Many people care about the category of spectra, or the derived category of a ring. These are examples of stable categories.
Stable ∞-categories#
Definition 1. An $\infty$-category is pointed if it has an object $0$ that is both initial and final.
In such a case, each mapping space $\operatorname{Hom}(X, Y)$ is pointed, just by looking at $X \to 0 \to Y$.
Definition 2. For $\mathcal{C}$ an $\infty$-category, a triangle is a diagram $$ \begin{CD} X @>{f}>> Y \br @VVV @V{g}VV \br 0 @>>> Z. \end{CD} $$ We say that $X \to Y \to Z$ is a fiber sequence if this is a pullback diagram, and a cofiber sequence if it is a pushout.
The restriction of this diagram to just $f$, or to just $g$, are trivial Kan fibrations. So cofibers and fibers are unique up to homotopy.
Definition 3. We say that an $\infty$-category $\mathcal{C}$ is stable if
- $\mathcal{C}$ is pointed,
- every morphism has both fibers and cofibers, and
- a triangle is a fiber sequence if and only if it is a cofiber sequence.
Triangulated categories#
The homotopy category of a stable $\infty$-category is what people call a triangulated category. This is a $1$-category $\mathcal{D}$ that
- is additive,
- has an equivalence $\mathcal{D} \to \mathcal{D}$ called the shift,
- a collection of distinguished triangles $X \to Y \to Z \to X[1]$, such that
- every morphism $X \to Y$ fits uniquely in a distinguished triangle, and for the identity morphism it gives $X \to X \to 0 \to X[1]$,
- distinguished triangles are stable under shifts, so $Y \to Z \to X[1] \to Y[1]$ is a distinguished triangle as well,
- a map between distinguished triangles on $X, Y, X[1]$ induces a map between $Z$,
- if $X \to Y \to Y/X \to X[1]$, $Y \to Z \to Z/Y \to Y[1]$, and $X \to Z \to Z/X \to X[1]$ are distinguished, then there is a distinguished $$ Y/X \to Z/X \to Z/Y \to Y/X[1] $$ for which the natural diagram commutes.
Theorem 4. If $\mathcal{C}$ is an $\infty$-category, there is a natural structure of a triangulated category on $\operatorname{h}\mathcal{C}$.
Let $\mathcal{M}^\Sigma$ denote the category of pushout diagrams $$ \begin{CD} X @>>> 0 \br @VVV @VVV \br 0 @>>> Y. \end{CD} $$ There is a functor $\mathcal{M}^\Sigma \to \mathcal{C}$ restricting to $X$, and this is a trivial Kan fibration by this fact from before. Then we can define a section $$ \Sigma \colon \mathcal{C} \to \mathcal{M}^\Sigma \to \mathcal{C}. $$ Similarly, we can define a loop space functor $\Omega$, and this will be an inverse to $\Sigma$. Denote $$ X[n] = \begin{cases} \Sigma^n X & n \ge 0, \br \Omega^{-n} X & n \le 0. \end{cases} $$
From construction, we have a homotopy equivalence $$ \operatorname{Map} _ \mathcal{C}(\Sigma^2 X, Y) \simeq \Omega^2 \operatorname{Map}(X, Y), $$ and then taking $\pi_0$ on both sides gives an abelian structure on all mapping spaces. All that is left for an additive category is the existence of coproducts, and you can constuct this as the cofiber of the zero map $X[-1] \to Y$.
Definition 5. A diagram $X \to Y \to Z \to \Sigma X$ in the homotopy diagram is a distinguished triangle if there is a pushout diagram $$ \begin{CD} X @>>> Y @>>> 0 \br @VVV @VVV @VVV \br 0 @>>> Z @>>> W \end{CD} $$ lifting the triangle in the homotopy category.
So let’s now verify the axioms. Let $\mathcal{E}$ denote the category of diagrams as above in $\mathcal{C}$. Then the restriction map to the $X \to Y$ part defines a Kan fibration $$ \mathcal{E} \to \mathsf{Fun}(\Delta^1, \mathcal{C}), $$ and taking connected components verifies the first axiom.
For the second claim, make another pushout and compare $$ \begin{CD} X @>>> Y @>>> 0 \br @VVV @VVV @VVV \br 0 @>>> Z @>>> W \br @. @VVV @VVV \br @. 0 @>>> V \end{CD} $$ with $X[1] \to Y[1]$.
For the third property, lift the maps $X \to Y$ and $X^\prime \to Y^\prime$ to a square in $\mathcal{C}$, then do pushouts to get an induced map $Z \to Z^\prime$. Then the next square will automatically commute.
For the octahedral axiom, what we do is stare at the diagram $$ \begin{CD} X @>>> Y @>>> Z @>>> 0 @. \br @VVV @VVV @VVV @VVV @. \br 0 @>>> Y/X @>>> Z/X @>>> X^\prime @>>> 0 \br @. @VVV @VVV @VVV @VVV \br @. 0 @>>> Z/Y @>>> Y^\prime @>>> (Y/X)^\prime. \end{CD} $$ Everything reduces to Proposition 4.3.2.15 of HTT.
The Dold–Kan correspondence#
In the classical case, this is what it says.
Theorem 6. If $\mathcal{A}$ is an abelian category, then the category of simplicial objects in $\mathcal{A}$ is canonically equivalent to non-negatively graded (homological) chain complex of $\mathcal{A}$.
There is an $\infty$-version of this.
Theorem 7. Let $\mathcal{C}$ be an $\infty$-category. Then there is a canonical equivalence between functors $N(\mathbb{Z} _ {\ge 0}) \to \mathcal{C}$ and functors $N(\Delta)^\mathrm{op} \to \mathcal{C}$.