A factorization system \((\mathcal {E},\mathcal {M})\) for which the class \(\mathcal {E}\) is closed for pullbacks is sometimes called an exact reflective factorization, see, e.g., [6]. This is equivalent to saying that the associated reflection functor is left exact (this is called a localization in the jargon of [6]). Dually, one characterizes co-localizations of a category Open image in new window with an initial object as co-exact coreflective factorizations where the right class \(\mathcal {M}\) is closed under pushouts. Therefore, in the stable \(\infty \)-case, we see that a \(\mathbb {Z}\)-fixed point in Open image in new window is a t-structure Open image in new window such that the truncation functors Open image in new window and Open image in new window respectively form a co-localizations and a localization of Open image in new window . In the terminology of [4] we therefore find that in the stable \(\infty \)-case \(\mathbb {Z}\)-fixed point in Open image in new window correspond to hereditary torsion pairs on Open image in new window . Since we have seen that for a \(\mathbb {Z}\)-fixed point in Open image in new window both Open image in new window and Open image in new window are stable \(\infty \)-categories, this result could be deduced also from [18, Prop. 1.1.4.1]: a left (resp., right) exact functor between stable \(\infty \)-categories is also right (resp., left) exact.

We can now precisely relate semi-orthogonal decompositions in a stable \(\infty \)-category Open image in new window to \([\mathbf k ]\)-slicings of Open image in new window . The only thing we still need is the following definition, which is an immediate adaptation to the stable setting of the classical definition given for triangulated categories (see, e.g., [3, 15]).

Definition 6.5

Let Open image in new window be a stable \(\infty \)-category. A semi-orthogonal decomposition with \(k+1\) classes on Open image in new window is the datum of \(k+1\) stable \(\infty \)-subcategories Open image in new window ,..., Open image in new window of Open image in new window such that
  1. (1)

    one has Open image in new window for \(h<i\) (semi-orthogonality);

     
  2. (2)
    for any object Y in Open image in new window there exists a unique Open image in new window -weaved tower, i.e., a factorization of the initial morphism \(0\longrightarrow Y\) as
    $$\begin{aligned} 0 \xrightarrow {f_{k}} Y_{k-1} \xrightarrow {f_{k-1}}Y_{{k-2}}\longrightarrow \cdots \longrightarrow Z_{{1}} \xrightarrow {f_1} Y_{0} \xrightarrow {f_0} Y, \end{aligned}$$
    with Open image in new window for any \(i=0,\ldots , k\).
     
Since Open image in new window -weaved towers are preserved by pullouts, one can equivalently require that any morphism \(f:X\longrightarrow Y\) in Open image in new window has a unique factorization of has a unique factorization
$$\begin{aligned} X \xrightarrow {f_{k}} Z_{k-1} \xrightarrow {f_{k-1}}Z_{{k-2}}\longrightarrow \cdots \longrightarrow Z_{{1}} \xrightarrow {f_1} Z_{0} \xrightarrow {f_0} Y, \end{aligned}$$
with Open image in new window , and this immediately leads to the following

Proposition 6.6

Let Open image in new window be a stable \(\infty \)-category. Then the datum of a semi-orthogonal decompositions with \(k+1\) classes on Open image in new window is equivalent to the datum of a \([\mathbf k ]\)-slicing on Open image in new window .

Proof

The only missing piece of information to show that a \([\mathbf k ]\)-slicing is a semi-orthogonal decompositions is the fact that the sub-\(\infty \)-categories Open image in new window are stable. But Open image in new window and both Open image in new window and Open image in new window are stable by Proposition 6.3. Therefore, also Open image in new window is stable (see [18]). Conversely, given a semi-orthogonal decomposition this defines a \([\mathbf k ]\)-slicing by means of the cofiber functors Open image in new window , by the same argument in the proof of Proposition 5.39. \(\square \)

Remark 6.7

By Remark 6.4, we recover in the stable \(\infty \)-setting the well known fact (see [4, IV.4]) that semi-orthogonal decompositions with a single class correspond to hereditary torsion pairs on the category.

Proposition 6.6 immediately suggests to generalize the definition of semi-orthogonal decomposition to the case of an arbitrary toset of indices, not necessarily finite.

Definition 6.8

Let I be a toset, and let \(I^\flat \) be the \(\mathbb {Z}\)-toset given by I endowed with the trivial \(\mathbb {Z}\)-action (see 2.30). An \(I^\flat \)-slicing of a stable \(\infty \)-category Open image in new window is called a I-semi-orthogonal decomposition of Open image in new window . The class of all I-semi-orthogonal decompositions of Open image in new window will be denoted by Open image in new window , i.e. Open image in new window .

