Examples of stable ∞-categories

• Derived categories
• dg-categories
• Unbounded derived categories
• Spectra

Derived categories#

Let $\mathcal{A}$ be an abelian category with enough projectives. Then there is a category $D^+(\mathcal{A})$, constructed as follows. Let $C \subseteq \mathcal{A}$ be the subcategory of projective objects, and consider the category of chain complexes $\operatorname{Ch}^+(C) \subseteq \operatorname{Ch}^+(\mathcal{A})$, meaning $C_m = 0$ for all sufficiently small $m$. (We still use the convention that $C_m \to C_{m-1}$.) Then we can define $D^+(\mathcal{A})$ as the homotopy category of this.

This is the $1$-categorical construction. The goal here is to upgrade it to a stable $\infty$-category. The idea is that $\operatorname{Ch}^+(\mathcal{A})$ is actually enriched in chain complexes. Given two chain complexes $A_\bullet$ and $B_\bullet$, we can define $$ \Hom(A_\bullet, B_\bullet)_ n = \prod_{l}^{} \Hom(A_l, B_{l+n}), $$ with the differential given by $$ (df)_ l = f_ {l+1} \circ d_B + (-1)^{n+1} d_B \circ f_l. $$ Then we see that $d^2 = 0$.

But more is true. A map of chain complexes is the same thing as a $0$-cycle, and $H_0(\Hom(A, B))$ will be the homotopy classes of maps. Recall that by Dold–Kan, we have that simplicial abelian groups are the same as non-negatively graded chain complexes of abelian groups. Since the mapping objects are chain complexes, we can use this to think of the derived category as a simplicially enriched category. Then we can use the homotopy coherent nerve to think of this as an $\infty$-category.

dg-categories#

There is a better and more direct way of doing this.

Definition 1. A dg-category is a category enriched $\operatorname{Ch}(\mathsf{Ab})$. For $\mathcal{C}$ a dg-category, we define its homotopy category $\operatorname{h}\mathcal{C}$ as the category with the same objects and $H^0$ as its morphisms.

Definition 2. For $\mathcal{C}$ a dg-category, we define is dg-nerve $N_\mathrm{dg}(\mathcal{C})$ as the simplicial set, where the $n$-simplices are given by the data of

• objects $X_i$ for $0 \le i \le n$,
• for $I = { i_- \lt i_1 \lt \dotsc \lt i_m \lt i_+ } \subset [n]$, an element $f_I \in \operatorname{Map}(X_{i_-}, X_{i_+})_m$,
• such that $$ df_I = \sum_{1 \le j \le m}^{} (-1)^i (f_{I-i_j} - f_{\lbrace i_j \lt \dotsb \lt i_+ \rbrace} \circ f_{\lbrace i_- \lt \dotsb \lt i_j \rbrace}). $$

For $\alpha \colon [m] \to [n]$ non-decreasing, the induced map is given by $$ (\lbrace X_i \rbrace, \lbrace f_I \rbrace) \mapsto (\lbrace X_{\alpha(i)} \rbrace, \lbrace g_J \rbrace) $$ where we define $$ g_J = \begin{cases} f_{\alpha(J)} & \alpha \vert_J \text{ is injective}, \br \id_ {X_1} & \text{if } J = \lbrace j_-, j_+ \rbrace, \alpha(j_-) = \alpha(j_+), \br 0 & \text{otherwise}. \end{cases} $$

You should think of this as, for every vertex of the $n$-simplex, there is a prescribed object, for every edge of the $n$-simplex, there is a morphism, and for every $2$-simplex, there is a homotopy between the two compositions, and so on.

Proposition 3. $N_\mathrm{dg}(\mathcal{C})$ is an $\infty$-category.

It also turns out that $$ \Hom_{N_\mathrm{dg}(\mathcal{C})}(X, Y) = \operatorname{DK}(\tau_{\ge 0} \operatorname{Map}_\mathcal{C}(X, Y)) $$ as Kan complexes.

Definition 4. Let $\mathcal{A}$ be an abelian category with enough projective objects. We define $$ D^+(\mathcal{A}) = N_\mathrm{dg}(\operatorname{Ch}^+(\mathcal{A}_\mathrm{proj})). $$

Theorem 5 (HA, 1.3.2.10). The category $D^+(\mathcal{A})$ is stable.

