and
nonabelian homological algebra
Any (infinity,1)-category $C$ can be flattened, by ignoring higher morphisms, into a 1-category $ho(C)$ called its homotopy category. The notion of a triangulated structure is designed to capture the additional structure canonically existing on $ho(C)$ when $C$ has the property of being stable. This structure can be described roughly as the data of an invertible suspension functor, together with a collection of sequences called distinguished triangles, which behave like shadows of homotopy (co)fibre sequences in stable (infinity,1)-categories, subject to various axioms.
A central class of examples of triangulated categories are the derived categories $D(\mathcal{A})$ of abelian categories $\mathcal{A}$. These are the homotopy categories of the stable (∞,1)-categories of chain complexes in $\mathcal{A}$. However the notion also encompasses important examples coming from nonabelian contexts, like the stable homotopy category, which is the homotopy category of the stable (infinity,1)-category of spectra. Generally, it seems that all triangulated categories appearing in nature arise as homotopy categories of stable (infinity,1)-categories (though examples of “exotic” triangulated categories probably exist).
By construction, passing from a stable (infinity,1)-category to its homotopy category represents a serious loss of information. In practice, endowing the homotopy category with a triangulated structure is often sufficient for many purposes. However, as soon as one needs to remember the homotopy colimits and homotopy limits that existed in the stable (infinity,1)-category, a triangulated structure is not enough. For example, even the mapping cone in a triangulated category is not functorial. Hence it is often necessary to work with some enhanced notion of triangulated category, like stable derivators, pretriangulated dg-categories, stable model categories or stable (infinity,1)-categories. See enhanced triangulated category for more details.
The notion of triangulated category was developed by Jean-Louis Verdier in his 1963 thesis under Alexandre Grothendieck. His motivation was to axiomatize the structure existing on the derived category of an abelian category. Axioms similar to Verdier’s were given by Albrecht Dold and Dieter Puppe in a 1961 paper. A notable difference is that Dold-Puppe did not impose the octahedral axiom (TR4).
The traditional definition of triangulated category is the following. But see remark 1 below.
A triangulated category is
equipped with a collection of triangles called distinguished triangles (dts)
such that the following axioms hold
TR0: every triangle isomorphic to a distinguished triangle is itself a distinguished triangle;
TR1: the triangle
is a distinguished triangle;
TR2: for all $f : X \to Y$, there exists a distinguished triangle
TR3: a triangle
is a distinguished triangle precisely if
is a distinguished triangle;
TR4: given two distinguished triangles
and
and given morphisms $\alpha$ and $\beta$ in
there exists a morphism $\gamma : Z \to Z'$ extending this to a morphism of distinguished triangles in that the diagram
commutes;
TR5: given three distinguished triangles of the form
there exists a distinguished triangle
such that the following big diagram commutes
This classical definition is actually redundant; TR4 and one direction of TR3 follow from the remaining axioms. See (May).
In the context of triangulated categories the translation functor $T : C \to C$ is often called the suspension functor and denoted $(-)[1] : X \mapsto X[1]$ (in an algebraic context) or $S$ or $\Sigma$ (in a topological context). However unlike “translation functor”, suspension functor is also the term used when the invertibility is not assumed, cf. suspended category.
If $(f,g,h)$ is a distinguished triangle, then $(f,g,-h)$ is not generally distinguished, although it is “exact” (induces long exact sequences in homology and cohomology). However, $(f,-g,-h)$ is always distinguished, since it is isomorphic to $(f,g,h)$:
The homotopy category of chain complexes $K(\mathcal{A})$ in an abelian category (the category of chain complexes modulo chain homotopy) is a triangulated category: the translation functor is the suspension of chain complexes and the distinguished triangles are those coming from the mapping cone construction $X \stackrel{f}{\to}Y \to Cone(f) \to T X$ in $Ch_\bullet(\mathcal{A})$.
The stable homotopy category (the homotopy category of the stable (∞,1)-category of spectra) is a triangulated category. This is also true for parametrized, equivariant, etc. spectra.
The stable category of a Quillen exact category is suspended category as exhibited by Bernhard Keller. If the exact category is Frobenius, i.e. has enough injectives and enough projective and these two classes coincide, then the suspension of the stable category is in fact invertible, hence the stable category is triangulated. A triangulated category equivalent to a triangulated categories is said to be an algebraic triangulated category.
As mentioned before, the homotopy category of a stable (∞,1)-category is a triangulated category. Slightly more generally, this applies also to a stable derivator, and slightly less generally, it applies to a stable model category. This includes both the preceding examples.
The localization $C/N$ of any triangulated category $C$ at a null system $N \hookrightarrow C$, i.e. the localization using the calculus of fractions given by the morphisms $f : X \to Y$ such that there exists distinguished triangles $X \to Y \to Z \to T X$ with $Z$ an object of a null system, is still naturally a triangulated category, with the distinguished triangles being the triangles isomorphic to an image of a distinguished triangle under $Q : C \to C/N$.
In algebraic geometry, important examples are given by the various triangulated categories of sheaves associated to a variety (e.g. bounded derived category of coherent sheaves, triangulated category of perfect complexes).
enhanced triangulated category, pretriangulated dg-category, stable (∞,1)-category
well-generated triangulated category, compactly generated triangulated category
triangulated categories of sheaves, Bondal-Orlov reconstruction theorem
The original reference is the thesis of Verdier:
Similar axioms were already given in
A comprehensive monograph is
and a survey is in section 10 of
section 3 of
Behrang Noohi, Lectures on derived and triangulated categories (arXiv).
A survey of formalisms used in stable homotopy theory to present the triangulated homotopy category of a stable (∞,1)-category is in
Discussion of the redundancy in the traditional definition of triangulated category is in
There was also some discussion at the nForum.