Stable ∞-categories · The Lumenary

MotivationH

What triangulated categories cannot do

Derived categories of rings and sheaves, and the stable homotopy category of spectra, are triangulated categories (see Triangulated categories): additive categories with a shift functor and a class of distinguished triangles $X \to Y \to Z \to X[1]$ satisfying Verdier's axioms. They have been the language of homological algebra since the 1960s. They also have well-known defects:

Cones are not functorial. Every morphism $f$ sits in a distinguished triangle with third term $\mathrm{Cone}(f)$ , and a commutative square between two morphisms extends to a map of triangles (axiom TR3) — but the map on cones is not unique, so $f \mapsto \mathrm{Cone}(f)$ is not a functor. We will see an explicit example.
Homotopy limits and colimits are invisible. A triangulated category has no good notion of limits of diagrams beyond biproducts; the colimit of a sequence of maps can be formed (Bökstedt–Neeman's homotopy colimit) only up to non-unique isomorphism.
Gluing and invariants. Triangulated categories do not glue along open covers, and algebraic K-theory is not an invariant of a triangulated category: Schlichting (2002) found two exact categories whose derived categories are equivalent as triangulated categories but whose K-theories differ.

All three defects have the same cause: passing to a triangulated category keeps morphisms up to homotopy but forgets the homotopies. A stable ∞-category keeps them. It is an ∞-category (see ∞-categories) with a zero object in which fibre sequences and cofibre sequences coincide; its homotopy category is automatically triangulated, and everything that was "up to non-unique isomorphism" becomes "up to a contractible space of choices".

IntuitionH

Linear algebra up to homotopy

Think of a stable ∞-category as a homotopy-theoretic version of an abelian group, or of linear algebra. Three intuitions, each made precise below:

Kernels are cokernels, shifted. For abelian groups, the kernel and cokernel of a map are different things. In a stable ∞-category the fibre of $g : Y \to Z$ and the cofibre of $f : X \to Y$ are related by a perfect duality: $X \to Y \to Z$ is a fibre sequence exactly when it is a cofibre sequence. The derived category explains why: the cofibre of a map of modules $f : M \to N$ is a complex whose $H_0$ is $\operatorname{coker} f$ and whose $H_1$ is $\ker f$ — kernels and cokernels are two homology groups of one object.
Suspension is invertible. The suspension $\Sigma X = 0 \sqcup_X 0$ and the loop object $\Omega X = 0 \times_X 0$ are inverse equivalences. For pointed spaces they are not ( $\Omega\Sigma S^0 \simeq \Omega S^1 \simeq \Z$ , a discrete space, which is not $S^0$ ); "stabilising" means formally inverting $\Sigma$ , which is how spectra arise.
Squares are symmetric. In a stable ∞-category a commutative square is a pullback if and only if it is a pushout. Since squares are where homotopies live (a square commutes via a chosen homotopy), this is a statement about homotopies, which is why it is invisible in an ordinary category.

DefinitionsH

Fibres, cofibres and stability

Throughout, $\mathcal{C}$ is an ∞-category (a quasi-category). A commutative square in $\mathcal{C}$ is a map $\Delta[1] \times \Delta[1] \to \mathcal{C}$ : four objects, four morphisms, a diagonal, and two 2-simplices witnessing that the two composites agree. Pullbacks and pushouts are ∞-categorical limits and colimits (Lurie, Higher Topos Theory, Chapter 4).

Definition— pointed ∞-category, triangles, fibres and cofibres

$\mathcal{C}$ is pointed if it has a zero object $0$ , both initial and terminal. A triangle in $\mathcal{C}$ is a commutative square

\begin{array}{ccc} X & \xrightarrow{f} & Y \\ \downarrow & & \downarrow{\scriptstyle g} \\ 0 & \longrightarrow & Z \end{array}

with a zero object in the lower left corner. It is a fibre sequence if the square is a pullback, and then $X$ is a fibre of $g$ , written $X = \mathrm{fib}(g)$ ; it is a cofibre sequence if the square is a pushout, and then $Z$ is a cofibre of $f$ , written $Z = \mathrm{cofib}(f)$ . The suspension of $X$ is $\Sigma X = \mathrm{cofib}(X \to 0)$ and the loop object is $\Omega X = \mathrm{fib}(0 \to X)$ .

