nLab triangulated category

Idea

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.

History

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).

Definition

The traditional definition of triangulated category is the following. But see remark below.

Definition

A triangulated category is

• an additive category;

• with (additive) translation;

• 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

$$X \stackrel{Id_X}{\to} X \to 0 \to T X$$

is a distinguished triangle;

TR2: for all $f : X \to Y$, there exists a distinguished triangle

$$X \stackrel{f}{\to} Y \to Z \to T X \,;$$

TR3: a triangle

$$X \stackrel{f}{\to} Y \stackrel{g}{\to} Z \stackrel{h}{\to} T X$$

is a distinguished triangle precisely if

$$Y \stackrel{g}{\to} Z \stackrel{h}{\to} T X \stackrel{-T(f)}{\to} T Y$$

is a distinguished triangle;

TR4: given two distinguished triangles

$$X \stackrel{f}{\to} Y \stackrel{g}{\to} Z \stackrel{h}{\to} T X$$

and

$$X' \stackrel{f'}{\to} Y' \stackrel{g'}{\to} Z' \stackrel{h'}{\to} T X'$$

and given morphisms $\alpha$ and $\beta$ in

$$\array{ X &\stackrel{f}{\to}& Y \\ \downarrow^\alpha && \downarrow^\beta \\ X' &\stackrel{f'}{\to}& Y' }$$

there exists a morphism $\gamma : Z \to Z'$ extending this to a morphism of distinguished triangles in that the diagram

$$\array{ X &\stackrel{f}{\to}& Y &\stackrel{g}{\to}& Z &\stackrel{h}{\to}& T X \\ \downarrow^\alpha && \downarrow^{\mathrlap{\beta}} && \downarrow^{\mathrlap{\exists \gamma}} && \downarrow^{\mathrlap{T(\alpha)}} \\ X' &\stackrel{f'}{\to}& Y' &\stackrel{g'}{\to}& Z' &\stackrel{h'}{\to}& T X' }$$

commutes;

TR5 (octahedral axiom): given three distinguished triangles of the form

$$\begin{aligned} & X \stackrel{f}{\to} Y \stackrel{h}{\to} Y/X \stackrel{}{\to} T X \\ & Y \stackrel{g}{\to} Z \stackrel{k}{\to} Z/Y \stackrel{}{\to} T Y \\ & X \stackrel{g \circ f}{\to} Z \stackrel{l}{\to} Z/X \stackrel{}{\to} T X \end{aligned}$$

there exists a distinguished triangle

$$Y/X \stackrel{u}{\to} Z/X \stackrel{v}{\to} Z/Y \stackrel{w}{\to} T (Y/X)$$

such that the following big diagram commutes

$$\array{ X &&\stackrel{g \circ f}{\to}&& Z &&\stackrel{k}{\to}&& Z/Y &&\stackrel{w}{\to}&& T (Y/X) \\ & {}_{f}\searrow && \nearrow_{g} && \searrow^{l} && \nearrow_{v} && \searrow^{} && \nearrow_{T(h)} \\ && Y &&&& Z/X &&&& T Y \\ &&& \searrow^{h} && \nearrow_{u} && \searrow^{} && \nearrow_{T(f)} \\ &&&& Y/X &&\stackrel{}{\to}&& T X }$$

If TR5 is not required, one speaks of a pretriangulated category.

Remark

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.

Remark

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.

Remark

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 $(f,g,h)$:

$$\array{ X & \xrightarrow{f} & Y & \xrightarrow{g} & Z & \xrightarrow{h} & T X\\ ^{id}\downarrow && ^{id} \downarrow && ^{-1} \downarrow && \downarrow^{id}\\ X & \xrightarrow{f} & Y & \xrightarrow{-g} & Z & \xrightarrow{-h} & T X}$$

Properties

Long fiber-cofiber sequences

The following prop. 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.

Lemma

Given a triangulated category, def. , and

$$A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} B/A \overset{h}{\longrightarrow} \Sigma A$$

a distinguished triangle, then

$$g\circ f = 0$$

is the zero morphism.

Proof

Consider the commuting diagram

$$\array{ A &\overset{id}{\longrightarrow}& A &\overset{}{\longrightarrow}& 0 &\overset{}{\longrightarrow}& \Sigma A \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}} \\ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A } \,.$$

Observe that the top part is a distinguished triangle by axioms T1 and T2 in def. . Hence by T3 there is an extension to a commuting diagram of the form

$$\array{ A &\overset{id}{\longrightarrow}& A &\overset{}{\longrightarrow}& 0 &\overset{}{\longrightarrow}& \Sigma A \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}} && \downarrow && \downarrow^{\mathrlap{\Sigma \mathrlap{id}}} \\ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A } \,.$$

Now the commutativity of the middle square proves the claim.

Proposition

Consider a triangulated category, def. , 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

$$A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} B/A \overset{h}{\longrightarrow} \Sigma A$$

the sequences of abelian groups

1. (long cofiber sequence)

   $$[\Sigma A, X]_\ast \overset{[h,X]_\ast}{\longrightarrow} [B/A,X]_\ast \overset{[g,X]_\ast}{\longrightarrow} [B,X]_\ast \overset{[f,X]_\ast}{\longrightarrow} [A,X]_\ast$$

2. (long fiber sequence)

   $$[X,A]_\ast \overset{[X,f]_\ast}{\longrightarrow} [X,B]_\ast \overset{[X,g]_\ast}{\longrightarrow} [X,B/A]_\ast \overset{[X,h]_\ast}{\longrightarrow} [X,\Sigma A]_\ast$$

are long exact sequences.