Remark 6. There is a notion of “inverting” morphisms in an ∞-category. It turns out that if we start with chain complexes in $\mathcal{A}$ and “invert” quasi-isomorphisms, then we actually get the same thing.

Unbounded derived categories#

Here, let $\mathcal{A}$ be a Grothendieck abelian category. This means that $\mathcal{A}$ has a generator, and small filtered colimits of monomorphisms are monomorphisms. For example, $R$-modules for $R$ a commutative ring works. Then there is a model structure on the category $\operatorname{Ch}(\mathcal{A})$, given by the following:

• cofibrations are degree-wise monomorphisms,
• weak equivalences are quasi-isomorphisms.

Proposition 7 (HA, 1.3.5.6). 1. If $M_n$ is injective for all $n$ and $M_n \cong 0$ for all large $n$, then $M$ is fibrant. 2. If $M$ is fibrant, then all $M_n$ are injective.

Once we have this, we can define the derived category as follows.

Definition 8. Let $\operatorname{Ch}(\mathcal{A})^0$ be the full subcategory of fibrant objects. Then define $$ D(\mathcal{A}) = N_\mathrm{dg}(\operatorname{Ch}(\mathcal{A})^0). $$

Proposition 9. The derived category $D(\mathcal{A})$ is stable.

Spectra#

We can think about $$ \varinjlim_n [\Sigma^n X, \Sigma^n Y] $$ which is some kind of homotopy classes of stable maps. It turns out that the stable homotopy category is trangulated. So we can try to upgrade this into a stable category.

There are two approaches to constructing this.

1. Construct this category “formally,” using object $(X, n)$ corresponding to $\Sigma^n X$, where $n$ can also be negative.
2. Use infinite loop spaces. Since there are loop space functors, we can look at the diagram $$ \dotsb \xrightarrow{\Omega} \mathcal{S} \xrightarrow{\Omega} \mathcal{S} $$ and take the limit.

Definition 10. If $F \colon \mathcal{C} \to \mathcal{D}$ is a functor of ∞-categories, we say that

• $F$ is excisive if pushout squares become pullback squares,
• $F$ is reduced if it sends the final object to the final object.

Definition 11. Suppose $\mathcal{C}$ admits finite limits. A spectrum object of $\mathcal{C}$ is a reduced and excisive functor $\mathcal{S}_ \ast^\mathrm{fin} \to \mathcal{C}$. We define $\operatorname{Sp}(\mathcal{C})$ as the full subcategory of $\operatorname{Fun}(\mathcal{S}_ \ast^\mathrm{fin}, \mathcal{C})$ given by the reduced excisive functors.

Here $\mathcal{S}_\ast^\mathrm{fin}$ is the pointed category of finite spaces (meaning having the homotopy type of a CW-complex with a finite number of cells). What does this have to infinite loop spaces? The point is that we have $$ \begin{CD} S^0 @>>> \ast \br @VVV @VVV \br \ast @>>> S^1 \end{CD} $$ as pushout square, and so we need that $$ \begin{CD} F(S^0) @>>> \ast \br @VVV @VVV \br \ast @>>> F(S^1) \end{CD} $$ is a pullback square. So $F(S^0)$ is the loop space of $F(S^1)$, and so on.

Proposition 12 (HA, 1.4.2.11). Suppose that $\mathcal{C}$ is a pointed ∞-category with finite limits and colimits. If $\Omega$ is an equivalence, then $\mathcal{C}$ is stable.

Proposition 13. If $\mathcal{C}$ has finite limits and colimits, then $\operatorname{Sp}(\mathcal{C})$ is stable.

Proposition 14. If $\mathcal{C}$ is a pointed ∞-category with finite limits, we have a tower of infinity categories $$ \dotsb \xrightarrow{\Omega} \mathcal{C} \xrightarrow{\Omega} \mathcal{C}. $$ Then $\operatorname{Sp}(\mathcal{C})$ can be identified with the homotopy limit of this diagram.

Here are two examples of spectra:

• the sphere spectrum $\mathbb{S}$ is presented by $\Sigma S^n \simeq S^{n+1}$;
• for an abelian group $A$, the Eilenberg–Mac Lane spectrum $HA$ is presented by $\Omega K(A, n+1) \simeq K(A, n)$.