The Lumenary · Research pathways

Stable ∞-categories

Name: ______________________

Date: ______________________

You will learn to

Define fibres, cofibres, suspension and loops in a pointed ∞-category, and define stable ∞-categories
Give examples — spectra, derived ∞-categories of rings, functor categories — and non-examples, and prove that no non-zero ordinary category is stable
Explain why the homotopy category of a stable ∞-category is additive and triangulated, proving the abelian group structure via Eckmann–Hilton
Show by an explicit example that cones in a triangulated category are not functorial, and explain how a stable ∞-category repairs this
Describe what stable ∞-categories add to triangulated categories, and what is known about enhancements

1.
Which of the following is the definition of a stable ∞-category?
- A.A pointed ∞-category in which every morphism has a fibre and a cofibre, and a triangle is a fibre sequence if and only if it is a cofibre sequence
- B.An ∞-category whose homotopy category is triangulated
- C.An ∞-category with finite limits and colimits
- D.An abelian category viewed as an ∞-category
2.
In $\mathcal{D}(\Z)$ , let $f : \Z \to \Z$ be multiplication by $6$ , with both copies of $\Z$ in degree $0$ . How many elements does $H_0(\mathrm{cofib}\, f)$ have?
3.
Pointed spaces form a pointed ∞-category with all limits and colimits. Which computation shows directly that it is not stable?
- A. $\Omega \Sigma S^0 \simeq \Omega S^1 \simeq \Z$ (a discrete space with infinitely many points), which is not equivalent to $S^0$
- B. $\pi_1(S^1) = \Z$ is not finite
- C.The zero object of pointed spaces is the empty set
- D.Pointed spaces have no cofibres
4.
In the proof that a pushout loop square forces $X \cong 0$ in an ordinary category, we show $\mathrm{id}_X = 0$ . Which property of the pushout is used, and how?
- A.Uniqueness: $\mathrm{id}_X$ and $0$ agree after both legs $0 \to X$ , so they are the same map out of the pushout
- B.Existence: the pushout gives a map $X \to 0$
- C.The pullback property of the same square
- D.Additivity: $\mathrm{id}_X - \mathrm{id}_X = 0$
5.
In stable_one_category_trivial, which hypothesis correctly formalises 'every fibre sequence is a cofibre sequence' for an ordinary category C with a zero object?
- A.∀ {P B Z : C} (snd : P ⟶ B) (g : B ⟶ Z), IsPullback (0 : P ⟶ 0) snd (0 : 0 ⟶ Z) g → IsPushout (0 : P ⟶ 0) snd (0 : 0 ⟶ Z) g
- B.∀ {X Y : C} (f : X ⟶ Y), IsZero X → IsZero Y
- C.∀ {P B Z : C} (snd : P ⟶ B) (g : B ⟶ Z), IsPushout (0 : P ⟶ 0) snd (0 : 0 ⟶ Z) g → IsPullback (0 : P ⟶ 0) snd (0 : 0 ⟶ Z) g
- D.∀ X : C, IsPullback (0 : (0 : C) ⟶ 0) 0 0 (0 : (0 : C) ⟶ X)
6.
In the worked example, work over the field $k = \mathbb{F}_5$ . How many morphisms of triangles $(a, b, c)$ from $T = (k \to 0 \to k[1] \to k[1])$ to $T' = (0 \to k[1] \to k[1] \to 0)$ extend the square $(a, b) = (0, 0)$ ?
7.
Why is $[X, Y] = \pi_0 \mathrm{Map}(X, Y)$ an abelian group in a stable ∞-category?
- A.Because $X \simeq \Sigma^2 \Omega^2 X$ and $[\Sigma^2 A, Y] = \pi_2 \mathrm{Map}(A, Y)$ , which is abelian by Eckmann–Hilton
- B.Because every stable ∞-category is the derived ∞-category of a ring
- C.Because additivity is part of the definition of stability
- D.Because $\pi_1$ of any space is abelian
8.
Which statements about enhancements of triangulated categories are true? (Choose all that apply.)
- A.Some triangulated categories are not the homotopy category of any stable ∞-category (or stable model category).
- B.The stable homotopy category has, up to Quillen equivalence, only one stable model category giving rise to it.
- C.The derived category of a ring is the homotopy category of a stable ∞-category.
- D.Enhancements are always unique when they exist.
9.
A linear map $f : \mathbb{Q}^m \to \mathbb{Q}^n$ is given by an $n \times m$ matrix A (a list of n rows). Viewed as a map of complexes in degree 0, its cofibre is the complex $\mathbb{Q}^m \xrightarrow{A} \mathbb{Q}^n$ in degrees 1 and 0. Using the provided rank, write cone_homology(A, m, n) returning (dim H_1, dim H_0) of the cofibre.
Computational — Practice

