Introduction

The point of stability is that fibre and cofibre sequences become the same data. In a stable category one can build a long exact sequence in either direction and the choice is forced — there is no “missing” octahedral axiom to postulate, no shift functor whose square has to be remembered. Cones are functorial; mapping fibres glue. The triangulated structure on the homotopy category is a theorem, not a definition.

The motivating example is spectra. Spectra are the natural home for stable phenomena precisely because the suspension equivalence $\Sigma$ is built in. Stable categories generalise this: any pointed category in which finite pushouts and pullbacks agree behaves like a category of spectra.

A consequence worth absorbing: every stable category is automatically enriched in spectra, and morphism objects $\operatorname{map}(X, Y)$ are spectra rather than anima. The triangulated homotopy category is a shadow.

Body

A stable category is a category $\mathcal{C}$ such that:

  1. $\mathcal{C}$ has a zero object,
  2. $\mathcal{C}$ admits all finite limits and finite colimits, and
  3. a square in $\mathcal{C}$ is a pullback iff it is a pushout.

Equivalently, $\mathcal{C}$ is pointed, has all pushouts, and the suspension functor $\Sigma\colon \mathcal{C} \to \mathcal{C}$ is an equivalence.

Examples

  • The category of spectra $\mathsf{Sp}$.
  • $\mathsf{Mod}_R$ for $R$ a ring spectrum.
  • The derived category $\mathsf{D}(\mathcal{A})$ of an abelian category, viewed as a stable category.
  • Quasi-coherent sheaves $\mathsf{QCoh}(X)$ on a scheme.

Inside $\mathsf{Pr}^L$

A Presentable category that is also stable is called a presentable stable category; the resulting full subcategory $\mathsf{Pr}^L_{\mathrm{st}} \subset \mathsf{Pr}^L$ is itself presentable and inherits a symmetric monoidal structure (the Lurie tensor product). $\mathsf{Pr}^L_{\mathrm{st}}$ is canonically equivalent to $\mathsf{Mod}_{\mathsf{Sp}}(\mathsf{Pr}^L)$.

Within $\mathsf{Pr}^L_{\mathrm{st}}$:

Stable vs triangulated

The homotopy category $h\mathcal{C}$ of a stable category is canonically triangulated, but $\mathcal{C}$ carries strictly more data: cones are functorial in the diagram (not just up to non-unique isomorphism), the octahedral axiom is automatic, and homotopy-coherent diagrams of fibre sequences glue. A classical (i.e. 1-categorical) triangulated category is a shadow of a stable one; it is not, in general, possible to recover the stable category from its homotopy category alone.

Pointers

  • Lurie, Higher Algebra, §1.1
  • Lurie, Higher Topos Theory, §5.5

See also: Presentable category, Compactly generated category, Compactly assembled category.