and
nonabelian homological algebra
Triangulated categories were introduced by Jean-Louis Verdier under the supervision of Grothendieck, motivated by the triangulated structure on derived categories.
A triangulated category is a category equipped with a notion of suspension objects/loop space objects for all of its objects such that in terms of these every morphism fits into a sequence that behaves like a homotopy fiber sequence.
More precisely, a triangulated category is a category that behaves like the homotopy category of a stable (∞,1)-category. Indeed, most examples of triangulated categories that arise in practice appear this way, and in fact often from stable model categories. Notice that the definition of stable (∞,1)-category is very simple and much simpler than the definition of triangulated category, def. 1 below.
Therefore, all the structure and properties of a triangulated category is best understood as a 1-categorical shadow (the decategorification) of the corresponding properties of stable (∞,1)-categories.
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 (∞,1)-categories of chain complexes in $\mathcal{A}$.
Triangulated categories are sufficient for some purposes, and can be easier to work with than the stable (∞,1)-categories that they come from, but – as with every quotient construction – often one needs more information than is present in the triangulated category, especially concerning the computation of homotopy limits and homotopy colimits: the ordinary limits and colimits and other universal constructions in a triangulated category generally have no useful interpretation.
Accordingly, there is a series on notions that refine that of a triangulated category, approximating more and more of the full structure of a stable (∞,1)-category:
triangulated category
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:
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.