Remark 6.9

If an I-semi-orthogonal decomposition of Open image in new window is given, then all the subcategories Open image in new window are stable, for any i in I.

Remark 6.10

Let J be a \({\mathbb {Z}}\)-toset, and let \(\iota (J)\) be the toset of equivalence classes of J, for the equivalence relation \(\sim \) of Lemma 2.27. Then every J-slicing of a stable \(\infty \)-category Open image in new window induces an \(\iota (J)\)-semi-orthogonal decomposition of Open image in new window . Namely, by Proposition 2.30, \(J\rightsquigarrow \iota (J)\) is the left adjoint of the fully faithful embedding Open image in new window , and the the projection to the quotient is a \(\mathbb {Z}\)-equivariant morphism
$$\begin{aligned} J\longrightarrow \iota (J)^\flat \end{aligned}$$
which is the unit of this adjunction. By functoriality of the slicings (Remark 3.12) we therefore have a natural map

7 Abelian slicings and tiltings

Quando si vuole uccidere un uomo bisogna colpirlo al cuore, e un Winchester è l’arma più adatta.

R. Rojo

We now review the abelian counterpart of the notion of J-slicing and relate slicings on hearts of a stable \(\infty \)-category Open image in new window with slicings of Open image in new window . First of all recall the notion of torsion pair on an abelian \(\infty \)-category, which is the abelian counterpart of the notion of t-structure on a stable \(\infty \)-category.

Definition 7.1

(torsion theory on an abelian\(\infty \)-category) Let Open image in new window be an abelian \(\infty \)-category. A torsion pair on an abelian \(\infty \)-category Open image in new window is a pair Open image in new window of full sub-\(\infty \)-subcategories of Open image in new window satisfying:
  1. (i)

    orthogonality: Open image in new window is contractible for each Open image in new window , Open image in new window ;

     
  2. (ii)
     
The subcategories Open image in new window and Open image in new window are called the torsion class and the torsion free class, respectively.

Notation 7.2

We denote by Open image in new window the set of torsion theories on Open image in new window ; this set has a natural choice for a partial order: Open image in new window if and only if Open image in new window , or equivalently Open image in new window .

The poset Open image in new window has a top and a bottom element, given by Open image in new window and Open image in new window , respectively. The following definition is directly inspired by [21].

Definition 7.3

(Abelian slicing) Let \((I, \le )\) be a poset. An abelianI-slicing on Open image in new window is a morphism of posets Open image in new window that preserve the top and bottom element. The image of \((\Lambda ,\Upsilon )\in {\mathcal {O}}(I)\) by \(\mathcal {T}\) will be denoted Open image in new window

Remark 7.4

Notice that, since there is no choice of a shift functor in an abelian \(\infty \)-category, there is no \(\mathbb {Z}\)-action on I or \(\mathbb {Z}\)-equivariancy condition involved in the above definition.

Remark 7.5

(The abelian slicings functor) By analogy with Remark 3.12, for any \(\infty \)-category \(\mathcal {A}\) we have a functor Open image in new window mapping a poset I to the poset of abelian I-slicings of Open image in new window .

Lemma 7.6

Let Open image in new window be a t-structure on Open image in new window with heart Open image in new window , and let \(\mathfrak {t}_1=\mathfrak {t}_0[1]\). Also let Open image in new window be another t-structure with \(\mathfrak {t}_0 \le \mathfrak {t}\le \mathfrak {t}_1\). Thenis a torsion theory on Open image in new window .

Proof

Clearly, Open image in new window and Open image in new window . Moreover, Open image in new window and Open image in new window , and so Open image in new window . Now, pick Open image in new window and consider the fiber sequenceinduced by the t-structure \(\mathfrak {t}\). From it we get the fiber sequenceWe have Open image in new window and Open image in new window . Since \(\mathcal {L}_1\) is closed by extensions (see Remark 3.18), this implies that Open image in new window . Therefore Open image in new window . An analogous argument shows that Open image in new window . \(\square \)

Proposition 7.7

Let \((J,\le )\) be a \({\mathbb {Z}}\)-toset, and let \(J^\heartsuit \) a heart of J. Then a J-slicing on Open image in new window induces a t-structure on Open image in new window together with an abelian \(J^\heartsuit \)-slicing on Open image in new window .