Definition— stable ∞-category

A pointed ∞-category $\mathcal{C}$ is stable if every morphism has a fibre and a cofibre, and a triangle in $\mathcal{C}$ is a fibre sequence if and only if it is a cofibre sequence (Lurie, Higher Algebra, Definition 1.1.1.9). A functor between stable ∞-categories is exact if it preserves zero objects and fibre (equivalently, cofibre) sequences.

Equivalent characterisations (Lurie, Higher Algebra, §1.1.3 and §1.4.2): a pointed ∞-category with finite limits and colimits is stable if and only if a commutative square is a pullback exactly when it is a pushout; and a pointed ∞-category with finite colimits is stable if and only if $\Sigma : \mathcal{C} \to \mathcal{C}$ is an equivalence. Note that stability is a property of an ∞-category, whereas a triangulation is extra structure on a category.

Definition— spectra

The ∞-category of spectra is the limit of the tower $\cdots \xrightarrow{\Omega} \mathcal{S}_* \xrightarrow{\Omega} \mathcal{S}_*$ of pointed spaces under the loop functor. Concretely, a spectrum is a sequence of pointed spaces $E_0, E_1, \dots$ with equivalences $E_n \simeq \Omega E_{n+1}$ . Its homotopy groups $\pi_k E = \pi_{k+n} E_n$ (for $n$ large) are defined for all $k \in \Z$ . Spectra $\mathrm{Sp}$ form the stabilisation of $\mathcal{S}_*$ : the universal stable ∞-category with a finite-limit-preserving functor $\Omega^\infty : \mathrm{Sp} \to \mathcal{S}_*$ .

Worked examplesH

Examples and non-examples

Example— Spectra

$\mathrm{Sp}$ is stable, and its homotopy category is the classical stable homotopy category. The sphere spectrum $\mathbb{S}$ has $\pi_k \mathbb{S} = \pi_k^s$ , the stable homotopy groups of spheres ( $\pi_0^s = \Z$ , $\pi_1^s = \Z/2$ , $\pi_2^s = \Z/2$ , $\pi_3^s = \Z/24$ ); an abelian group $A$ gives the Eilenberg–MacLane spectrum $HA$ with $\pi_0 HA = A$ and all other homotopy groups zero. Fibre sequences of spectra give long exact sequences of homotopy groups in both directions, which is the algebraic heart of stability.

Example— Derived ∞-categories

For a ring $R$ , the derived ∞-category $\mathcal{D}(R)$ is obtained from chain complexes of $R$ -modules by inverting quasi-isomorphisms in the ∞-categorical sense (for instance as the dg-nerve of the dg-category of K-projective complexes). It is stable; its homotopy category is the classical derived category $D(R)$ ; the cofibre of a chain map is its mapping cone, and $\Sigma$ is the shift $[1]$ . The same construction works for any abelian category with enough projectives or injectives, for quasi-coherent sheaves on a scheme, and for modules over a ring spectrum.

Example— Constructions that preserve stability

If $\mathcal{C}$ is stable and $K$ is any simplicial set, the functor ∞-category $\mathrm{Fun}(K, \mathcal{C})$ is stable (fibres and cofibres are computed pointwise). In particular $\mathrm{Fun}(\Delta[1], \mathcal{C})$ , the ∞-category of morphisms of $\mathcal{C}$ , is stable — and the cofibre is a functor on it, as we explain below.

Example— Non-examples

Pointed spaces $\mathcal{S}_*$ : pointed, with all limits and colimits, but $\Sigma$ is not an equivalence ( $\Omega \Sigma S^0 \simeq \Z \not\simeq S^0$ ).
Any non-zero ordinary category, such as abelian groups or even the triangulated category $D(R)$ regarded as an ordinary category. Theorem 1 below shows that in an ordinary category with a zero object the 1-categorical loop object $0 \times_X 0$ is always $0$ , so if fibre sequences were cofibre sequences every object would be zero. (In $D(R)$ the square with corners $X[-1], 0, 0, X$ commutes, but it is not a pullback in the ordinary category $D(R)$ ; it is one only in the ∞-category $\mathcal{D}(R)$ .)

