nLab
triangulated category

Idea

A triangulated category is a category that behaves like the homotopy category of a stable (infinity,1)-category. Indeed, most examples of triangulated categories that arise in practice appear this way, and in fact often from stable model categories.

Therefore, all the structure and properties of a triangulated category is best understood as a 1-categorical shadow of the corresponding properties of stable (infinity,1)-categories, which are more natural and correct. Triangulated categories are sometimes sufficient, and can be easier to work with since they are only 1-categorical rather than higher-categorical, but often one needs more information than is present in the triangulated category, usually relating to homotopy limits and colimits. An intermediate notion, which contains a good deal of this information, is a stable derivator derivator.

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

$$X\stackrel{{\mathrm{Id}}_X}{\to }X\to 0\to TX$$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

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

TR3: a triangle

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

is a distinguished triangle precisely if

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

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

and

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

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

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

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

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

commutes;

TR5: given three distinguished triangles of the form

$$\begin{array}{rl}& X\stackrel{f}{\to }Y\stackrel{h}{\to }Y/X\stackrel{}{\to }TX\\ & Y\stackrel{g}{\to }Z\stackrel{k}{\to }Z/Y\stackrel{}{\to }TY\\ & X\stackrel{g\circ f}{\to }Z\stackrel{l}{\to }Z/X\stackrel{}{\to }TX\end{array}$$\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

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

$$\begin{array}{ccccccccccccc}X& & \stackrel{g\circ f}{\to }& & Z& & \stackrel{k}{\to }& & Z/Y& & \stackrel{k}{\to }& & T(Y/X)\\ & {}_f\searrow & & {\nearrow }_g& & {\searrow }^l& & {\nearrow }_v& & {\searrow }^{}& & {\nearrow }_{T(h)}\\ & & Y& & & & Z/X& & & & TY\\ & & & {\searrow }^h& & {\nearrow }_u& & {\searrow }^{}& & {\nearrow }_{T(f)}\\ & & & & Y/X& & \stackrel{}{\to }& & TX\end{array}$$\array{ X &&\stackrel{g \circ f}{\to}&& Z &&\stackrel{k}{\to}&& Z/Y &&\stackrel{k}{\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 }

Mike: This classical definition is actually redundant; TR4 and one direction of TR3 follow from the remaining axioms. See J.P. May, The additivity of traces in triangulated categories, pdf.

Remarks

• 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 $(f,g,h)$:

  $$\begin{array}{ccccccc}X& \stackrel{f}{\to }& Y& \stackrel{g}{\to }& Z& \stackrel{h}{\to }& TX\\ {}^{\mathrm{id}}\downarrow & & {}^{\mathrm{id}}\downarrow & & {}^{-1}\downarrow & & {\downarrow }^{\mathrm{id}}\\ X& \stackrel{f}{\to }& Y& \stackrel{-g}{\to }& Z& \stackrel{-h}{\to }& TX\end{array}$$\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}

Examples

• The homotopy category of chain complexes in an abelian category (the category of chain complexes modulo chain homotopy) is a triangulated category: the translation functor is the shift functor on chain complexes and the distinguished triangles are those coming from the mapping cone construction $X\stackrel{f}{\to }Y\to \mathrm{Cone}(f)\to TX$.

• The stable homotopy category (the homotopy 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 (infinity,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 preceeding 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 $X\to Y\to Z\to TX$ 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 $(\mathrm{\infty },1)$-category, and usually of a stable model category.

Discussion

The original definition of triangulated categories is apparently due to Verdier, who developed the theory upon guidelines by Grothendieck; Dold and Puppe developed independently a version without octahedron axiom with motivation in algebraic topology. In the manuscript Pursuing Stacks, Grothendieck mentions that the usual definition of triangulated categories and the corresponding derived categories seemed to be inadequate for some of the developments that he wished for. He also says something to the effect that he had tried to interest various of his ex-students in doing a thorough treatment of the ideas, which he considered to be necessary for future development, and which he then proceeds to sketch out.

Zoran Skoda: I am not quite sure if this is entirely correct. Grothendieck indeed wanted more flexibility in homotopical algebra and went to develop these things; but if one talks only very specifically about the concept of triangulated category itself (not wider context) than the main complaint of everybody was about the crudeness of localization at quasiisomorphisms; the thing which for example Drinfel’d’s “quotients of dg-categories” paper successfully rectifies (and then again Lyubashenko in quotients of $A_{\mathrm{\infty }}$-categories).

That led to the theory of derivators, where the idea is that in addition to looking at a basic category of ‘things’ such as chain complexes, you should also look at all categories of diagrams of such things, and the derived / homotopy Kan extensions between the corresponding derived categories that correspond to a change of the indexing category. The basic idea behind this was also explored slightly later by Alex Heller (1988). See the references on the pages derivator, pointed derivator, and stable derivator.

References

• A. Neeman, 2001, Triangulated Categories , Princeton University Press.

• Kashiwara-Schapira, Categories and Sheaves, section 10.

• J. Lurie, Stable Infinity-Categories, section 3.

• B. Noohi, Lectures on derived and triangulated categories (arXiv).

• F. Muro’s complementary slides surveying triangulated categories: htag.pdf.

• Neil Strickland, Axiomatic stable homotopy - a survey (arXiv:math.AT/0307143) is a survey of formalisms used in stable homotopy theory to present the triangulated homotopy category of a stable (infinity,1)-category.