On Stable (∞, 1)-Categories

Fosco Loregian t-structures on stable (∞, 1)-categories

a thesis in Mathematics with 105 diagrams coded in koDi

Scuola Internazionale Superiore di Studi Avanzati 2016

Contents

Introduction i

1 Factorization Systems 1 1.1 Overview of factorization systems...... 1 1.2 Markings and prefactorizations...... 2 1.3 Factorization systems...... 6 1.3.1 Weak factorization systems...... 8 1.4 Closure properties...... 9 1.5 A second glance at factorization...... 12 1.5.1 A factor-y of examples...... 13 1.5.2 Chains of factorization systems...... 15

2 Reflectivity and Normality 19 2.1 The fundamental connection...... 21 2.2 Semiexactness and simplicity...... 22 2.3 Normal torsion theories...... 24

3 The “Rosetta stone” 33 3.1 t-structures are factorization systems...... 33 3.2 Examples...... 39 3.3 The Rosetta stone is model independent ...... 42 3.3.1 Enriched factorization systems...... 43 3.3.2 Homotopy factorization systems...... 44

4 Hearts and towers 45 4.1 Posets with Z-actions...... 46 4.1.1 J-families of t-structures...... 49 4.2 Towers of morphisms...... 53 4.3 Hearts of t-structures...... 58 4.3.1 Abelianity of the heart...... 60 4.3.2 Abelian subcategories as hearts...... 65 4.4 Semiorthogonal decompositions...... 68 Contents

5 Recollements 73 5.1 Introduction...... 73 5.2 Classical Recollements...... 74 5.2.1 The classical gluing of t-structures...... 80 5.3 Stable Recollements...... 84 5.3.1 The Jacob’s ladder: building co/reflections...... 84 5.3.2 The ntt of a recollement...... 87 5.4 Properties of recollements...... 91 5.4.1 Geometric associativity of the gluing...... 91 5.4.2 Abstract associativity of the gluing...... 96 5.4.3 Gluing J-families...... 97

6 Operations on t-structures 101 6.1 Basic constructions...... 102 6.2 The poset of t-structures...... 103 6.3 Tensor product of t-structures...... 104 6.4 Tilting of t-structures...... 106 6.5 Algebras for a monad ...... 110

7 Stability Conditions 113 7.1 Introduction ...... 113 7.2 Slicings ...... 115 7.2.1 A topology on slicings ...... 119 7.3 Stability conditions ...... 122 7.4 Hearts and endocardia ...... 123 7.5 Deformation of stability conditions ...... 126

A Stable ∞-categories 139 A.1 Triangulated higher categories...... 139 A.2 Building stable categories...... 144 A.2.1 Stable ∞-categories...... 145 A.3 t-structures...... 151 A.4 Spanier-Whitehead stabilization...... 154 A.4.1 Construction via monads...... 154 A.5 Stability in different models...... 156 A.5.1 Stable Model categories...... 156 В родстве со всем, что есть, уверясь И знаясь с будущим в быту, Нельзя не впасть к концу, как в ересь, В неслыханную простоту.

Но мы пошажены не будем, Когда ее не утаим. Она всего нужнее людям, Но сложное понятней им…

Б. Л. Пастернак 四拳波羅蜜大光明. Introduction

The present work re-enacts the classical theory of t-structures reducing the classical definition given in [BBD82, KS] to a rather primitive categorical gadget: suitable reflective factorization systems (Def. 2.3.1, 2.3.9), which we call normal torsion theories following [CHK85, RT07]. A relation between these two objects has previously been noticed by other au- thors [RT07, HPS97, BR07] on the level of homotopy categories. The main achievement of the present thesis is to observe and prove that this relation exists genuinely when the definition is lifted to the higher-dimensional world where the notion of triangulated category comes from, i.e. stable (∞, 1)-categories. Stable (∞, 1)-categories provide a far more natural setting to interpret the language of homological algebra: the main conceptual aim of the present work is to give explicit examples of this meta-principle. To achieve this result, it seemed unavoidable to adopt a preferential model for (∞, 1)-category theory: instead of working in a ‘model-free’ setting, we choose the ubiquitous dialect of Lurie’s stable quasicategories; dis- cussing to which extent (if any) the results we prove are affected by this choice, and establishing a meaningful dictionary between the validity of the general statement 3.1.1 in various different flavours on ∞-category theory occupies sections A.5 and 3.3; despite the fact that this is one of the most important issues from a categorical point of view, a rapid convergence of the present thesis into its final form has to be ensured; hence, we will defer a torough examination of the topic of model (in)dependence to subsequent works. The first part of the thesis (Ch. 1–3) builds (or rather, ‘reinterprets’) the calculus of factorization in the setting of ∞-categories. The desire to link this calculus with homological algebra and higher algebra deserves further explanation. The language of factorization systems proved to be ubiquitous inside and outside category theory (among various different applications now es- tablished in the mathematical practice, the ‘modern view’ in algebraic topology revolves around the notion of orthogonality and lifting/extension problem, as it is said in the first pages of [?]. The modern ‘synthetic’ approach to homotopy theory inescapably relies on the notion of a (weak) ii factorization system ([Qui67, DS95, Rie11]). In light of this, finding ‘concrete’ means of application for the calculus of factorization should be a natural step towards a popularization of this per- vasive and deep language. And among all the various fields of application, homological algebra, a notable kind of ‘abelian’ homotopy theory, should be the most natural test bench to measure the validity of this effort. Despite the intrinsic simplicity, almost a triviality, of Thm. 3.1.1, and despite the fact that the author feels he had failed at such an ambitious task, the pages you’re about to read should be interpreted in this spirit.