ProofH

Central theorems

Theorem— Theorem 1 (stability is invisible to ordinary categories)

Let $\mathcal{C}$ be an ordinary category with a zero object. Then for every object $X$ :

the square with corners $0, 0, 0, X$ is a pullback, so the 1-categorical loop object $0 \times_X 0$ is $0$ ; dually the 1-categorical suspension $0 \sqcup_X 0$ is $0$ ;
if that square is also a pushout, then $X \cong 0$ .

Consequently, if every fibre sequence in $\mathcal{C}$ is a cofibre sequence, then every object of $\mathcal{C}$ is zero. Since limits and colimits in the nerve $N(\mathcal{C})$ are ordinary limits and colimits, no non-zero ordinary category is a stable ∞-category.

Proof

(1) A cone over $0 \to X \leftarrow 0$ is an object $W$ with two maps $W \to 0$ . There is exactly one map $W \to 0$ , and it factors both legs, because any two maps into a zero object are equal; so $0$ with its identity legs is a limit. Dually, a cocone under $0 \leftarrow X \to 0$ is a pair of maps $0 \to W$ , and the unique map $0 \to W$ is the unique factorisation.

(2) Suppose the square $0 \to 0$ , $0 \to X$ is a pushout. The maps $\mathrm{id}_X$ and $0 : X \to X$ agree after composing with each leg $0 \to X$ , because any two maps out of a zero object are equal. By the uniqueness part of the universal property of the pushout, $\mathrm{id}_X = 0$ , so $X$ is a zero object.

For the consequence, apply the hypothesis to the triangle with $X$ replaced by $0$ , $Y = 0$ , $Z = X$ : it is a fibre sequence by (1), hence a cofibre sequence, hence $X \cong 0$ by (2).

∎

In a stable ∞-category the same square, with corners $\Omega X, 0, 0, X$ , is both a pullback and a pushout, and $\Omega X$ is usually far from $0$ : in $\mathcal{D}(R)$ it is $X[-1]$ . The information lives in the homotopy that makes the square commute — two null-homotopies of the zero map $\Omega X \to X$ , which together form a "loop" — and an ordinary category has no room for it.

Theorem— Theorem 2 (the homotopy category is additive; Lurie, Higher Algebra §1.1.2)

If $\mathcal{C}$ is stable, then $h\mathcal{C}$ is additive: each $\mathrm{Hom}_{h\mathcal{C}}(X, Y)$ is an abelian group, composition is bilinear, and finite products and coproducts exist and agree.

Proof

We prove the key point, that $[X, Y] := \mathrm{Hom}_{h\mathcal{C}}(X, Y) = \pi_0 \mathrm{Map}(X, Y)$ is naturally an abelian group, and refer to Lurie for bilinearity and biproducts.

The functor $\mathrm{Map}(-, Y)$ turns colimits in $\mathcal{C}$ into limits of spaces. Applied to the pushout $\Sigma A = 0 \sqcup_A 0$ it gives a pullback of spaces $\mathrm{Map}(\Sigma A, Y) \simeq \mathrm{Map}(0, Y) \times_{\mathrm{Map}(A, Y)} \mathrm{Map}(0, Y)$ . Since $\mathrm{Map}(0, Y)$ is contractible (with image the zero map), this is the loop space $\Omega(\mathrm{Map}(A, Y), 0)$ . Applying this twice,

[\Sigma^2 A, Y] = \pi_0 \Omega^2 \mathrm{Map}(A, Y) = \pi_2\big(\mathrm{Map}(A, Y), 0\big),

which is an abelian group by the Eckmann–Hilton argument (see Higher categories). In a stable ∞-category $\Sigma$ is an equivalence with inverse $\Omega$ , so every $X$ is $\Sigma^2 A$ for $A = \Omega^2 X$ , and $[X, Y] \cong \pi_2(\mathrm{Map}(\Omega^2 X, Y), 0)$ is an abelian group, naturally in $X$ and $Y$ .

