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 (octahedral axiom): given three distinguished triangles of the form
there exists a distinguished triangle
such that the following big diagram commutes
If TR5 is not required, one speaks of a pretriangulated category.
This classical definition is redundant; TR4 and one direction of TR3 follow from the remaining axioms. See (May).
The octahedral axiom has many equivalent formulations, a concise account is in (Hubery). Notice that one of the equivalent axioms, called axiom B, there, is essentially just an axiomatization of the existence of homotopy pushouts.
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 following prop. 1 is the incarnation in the axiomatics of triangulated categories of the long exact sequences of homotopy groups induced by homotopy fiber sequences in homotopy theory.
Given a triangulated category, def. 1, and
a distinguished triangle, then
is the zero morphism.
Consider the commuting diagram
Observe that the top part is a distinguished triangle by axioms T1 and T2 in def. 1. Hence by T3 there is an extension to a commuting diagram of the form
Now the commutativity of the middle square proves the claim.
Consider a triangulated category, def. 1, with shift functor denoted $\Sigma$ and with hom-functor denoted $[-,-]_\ast \colon Ho^{op}\times Ho \to Ab$. Then for $X$ any object, and for any distinguished triangle
the sequences of abelian groups
(long cofiber sequence)
(long fiber sequence)
are long exact sequences.
Regarding the first case:
Since $g \circ f = 0$ by lemma 1, we have an inclusion $im([g,X]_\ast) \subset ker([f,X]_\ast)$. Hence it is sufficient to show that if $\psi \colon B \to X$ is in the kernel of $[f,X]_\ast$ in that $\psi \circ f = 0$, then there is $\phi \colon C \to X$ with $\phi \circ g = \psi$. To that end, consider the commuting diagram
where the commutativity of the left square exhibits our assumption.
The top part of this diagram is a distinguished triangle by assumption, and the bottom part is by condition $T1$ in def. 1. Hence by condition T3 there exists $\phi$ fitting into a commuting diagram of the form
Here the commutativity of the middle square exhibits the desired conclusion.
This shows that the first sequence in question is exact at $[B,X]_\ast$. Applying the same reasoning to the distinguished traingle $(g,h,-\Sigma f)$ provided by T2 yields exactness at $[C,X]_\ast$.
Regarding the second case:
Again, from lemma 1 it is immediate that
so that we need to show that for $\psi \colon X \to B$ in the kernel of $[X,g]_\ast$, hence such that $g\circ \psi = 0$, then there exists $\phi \colon X \to A$ with $f \circ \phi = \psi$.
To that end, consider the commuting diagram
where the commutativity of the left square exhibits our assumption.
Now the top part of this diagram is a distinguished triangle by conditions T1 and T2 in def. 1, while the bottom part is a distinguished triangle by applying T2 to the given distinguished triangle. Hence by T3 there exists $\tilde \phi \colon \Sigma X \to \Sigma A$ such as to extend to a commuting diagram of the form
At this point we appeal to the condition in def. 1 that $\Sigma \colon Ho \to Ho$ is an equivalence of categories, so that in particular it is a fully faithful functor. It being a full functor implies that there exists $\phi \colon X \to A$ with $\tilde \phi = \Sigma \phi$. It being faithful then implies that the whole commuting square on the right is the image under $\Sigma$ of a commuting square
This exhibits the claim to be shown.
A pointed model category $\mathcal{C}$ is called a stable model category if the canonically induced reduced suspension-functor on its homotopy category
is an equivalence of categories.
In this case $(Ho(\mathcal{C}),\Sigma)$ is a triangulated category. (Hovey 99, section 7, for review see also Schwede, section 2).
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. Also the full subcategory called the Spanier-Whitehead category is triangulated.
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 concept orignates in the thesis
Similar axioms were already given in
A comprehensive monograph is
Discussion of the relation to stable model categories originates in
Other surveys include
Andrew Hubery, Notes on the octahedral axiom, (pdf)
Masaki Kashiwara, Pierre Schapira, section 10 Categories and Sheaves,
Jacob Lurie, section 3 of Stable Infinity-Categories,
Behrang Noohi, Lectures on derived and triangulated categories (arXiv:0704.1009).
Stefan Schwede, Triangulated categories (pdf)
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.