Proof

Let Open image in new window be a fixed J-slicing on Open image in new window , and let \(J^\heartsuit =U_0\cap L_1\) for some (unique) upper set \(U_0\) and lower set \(L_1\) in J. Finally, let \(\mathfrak {t}_0\) be the t-structure on Open image in new window corresponding to the slicing \((L_0,U_0)\) of J. Then we know from Remark 5.15 that Open image in new window is the standard heart of \(\mathfrak {t}_0\). Let \(\mathfrak {t}_1\) be the t-structure on Open image in new window corresponding to the slicing \((L_1,U_1)\) of J. By Lemma 5.7 we know that \(\mathfrak {t}_1=\mathfrak {t}_0[1]\). Moreover we know from Remark 3.33 that every upper set \(\Upsilon \) of \(J^\heartsuit \) is of the form \(\Upsilon =U\cap J^\heartsuit \) for a unique upper set U in J with \(U_0\le U\le U_1\). Let (LU) be the slicing of J determined by U. By Lemma 7.6,defines an abelian \(J^\heartsuit \)-slicing on Open image in new window . \(\square \)

As we are going to show, in the bounded case we also have the converse of the above proposition.

Lemma 7.8

Suppose that \(\mathfrak {t}\) is a bounded t-structure on Open image in new window with heart Open image in new window . Then a torsion theory Open image in new window on Open image in new window induces a bounded \({\mathbb {Z}} \times _{\text {lex}} [\mathbf 1 ]\)-slicing on Open image in new window .

Proof

Since every interval of the form \([n_0,n_1]\) in \({\mathbb {Z}} \times _{\text {lex}} [\mathbf 1 ]\) is finite, a bounded \({\mathbb {Z}} \times _{\text {lex}} [\mathbf 1 ]\)-slicing is discrete of finite type. Therefore, by Proposition 4.26 we are reduced to showing that a torsion theory \(\mathcal {T}\) on Open image in new window induces a Bridgeland \({\mathbb {Z}} \times _{\text {lex}} [\mathbf 1 ]\)-slicing on Open image in new window . Since Open image in new window is a torsion theory of Open image in new window , we have that Open image in new window is a torsion theory of Open image in new window for any \(n\in {\mathbb {Z}}\). Consider the full subcategoriesSince the \(\mathbb {Z}\)-action on \(\mathbb {Z}\times _{\text {lex}}[\mathbf 1 ]\) is \((n,i)+1=(n+1,i)\), it is immediate to see that Open image in new window for every (ni) in \(\mathbb {Z}\times _{\text {lex}}[\mathbf 1 ]\). Let now Open image in new window and Open image in new window with \((n_1,i_1)>(n_2,i_2)\). Since the order is the lexicographic one, we either have \(n_1>n_2\) or \(n_1=n_2\) and \(i_1=1\) and \(i_2=0\). In the first case Open image in new window and Open image in new window with \(n_1>n_2\) and so Open image in new window ; in the second case Open image in new window and Open image in new window and so again Open image in new window . Finally, we have to show that for every object X of Open image in new window we have a factorization of the initial morphism \(0\longrightarrow X\) into morphisms whose cofibers are in Open image in new window for a decreasing sequence of indices (ni)’s in the lexicographic order on \({\mathbb {Z}} \times [\mathbf 1 ]\). To see this, let X be an object in Open image in new window and consider the Open image in new window -weaved tower of its initial morphism. Keeping only the nontrivial morphisms in this tower we are reduced to a finite factorization of the form
$$\begin{aligned} 0=X_0 \xrightarrow {\alpha _1} \cdots \xrightarrow {\alpha _k} X_k=X \end{aligned}$$
with Open image in new window for a suitable sequence of decreasing integers \(n_0>n_1>\cdots > n_k\). Since Open image in new window is a torsion theory on Open image in new window , for each l we have a pullout diagramin Open image in new window . By Proposition 5.34, this is a pullout diagram in Open image in new window and so we can consider the commutative diagramwhere each square is a pullout in Open image in new window . As l ranges from 0 to k the sequence
$$\begin{aligned} 0=X_0 \xrightarrow {\alpha _1,+}{\tilde{X}}_0 \xrightarrow {\alpha _1,-}X_1\xrightarrow {\alpha _2,+} \cdots \xrightarrow {\alpha _k,-} X_k=X \end{aligned}$$
gives a factorization of \(0\longrightarrow X\) into morphisms whose cofibers are in Open image in new window for a decreasing sequence of indices (ni)’s. \(\square \)

