Triangulated Category

A triangulated category is the axiomatic framework for the homotopy category $K(\mathcal{A})$ and the derived category $D(\mathcal{A})$ . It captures the essential features of these categories: a shift functor and a class of distinguished triangles replacing short exact sequences, without requiring the full abelian structure.

Definition

Definition5.1Triangulated Category

A triangulated category is an additive category $\mathcal{T}$ equipped with:

An additive autoequivalence $[1] : \mathcal{T} \to \mathcal{T}$ called the shift functor (or translation functor, or suspension).
A class of distinguished triangles $X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} X[1]$ , subject to the axioms (TR1)-(TR4) below.

Definition5.2Axioms TR1-TR4

(TR1) (a) Every morphism $u : X \to Y$ can be completed to a distinguished triangle $X \to Y \to Z \to X[1]$ . (b) The triangle $X \xrightarrow{\mathrm{id}} X \to 0 \to X[1]$ is distinguished. (c) Any triangle isomorphic to a distinguished triangle is distinguished.

(TR2) Rotation. $X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} X[1]$ is distinguished if and only if $Y \xrightarrow{v} Z \xrightarrow{w} X[1] \xrightarrow{-u[1]} Y[1]$ is distinguished.

(TR3) Morphisms of triangles. Given distinguished triangles and a commutative square $(\alpha, \beta)$ on the first two terms, there exists (not necessarily uniquely) a morphism $\gamma$ completing the morphism of triangles.

(TR4) Octahedral axiom. See the octahedral axiom page.

Examples

ExampleK(A) is triangulated

For an abelian category $\mathcal{A}$ , $K(\mathcal{A})$ is triangulated with shift $A^\bullet \mapsto A^\bullet[1]$ and distinguished triangles $A \xrightarrow{f} B \to \mathrm{Cone}(f) \to A[1]$ .

ExampleD(A) is triangulated

The derived category $D(\mathcal{A})$ inherits a triangulated structure from $K(\mathcal{A})$ via Verdier localization. Distinguished triangles in $D(\mathcal{A})$ are images of distinguished triangles in $K(\mathcal{A})$ .

ExampleStable homotopy category

The stable homotopy category $\mathbf{SHC}$ of spectra is triangulated with suspension as the shift functor and cofiber sequences as distinguished triangles.

ExampleStable module category

For a self-injective algebra $A$ , the stable module category $\underline{\mathrm{mod}}\text{-}A$ (modding out maps factoring through projectives) is triangulated. The shift functor is $\Omega^{-1}$ (the cosyzygy).

ExampleNot triangulated: Ab

The abelian category $\mathbf{Ab}$ itself is not triangulated. The notion of distinguished triangle does not make sense for abelian categories; instead, abelian categories have short exact sequences.

ExampleSingularity category

For a Noetherian ring $R$ , the singularity category $D_{\mathrm{sg}}(R) = D^b(\mathrm{mod}\text{-}R) / \mathrm{Perf}(R)$ (bounded derived category modulo perfect complexes) is triangulated and measures the singularity of $R$ .

ExampleTriangulated structure on K(Inj)

$K(\mathrm{Inj}\, \mathcal{A})$ (homotopy category of injective complexes) is triangulated and equivalent to $D(\mathcal{A})$ when $\mathcal{A}$ has enough injectives.

ExampleExact triangle = long exact sequence

In a triangulated category, applying a cohomological functor to a distinguished triangle gives a long exact sequence. This is how long exact sequences arise in derived categories.

ExampleDistinguished triangles are not short exact sequences

A distinguished triangle $A \to B \to C \to A[1]$ is NOT a short exact sequence. The third map $C \to A[1]$ has no analogue in the abelian setting. Moreover, the "third vertex" $C$ (the cone) is determined only up to non-unique isomorphism.

ExampleTriangulated categories from DG categories

The homotopy category $H^0(\mathcal{C})$ of a pretriangulated DG category $\mathcal{C}$ is triangulated. This provides a source of triangulated categories beyond homotopy/derived categories.

ExampleCompact objects

In a triangulated category with coproducts, an object $X$ is compact if $\mathrm{Hom}(X, \bigoplus_i Y_i) \cong \bigoplus_i \mathrm{Hom}(X, Y_i)$ . In $D(R)$ , the compact objects are the perfect complexes.

ExampleSemi-orthogonal decomposition

A triangulated category $\mathcal{T}$ has a semi-orthogonal decomposition $\mathcal{T} = \langle \mathcal{A}, \mathcal{B} \rangle$ if every object fits into a distinguished triangle $B \to X \to A \to B[1]$ with $A \in \mathcal{A}$ , $B \in \mathcal{B}$ , and $\mathrm{Hom}(\mathcal{A}, \mathcal{B}) = 0$ . This is a key tool in algebraic geometry.

Deficiencies of Triangulated Categories

RemarkNon-functoriality of cones

The cone of a morphism in a triangulated category is not functorial: it is determined only up to non-unique isomorphism. This is a fundamental deficiency that motivates the use of stable $\infty$ -categories and DG enhancements.

RemarkLooking ahead

The deficiencies of triangulated categories are addressed by:

DG categories (algebraic enhancement)
Stable $\infty$ -categories (homotopical enhancement)
t-structures (recovering abelian structure from triangulated categories)