Answer key — Stable ∞-categories

1.
A stable ∞-category is a pointed ∞-category in which every morphism has a fibre and a cofibre and fibre sequences coincide with cofibre sequences. Equivalently (with finite limits and colimits), squares are pullbacks exactly when they are pushouts; or (with finite colimits) $\Sigma$ is an equivalence.
2.
$\mathrm{cofib}(f)$ is the complex $\Z \xrightarrow{6} \Z$ in degrees $1, 0$ . Its $H_0$ is $\Z/6\Z$ , with $6$ elements, and its $H_1 = \ker f = 0$ . The cofibre sequence $\Z \xrightarrow{6} \Z \to \Z/6$ is the familiar short exact sequence.
3.
$\Sigma S^0 = S^1$ and $\Omega S^1 \simeq \Z$ , a discrete space with infinitely many points, which is not $S^0$ . So $\Sigma$ is not an equivalence and pointed spaces are not stable. Spectra are obtained by formally inverting $\Sigma$ .
4.
The uniqueness half of the pushout property: two maps out of the pushout $X$ that agree after composing with both legs are equal. Here the legs are maps $0 \to X$ , so $\mathrm{id}_X$ and $0$ agree after each leg (any two maps out of $0$ coincide). Hence $\mathrm{id}_X = 0$ and $X$ is a zero object. In Lean this is h.hom_ext applied with (isZero_zero C).eq_of_src.
5.
Option (a). The proof applies it to the loop square of X, which is a pullback by loop_square_isPullback, to conclude that it is a pushout, and then isZero_of_loop_square_isPushout gives IsZero X.
6.
Both conditions read $0 = 0$ , so every $c \in \mathrm{Hom}(k[1], k[1]) \cong k$ works: $5$ fills over $\mathbb{F}_5$ . In the stable ∞-category $\mathcal{D}(k)$ , the square comes with a homotopy, and the homotopies (up to homotopy) are also in bijection with $k$ ; each picks out exactly one fill.
7.
$\mathrm{Map}(\Sigma A, Y) \simeq \Omega \mathrm{Map}(A, Y)$ , so $[\Sigma^2 A, Y] = \pi_2(\mathrm{Map}(A, Y), 0)$ , abelian by Eckmann–Hilton. Since $\Sigma$ is an equivalence, every $X$ is $\Sigma^2 A$ with $A = \Omega^2 X$ .
8.
The first three are true (Muro–Schwede–Strickland; Schwede; Lurie's derived ∞-categories). The fourth is false: uniqueness of enhancements holds in important cases (Lunts–Orlov, Canonaco–Stellari, Schwede) but not always.

from fractions import Fraction


def rank(rows):
    rows = [[Fraction(x) for x in r] for r in rows]
    r = 0
    cols = len(rows[0]) if rows else 0
    for c in range(cols):
        p = next((i for i in range(r, len(rows)) if rows[i][c] != 0), None)
        if p is None:
            continue
        rows[r], rows[p] = rows[p], rows[r]
        for i in range(len(rows)):
            if i != r and rows[i][c] != 0:
                t = rows[i][c] / rows[r][c]
                rows[i] = [x - t * y for x, y in zip(rows[i], rows[r])]
        r += 1
    return r


def cone_homology(A, m, n):
    r = rank(A)
    return (m - r, n - r)

Only the last function is new; rank is the helper from the starter. The cofibre of $f$ has $H_1 = \ker f$ and $H_0 = \operatorname{coker} f$ : kernels and cokernels are two homology groups of one object. In particular $f$ is an isomorphism exactly when its cofibre is zero, and the Euler characteristic of the cofibre is $n - m$ .