∎

Theorem— Theorem 3 (the homotopy category is triangulated; Lurie, Higher Algebra Theorem 1.1.2.14; stated)

Let $\mathcal{C}$ be stable. Then $h\mathcal{C}$ , with shift functor $\Sigma$ and distinguished triangles the diagrams $X \xrightarrow{f} Y \xrightarrow{g} Z \xrightarrow{h} \Sigma X$ arising from a pair of pushout squares

\begin{array}{ccccc} X & \xrightarrow{f} & Y & \longrightarrow & 0 \\ \downarrow & & \downarrow{\scriptstyle g} & & \downarrow \\ 0 & \longrightarrow & Z & \xrightarrow{h} & W \end{array}

(the outer rectangle is a pushout, so $W \simeq \Sigma X$ ), is a triangulated category: it satisfies Verdier's axioms TR1–TR4, including the octahedral axiom.

The axioms are inherited from the ∞-category: TR1 because cofibres exist; rotation (TR2) by pasting pushout squares; TR3 because any square in $h\mathcal{C}$ lifts to a square in $\mathcal{C}$ and cofibres are functorial there; the octahedral axiom (TR4) by taking cofibres of a composable pair, a diagram of shape $\Delta[2]$ , all at once. The long exact sequences of a triangulated category are the shadows of the long exact sequences of homotopy groups of the fibre sequences of mapping spaces $\mathrm{Map}(W, X) \to \mathrm{Map}(W, Y) \to \mathrm{Map}(W, Z)$ .

Theorem— Theorem 4 (cofibres are functorial; stated)

In a stable ∞-category (indeed in any pointed ∞-category with cofibres) there is a functor $\mathrm{cofib} : \mathrm{Fun}(\Delta[1], \mathcal{C}) \to \mathcal{C}$ sending a morphism to its cofibre, well defined up to a contractible space of choices.

The reason: cofibre sequences are left Kan extensions (colimits), and the ∞-category of cofibre sequences maps to $\mathrm{Fun}(\Delta[1], \mathcal{C})$ by a trivial fibration (Lurie, Higher Topos Theory, §4.3.2). The crucial difference from TR3 is the source category: a morphism in $\mathrm{Fun}(\Delta[1], \mathcal{C})$ from $f : X \to Y$ to $f' : X' \to Y'$ is not just a pair $(a, b)$ with $b f = f' a$ in $h\mathcal{C}$ , but a pair together with a chosen homotopy $H : b f \simeq f' a$ . Different homotopies induce different maps on cofibres.

Example— Cones are not functorial in a triangulated category — and how the homotopy fixes it

Let $k$ be a field and work in $D(k)$ , with the distinguished triangles

T : \; k \to 0 \to k[1] \xrightarrow{-\mathrm{id}} k[1], \qquad T' : \; 0 \to k[1] \xrightarrow{\mathrm{id}} k[1] \to 0.

( $T$ is a rotation of the contractible triangle $k \to k \to 0 \to k[1]$ ; $T'$ is the cone triangle of $0 \to k[1]$ .) Take $a = 0 : k \to 0$ and $b = 0 : 0 \to k[1]$ ; the square $b \circ f = f' \circ a$ commutes trivially. A morphism of triangles $(a, b, c)$ needs $c : k[1] \to k[1]$ with $c \circ g = g' \circ b$ and $h' \circ c = a[1] \circ h$ . Here $g : 0 \to k[1]$ and $h' : k[1] \to 0$ , so both conditions say $0 = 0$ : every $c = \lambda \cdot \mathrm{id}$ , $\lambda \in k$ , completes the square. The map on cones is not determined.

In $\mathcal{D}(k)$ , the square $(a, b)$ comes with a homotopy $H : b f \simeq f' a$ between two maps $k \to k[1]$ (both zero). Homotopies from the zero map to itself form the loop space $\Omega \mathrm{Map}(k, k[1])$ , whose components are $[k, \Omega k[1]] = [k, k] = k$ . At the level of chain complexes, $H$ is a degree-one map $h : k \to k$ (from $X_0$ to $Y'_1$ ), the induced map on mapping cones is $(u, v) \mapsto (a u, b v + h u)$ , and on $\mathrm{Cone}(f) = k[1] \to k[1] = \mathrm{Cone}(f')$ it is multiplication by $h = \lambda$ . So each choice of homotopy picks out exactly one of the many TR3 fillers — the experiment below computes this with exact arithmetic.