Remark 7.9

The nontrivial upper sets in \(\mathbb {Z}\times _{\text {lex}} [\mathbf 1 ]\) are easily described: they are all of the form \([(n,i),+\infty )\) for some \(n\in \mathbb {Z}\) and \(i\in [\mathbf 1 ]\). In the notation of Lemma 7.8, the t-structures on Open image in new window corresponding to these upper sets are easily described by means of Remark 4.27. We haveEquivalently,

Definition 7.10

Let Open image in new window be a bounded t-structure on Open image in new window and let Open image in new window be its standard heart. For every torsion theory \(\mathcal {T}\) on Open image in new window , the t-structure Open image in new window is said to be obtained tilting (it’s a verb) Open image in new window with \(\mathcal {T}\) (see [4]).

Remark 7.11

One immediately sees that the tilting of a t-structure Open image in new window by the bottom torsion theory Open image in new window is the trivial tilting, while tilting by the the top torsion theory Open image in new window correspond to shifting by 1:

Remark 7.12

The construction of Lemma 7.8 gives a map
and the explicit description in Remark 7.9 show that this is a morphism of posets. Namely, if \(\mathcal {T}_1\le \mathcal {T}_2\) then Open image in new window and so Open image in new window for every (ni). In particular, tilting defines a morphism of posetsThis construction can be seen as a byproduct of the functoriality of slicings as follows. The \({\mathbb {Z}}\)-toset \({\mathbb {Z}} \times _{\text {lex}} [\mathbf 1 ]\) has an obvious \({\mathbb {Z}}\)-equivariant morphism of posets to \({\mathbb {Z}}\) given by the projection \(\pi \) on the first factor. However, and remarkably, there is also another less trivial \({\mathbb {Z}}\)-equivariant morphism
$$\begin{aligned} \pi \circ \text {tilt}:{\mathbb {Z}} \times _{\text {lex}} [\mathbf 1 ] \longrightarrow {\mathbb {Z}} \end{aligned}$$
given by the composition of the projection \(\pi \) with the \({\mathbb {Z}}\)-equivariant automorphism of the poset \({\mathbb {Z}} \times _{\text {lex}} [\mathbf 1 ]\) defined by
$$\begin{aligned} {\left\{ \begin{array}{ll} \text {tilt}(n,0)=(n-1,1) \\ \text {tilt}(n,1)=(n,0) \end{array}\right. }. \end{aligned}$$
Since taking slicings of a fixed stable \(\infty \)-category with respect to a \(\mathbb {Z}\)-toset J is functorial in J (Remark 3.12), we have a morphism of \(\mathbb {Z}\)-tosets

Composing this with the morphism of posets Open image in new window gives the tilting map.

The proposition below can be found in [4] for \(J={\mathbb {Z}} \times _{\text {{lex}}} [\mathbf 1 ]\) and in [5] for \(J={\mathbb {R}}\).

Proposition 7.13

Let J be a \({\mathbb {Z}}\)-toset, and let \(J^\heartsuit \) be a heart of J. Then giving a bounded J-slicing on Open image in new window is equivalent to giving a bounded t-structure Open image in new window on Open image in new window together with an abelian \(J^\heartsuit \)-slicing on the standard heart Open image in new window of Open image in new window . Moreover the bounded J-slicing on Open image in new window is discrete if and only if the abelian \(J^\heartsuit \)-slicing of Open image in new window is discrete.

Proof

In one direction this is the content of Proposition 7.7. Vice versa, assume we have a bounded t-structure Open image in new window on Open image in new window and an abelian \(J^\heartsuit \)-slicing \(\mathcal {T}\) on its standard heart Open image in new window . By definition this is a morphism of posets Open image in new window . By Remark 7.12, tilting a fixed t-structure gives a morphism of posets Open image in new window , and so by composition we get a morphism of posetsRecalling the identification of \({\mathcal {O}}(J^\heartsuit )\) with the interval \([U_0,U_1]\) of \({\mathcal {O}}(J)\) from Remark 3.33, and that \(U_1=U_0+1\) from Lemma 5.7, this is a morphism of posets Open image in new window and so it induces a uniquely determined \(\mathbb {Z}\)-equivariant morphism of \(\mathbb {Z}\)-tosetsBy Remark 7.11, \(\mathfrak {t}_{(1,U_0)}=\mathfrak {t}_{(0,U_0)}[1]=\mathfrak {t}_{(0,U_0+1)}\) and so \(\mathfrak {t}\) factors through the natural morphism of \(\mathbb {Z}\)-tosets
$$\begin{aligned} \mathbb {Z}\times _{\text {lex}}[U_0,U_0+1]\longrightarrow {\mathcal {O}}(J) \end{aligned}$$
given by \((n,U)\mapsto U+n\). In other words, \(\mathfrak {t}\) uniquely defines a J-slicing on Open image in new window , which is bounded since the t-structure Open image in new window is, by Remark 5.17. Finally, the construction manifestly preserves finite types. \(\square \)

8 Concluding remarks

That’s all, folks!