Structure of the thesis

The thesis is the results of a re-organization and methodical arrangement of the papers [FL16b, FL15a, FL15b, FL16a] (all written having my advisor as co-author) that have appeared on the arXiv since August 2014; the con- tent is essentially unchanged; some sections and subsections (like e.g. 1.5, 1.5.2.1, 3.2, a renewed proof of 4.3.20, and Ch. 6) do not appear anywhere at the moment of writing(1), but contain little new material and serve as linking sections making the discussion more complete and streamlined, de- veloping certain natural derivations of the basic theory which would have easily exceeded the average length of a research paper. Figure (1) below depicts the dependencies among the various chapters: a dashed line indicates a feeble logical dependence, whereas a thick line indicates a stronger one, unavoidable at first reading. The first three chapters outline the main result of the present work, summarized as follows:

For each stable ∞-category C there is a bijective correspondence between t-structures on the triangulated homotopy category Ho(C) and suitable orthogonal factorization systems on C called normal torsion theories.

This constitutes the backbone and the basic environment in which every subsequent application (the theory of recollements in stable ∞-categories in Ch. 5, and Bridgeland’s theory of stability conditions in Ch. 7) takes place. The main original contribution given in the present work is the ‘Rosetta stone’ theorem proving the quoted remark above; this is the main result of [FL16b], the only preprint that, at the moment of writing, has also been published by a peer-reviewed journal. There are several minor results following from the ‘Rosetta stone’, like the fact that constructions one can perform on normal torsion theories are (at least to the categorically-minded) more natural and canonical than the corresponding construction in homological algebra, done on bare t-structures.

(1)May 28, 2020 iii

A word on model dependency

Ideally speaking, if there is an equivalence between two models for ∞-categories (say, red and blue ∞-categories), these two models both possess a notion of factorization systems and a calculus(2) thereof; moreover, these two notions of factorization system correspond to each other under the equivalence of models. Turning this principle of equivalence and correspondence into a genuine theorem is often a subtle matter (apart from being inherently difficult and a delicate issue, this is perhaps due to the fact that theauthor is ignorant of how to retrieve such a result in the existing literature): it is however possible to recognize at least three different settings having each its own ‘calculus of factorization’: • stable model categories, where one can speak about homotopy factorization systems following [Bou77, Joy08]; this leads to the definition of a homotopy t-structures on stable model categories as suitable analogues of normal torsion theories in the set hfs(M) of homotopy factorization systems on a model category M. • dg-categories, where we speak about enriched (over Ch(k)) factorization systems (see [DK74]); this leads to the definition of dg-t-structures as enriched analogues of normal torsion theories in the set of dg-fs(D) of enriched factorization systems on a dg-category D. • derivators, where we can define t-derivators via a (genuinely new) notion of factorization system on a derivator, and recognize the analogue of normal torsion theory in this setting. At the moment of writing, all these points are being studied, and will hope- fully appear as separate results in the near future.

A word on the state of the art

Drawing equally from homological algebra, algebraic geometry, topology and category theory, the present work has not a single, well-defined flavour. Several sources of inspirations came from classical literature in algebraic topology [HPS97, Tie69, Hel68]; several others belong to the classical and less classical literature on algebraic geometry [Ver96, Bri07, Bri09, BO95]; others belong to pure category theory [RT07, CHK85, JM09, KT93, LW, Zan04], and others (see below) do not even belong to what is canonically recognized as mathematical literature. The approach to the theory of ∞-categories taken here will certainly appear rather unorthodox to some readers: [Lur09, Lur17] have taught the author more about 1-categories than he did about ∞-categories. This, again, must be attributed to the ignorance of the author, which is more comfort- able with the language of categories rather than with homotopy theory.

(2)By a ‘calculus’ of factorization systems we naïvely mean an analogue of the major results expressed in Ch. 1, translated from the red to the blue model. iv

Notation and Conventions

Categories (in the broad sense of ‘categories and ∞-categories’) are denoted as boldface letters C, D and suchlike, opposed to generic, variable simplicial sets which are denoted by capital Latin letters (this creates an extremely rare, harmless conflict with the same notation adopted for objects ina category: the context always allows us to avoid confusion); functors between categories are always denoted as capital Latin letters in a sufficiently large neighbourhood(3) of F, G, H, K and suchlike; the category of functors C → D is denoted as Fun(C, D), DC, [C, D] (or, at the risk of being pedantic, as (Q)Cat(C, D)); morphisms in Fun(C, D) (i.e. natural transformations between functors) are often written in the Greek alphabet; the simplex category ∆ is the topologist’s delta, having objects nonempty finite ordinals ∆[n] := {0 < 1 ··· < n} regarded as categories in the obvious way; we adopt [Lur09] as a main reference for ∞-category theory, even if we can’t help but confess that we profited from every single opportunity to deviate from the aesthetic of that book; in particular, we accept the (alas!) settled abuse to treat ‘quasicategory’ and ‘∞-category’