Proof

Regarding the first case:

Since $g \circ f = 0$ by lemma , 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

$$\array{ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A \\ \downarrow && {}^{\mathllap{\psi}}\downarrow \\ 0 &\overset{}{\longrightarrow}& X &\overset{id}{\longrightarrow}& X &\overset{}{\longrightarrow}& 0 } \,,$$

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. . Hence by condition T3 there exists $\phi$ fitting into a commuting diagram of the form

$$\array{ A &\overset{f}{\longrightarrow}& B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A \\ \downarrow && {}^{\mathllap{\psi}}\downarrow && \downarrow^{\mathrlap{\phi}} && \downarrow \\ 0 &\overset{}{\longrightarrow}& X &\overset{id}{\longrightarrow}& X &\overset{}{\longrightarrow}& 0 } \,.$$

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 it is immediate that

$$im([X,f]_\ast) \subset ker([X,g]_\ast)$$

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

$$\array{ X &\longrightarrow& 0 &\longrightarrow& \Sigma X &\overset{- id}{\longrightarrow}& \Sigma X \\ \downarrow^{\mathrlap{\psi}} && \downarrow \\ B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A &\overset{-\Sigma f}{\longrightarrow}& \Sigma B } \,,$$

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. , 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

$$\array{ X &\longrightarrow& 0 &\longrightarrow& \Sigma X &\overset{- id}{\longrightarrow}& \Sigma X \\ \downarrow^{\mathrlap{\psi}} && \downarrow && \downarrow^{\mathrlap{\tilde \phi}} && \downarrow^{\mathrlap{\Sigma \psi}} \\ B &\overset{g}{\longrightarrow}& B/A &\overset{h}{\longrightarrow}& \Sigma A &\overset{-\Sigma f}{\longrightarrow}& \Sigma B } \,,$$

At this point we appeal to the condition in def. 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

$$\array{ X &\overset{-id}{\longrightarrow}& X \\ {}^{\mathllap{\phi}}\downarrow && \downarrow^{\mathrlap{\psi}} \\ A &\underset{-f}{\longrightarrow}& B } \,.$$

This exhibits the claim to be shown.

From stable model categories and stable $\infty$-categories

A pointed model category $\mathcal{C}$ is called a stable model category if the canonically induced reduced suspension-functor on its homotopy category

$$\Sigma \;\colon\; Ho(\mathcal{C}) \longrightarrow Ho(\mathcal{C})$$

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).

Similarly, the homotopy category of a stable (∞,1)-category is a triangulated category, see there.

Examples

• 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 stable category of a Frobenius exact category 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 particular, therefore, the derived category of any abelian category is a triangulated category, since it is the localization of the homotopy category at the null system of acyclic complexes. This example is also the homotopy category of a stable $(\infty,1)$-category, and usually of a stable model category.

• 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).

Related concepts

• exact triangle, distinguished triangle

• enhanced triangulated category, pretriangulated dg-category, stable (∞,1)-category

• tensor triangulated category

• derived category

• t-structure, Bridgeland stability condition

• suspended category, n-angulated category

• Bousfield lattice, thick subcategory, additive category,

• exceptional collection

• well-generated triangulated category, compactly generated triangulated category

• spectrum of a triangulated category

• triangulated categories of sheaves, Bondal-Orlov reconstruction theorem

• triangulated subcategory, full triangulated subcategory, periodic triangulated category

References

The concept orignates in the thesis

• Verdier, Jean-Louis, Des Catégories Dérivées des Catégories Abéliennes, Astérisque 239 (1996) [doi:10.24033/ast.364, numdam:AST_1996__239__R1_0, pdf]

Similar axioms were already given in

• Albrecht Dold, Dieter Puppe, Homologie nicht-additiver Funktoren, Annales de l’Institut Fourier (Université de Grenoble) 11: 201–312, 1961, eudml.

Monographs:

• Sergei Gelfand, Yuri Manin, Section IV of: Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [doi:10.1007/978-3-662-12492-5]

• Amnon Neeman, Triangulated Categories, Annals of Mathematics Studies 213, Princeton University Press (2001) [ISBN:9780691086866, pdf]

Discussion of the relation to stable model categories originates in

• Mark Hovey, section 7 of Model Categories Mathematical Surveys and Monographs, Volume 63, AMS (1999) (pdf, Google books)

Other surveys:

• Masaki Kashiwara, Pierre Schapira, Section 10 of: Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006) [doi:10.1007/3-540-27950-4, pdf]

• 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, pp. 383-418 in Masoud Khalkhali, Matilde Marcolli (eds.), An Invitation to Noncommutative Geometry, World Scientific (2008) (doi:10.1142/9789812814333_0006 arXiv:0704.1009).

• Fernando Muro (pdf)

• Stefan Schwede, Triangulated categories (pdf)

• Introduction to Stable homotopy theory – Triangulated structure

A survey of formalisms used in stable homotopy theory to present the triangulated homotopy category of a stable (∞,1)-category is in

• Neil Strickland, Axiomatic stable homotopy - a survey (arXiv:math.AT/0307143)

Discussion of the redundancy in the traditional definition of triangulated category is in

• Peter May, The additivity of traces in triangulated categories, (pdf)

There was also some discussion at the nForum.

On localization of triangulated categories:

• Henning Krause, Localization theory for triangulated categories, in proceedings of Workshop on Triangulated Categories, Leeds 2006 [arXiv:0806.1324]

category: triangulated categories