Theorem— Theorem 5 (enhancements; stated)

For a ring $R$ (more generally a Grothendieck abelian category), the classical derived category is the homotopy category of the stable ∞-category $\mathcal{D}(R)$ (Lurie, Higher Algebra, §1.3).
Not every triangulated category is the homotopy category of a stable ∞-category: Muro, Schwede and Strickland (2007) constructed triangulated categories with no model.
Enhancements can be unique: Schwede's rigidity theorem (2007) shows that any stable model category whose homotopy category is equivalent to the stable homotopy category, as a triangulated category, is Quillen equivalent to spectra; Lunts and Orlov (2010) and Canonaco and Stellari (2018) proved uniqueness of dg-enhancements for many derived categories of geometric origin.

ExplanationH

The landscape: what is known and what is open

Established. Stable ∞-categories are the standard setting for modern homological algebra and stable homotopy theory (Lurie, Higher Algebra, Chapter 1): t-structures and their abelian hearts, the derived ∞-categories of abelian categories, Brown representability, and the comparison with pretriangulated dg-categories, which over a field $k$ are equivalent, as a homotopy theory, to $k$ -linear stable ∞-categories (Cohn, 2013). Algebraic K-theory has a universal characterisation as an invariant of small stable ∞-categories (Blumberg, Gepner and Tabuada, 2013), resolving the third defect of the motivation. Spectral algebraic geometry, noncommutative motives and the study of localising invariants (K-theory, topological Hochschild and cyclic homology) are built on this foundation.

Known limits. There are triangulated categories without enhancements (Theorem 5), and triangulated categories with several inequivalent enhancements are known; when an enhancement is unique is understood in important cases but not in general.

Open problems that live here. Many central open problems of stable homotopy theory are statements about the stable ∞-category of spectra: the structure of the stable homotopy groups of spheres (computed only in a finite range), and questions in chromatic homotopy theory about localisations of $\mathrm{Sp}$ . Classifying the "thick subcategories" of a stable ∞-category, as Hopkins and Smith did for finite spectra (1998), is open for most examples.

Formal mathematicsF

What Lean can say: the 1-categorical shadow

Mathlib has no stable ∞-categories (and no spectra as an ∞-category), but it has substantial triangulated-category theory: pretriangulated and triangulated categories with Verdier's axioms, the homotopy category of complexes and the derived category of an abelian category as triangulated categories, and t-structures. The formal side of this lesson therefore does two things: it proves Theorem 1 — why stability cannot happen in an ordinary category — and it records the triangulated shadow.

Theorem 1, from the definitions of pullbacks and pushouts in Mathlib (IsPullback, IsPushout). The proofs are those of the lesson: maps into or out of a zero object are unique (IsZero.eq_of_tgt, IsZero.eq_of_src), and the uniqueness half of the pushout property (IsPushout.hom_ext) forces 𝟙 X = 0:

loop_square_isPullbackLean 4Proved here from basic factsFormally verified

/-- In an ordinary category with a zero object, the **loop object** `ΩX = 0 ×_X 0` is zero:
the square with three corners `0` and fourth corner `X` is a pullback. Proof: a cone over
`0 ⟶ X ⟵ 0` is an object `W` with two maps `W ⟶ 0`; there is exactly one map `W ⟶ 0`, and it
factors both legs because all maps into a zero object are equal. In a stable ∞-category,
by contrast, `ΩX` is the shift `X[-1]`, and the square commutes only *via a homotopy*, which
is where the information lives. -/
theorem loop_square_isPullback (X : C) :
    IsPullback (0 : (0 : C) ⟶ 0) (0 : (0 : C) ⟶ 0) (0 : (0 : C) ⟶ X) (0 : (0 : C) ⟶ X) := by
  refine IsPullback.of_isLimit (c := PullbackCone.mk (0 : (0 : C) ⟶ 0) (0 : (0 : C) ⟶ 0)
    (by simp)) ?_
  refine PullbackCone.IsLimit.mk _ (fun _ => 0) ?_ ?_ ?_
  · intro s; exact (isZero_zero C).eq_of_tgt _ _
  · intro s; exact (isZero_zero C).eq_of_tgt _ _
  · intro s m _ _; exact (isZero_zero C).eq_of_tgt _ _