Bosko

We have explored two classes of J-slicings so far: those for which J has a heart, and those for which \({\mathbb {Z}}\) acts trivially on J. In this section, we show how these two cases are fundamental building blocks for all other J-slicings.

Lemma 8.1

Let \(\mathfrak {t}\) be a J-slicing on a stable \(\infty \)-category Open image in new window , and let \(I_i\subseteq J\) be the equivalence class of \(i\in J\) with respect to the equivalence relation \(\sim \) of Lemma 2.27. For every \((\Lambda ,\Upsilon )=(L\cap I_{i},U\cap I_{i})\) in \({\mathcal {O}}(I_i)\), let Open image in new window . Then \(\mathfrak {t}_i:(\Lambda ,\Upsilon )\longrightarrow \mathfrak {t}_{i;\Lambda ,\Upsilon }\) is a \(I_i\)-slicing of Open image in new window .

Proof

By Remark 6.10, the J-slicing \(\mathfrak {t}\) induces an \(\iota (J)\)-semi-orthogonal decomposition of Open image in new window : for every equivalence class [i] in \(\iota (J)\) the corresponding slice in this semi-orthogonal decomposition is the subcategory Open image in new window of Open image in new window determined by the J-slicing \(\mathfrak {t}\), where \(I_i\subseteq J\) is the equivalence class of i with respect to the equivalence reltion \(\sim \), as a subset of J. As they are the slices of a semi-orthogonal decomposition, the subcategories Open image in new window are stable (this can also be seen directly from the definition of the \(\mathcal {C}_{I_i}\)’s). As shown in Example 3.23, \(I_i\) is an interval of J and a sub-\(\mathbb {Z}\)-toset of J, simply by definition of the equivalence relation. For every i, we can therefore write \(I_i=U_{i;0}\cap L_{i;1}\). By Remark 3.33 every slicing \((\Lambda ,\Upsilon )\) of \(I_i\) is of the form \(\Lambda =L\cap I_{i}\) and \(\Upsilon =U\cap I_{i}\) for a unique slicing (LU) of J with \(U_{i;0}\le U\le U_{i;1}\). This gives an isomorphism of tosets between \({\mathcal {O}}(I_i)\) and the interval \([U_{i;0},U_{i;1}]\) in \({\mathcal {O}}(J)\). Now, to show that \(\mathfrak {t}_{i;\Lambda ,\Upsilon }\) is a t-structure on Open image in new window one verbatim repeats the proof of Lemma 7.6 to get orthogonality of the classes and the existence of the relevant fiber sequences. Next, to show that Open image in new window notice that, since \(I_i\) is an equivalence class, we have \(I_i+1=I_i\) and so
$$\begin{aligned} (U\cap I_{i})+1=(U+1)\cap (I_i+1)\subseteq U\cap I_i. \end{aligned}$$
This shows that \(\mathfrak {t}_i\) is a morphism of sets Open image in new window and it is immediate to see that this is actually a morphism of tosets. Finally, \(\mathbb {Z}\) equivariance is obtained by noticing that \(\Upsilon +1=(U\cap I_i)+1=(U+1)\cap I_i\) and \(\Lambda +1=(L+1)\cap I_i\), so thatwhere we used that Open image in new window is a stable subcategory of Open image in new window and so Open image in new window . \(\square \)

We can now state and prove our main result, summarising and putting together the various pieces constructed so far. To make a self-standing statement, we explicitly recall the definition of the equivalence relation \(\sim \) from Lemma 2.27 in the statement of the theorem below.

Theorem 8.2

Let \((J,\le )\) a \(\mathbb {Z}\)-toset. A finite type J-slicing on a stable \(\infty \)-category Open image in new window is equivalent to the following data:
  1. (i)

    a finite type semi-orthogonal decomposition of Open image in new window whose slices Open image in new window are stable \(\infty \)-subcategories of Open image in new window indexed by equivalence classes in J with respect to the equivalence relation \(x\sim y\) if and only if there exist integers \(n_1\) and \(n_2\) with \(x+n_1\le y\le x+n_2\);

     
  2. (ii)

    a bounded t-structure \(\mathfrak {t}_{[x]}\) on Open image in new window for each [x] in \(J/_{\!\sim }\);

     
  3. (iii)

    a finite type abelian \([x,x+1)\)-slicing on Open image in new window for every [x] in \(J/_{\!\sim }\) such that x is not a fixed point of the \(\mathbb {Z}\)-action on J.

     

Proof

By Lemma 8.1, a J-slicing on a stable \(\infty \)-category induces semi-orthogonal decomposition of Open image in new window whose slices Open image in new window are stable \(\infty \)-subcategories of Open image in new window indexed by equivalence classes \(I_i\) in J with respect to the equivalence relation \(\sim \), together with \(I_i\)-slicings of these subcategories. If i is a fixed point for the \(\mathbb {Z}\)-action on J, then \(I_i=\{i\}\) and \({\mathcal {O}}(I_i)=[\mathbf 1 ]\) so that a \(I_j\)-slicing is trivial. On the other hand, by Example 5.6, precisely when i is not a fixed point of the \(\mathbb {Z}\)-action the interval \(I_i\) has a heart \(I_i^\heartsuit \) which can be identified with the interval \([i,i+1)\) of J. Therefore, by Proposition 7.7 an \(I_i\)-slicing on Open image in new window induces a t-structure on Open image in new window together with an abelian \([i,i+1)\)-slicing on the standard heart Open image in new window . Moreover, by Proposition 7.13, in the finite type case an \(I_i\)-slicing on Open image in new window is precisely equivalent to this datum of a bounded t-structure on Open image in new window with an abelian \(I_i^\heartsuit \)-slicing on Open image in new window .

Vice versa, given the data (i)-(iii), we want to define a J-slicing of Open image in new window . For every upper set U of J, we can write
$$\begin{aligned} U=\bigcup _{[i]\in J/\sim }U\cap I_i \end{aligned}$$
and, by Proposition 7.13 again, the data (i)-(iii) define \(\infty \)-subcategories Open image in new window of Open image in new window . Defining Open image in new window as the extension closed \(\infty \)-subcategory of Open image in new window generated by the Open image in new window ’s provides a map Open image in new window which is immediate to see it is montone and \({\mathbb {Z}}\)-equivariant and so is a J-slicing of Open image in new window . Moreover, this J-slicing is of finite type as all the slicings provided by the data are of finite type, and manifestly this way of constructing finite type J-slicings out of data (i)-(iii) provides an inverse to the construction described in the first part of the proof. \(\square \)

8.1 Translations of the opening quotes.

  1. §2.)

    These are the first three verses of Tao te Ching’s chapter 63: “Act without action / work without work / taste without taste.”

     
  2. §3.)

    This is a quote from Histoire d’O: “her freedom was worse than any chain.”

     
  3. §4.)

    This is Genesis 11:7: “let us go down, and there confound their language, that they may not understand one another’s speech.”

     
  4. §6.)

    This is a quote from E. Cioran’s Brief history of decay: “Life is what decomposes at every moment; it is a monotonous loss of light, an insipid dissolution in the darkness, without scepters, without halos.”

     
  5. §7.)

    This is a famous line Ramon Rojo says in S. Leone’s movie A fistful of dollars: “When you want to kill a man you must shoot for his heart and a Winchester is the best weapon.”

     

Footnotes

  1. 1.

    Here and in the rest of the paper we are implicitly using the equivalence between t-structures and normal torsion theories: if Open image in new window is a stable \(\infty \)-category with a terminal object, there exists an antitone Galois connection between the poset Open image in new window of reflective subcategories of Open image in new window and the poset \(\text {pf}(\mathcal {C})\) of prefactorization systems on Open image in new window such that \(r(\mathbb {F})\) 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) Open image in new window : 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 \((\infty ,1)\)-categories.

  2. 2.

    The Japanese verb Open image in new window (“kiru”, to cut) contains the radical Open image in new window , the same of katana.

Notes

Acknowledgements

Open access funding provided by Max Planck Society. We thank the anonymous referee for useful comments that helped us to improve the overall exposition.

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Domenico Fiorenza
    • 1
  • Fosco Loregian
    • 2
    Email author
  • Giovanni Luca Marchetti
    • 3
  1. 1.Dipartimento di Matematica “Guido Castelnuovo”Università degli Studi di Roma “la Sapienza”RomeItaly
  2. 2.Max Planck Institute for MathematicsBonnGermany
  3. 3.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK

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