# nLab triangulated category

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

### Theorems

#### Stable homotopy theory

stable homotopy theory

# Contents

## Idea

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 $D\left(𝒜\right)$ of abelian categories $𝒜$. These are the homotopy categories of the (∞,1)-categories of chain complexes in $𝒜$.

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:

## Definition

The traditional definition of triangulated category is the following. But see remark 1 below.

###### Definition

A triangulated category is

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

$X\stackrel{f}{\to }Y\to Z\to \mathrm{TX}\phantom{\rule{thinmathspace}{0ex}};$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

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

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

$\begin{array}{ccccccc}X& \stackrel{f}{\to }& Y& \stackrel{g}{\to }& Z& \stackrel{h}{\to }& TX\\ {↓}^{\alpha }& & {↓}^{\beta }& & {↓}^{\exists \gamma }& & {↓}^{T\left(\alpha \right)}\\ X\prime & \stackrel{f\prime }{\to }& Y\prime & \stackrel{g\prime }{\to }& Z\prime & \stackrel{h\prime }{\to }& TX\prime \end{array}$\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

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

$\begin{array}{ccccccccccccc}X& & \stackrel{g\circ f}{\to }& & Z& & \stackrel{k}{\to }& & Z/Y& & \stackrel{k}{\to }& & T\left(Y/X\right)\\ & {}_{f}↘& & {↗}_{g}& & {↘}^{l}& & {↗}_{v}& & {↘}^{}& & {↗}_{T\left(h\right)}\\ & & Y& & & & Z/X& & & & TY\\ & & & {↘}^{h}& & {↗}_{u}& & {↘}^{}& & {↗}_{T\left(f\right)}\\ & & & & Y/X& & \stackrel{}{\to }& & TX\end{array}$\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 }
###### Remark

This classical definition is actually redundant; TR4 and one direction of TR3 follow from the remaining axioms. See (May).

###### Remark

In the context of triangulated categories the translation functor $T:C\to C$ is often called the suspension functor and denoted $\left(-\right)\left[1\right]:X↦X\left[1\right]$ (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 $\left(f,g,h\right)$ is a distinguished triangle, then $\left(f,g,-h\right)$ is not generally distinguished, although it is “exact” (induces long exact sequences in homology and cohomology). However, $\left(f,-g,-h\right)$ is always distinguished, since it is isomorphic to $\left(f,g,h\right)$:

$\begin{array}{ccccccc}X& \stackrel{f}{\to }& Y& \stackrel{g}{\to }& Z& \stackrel{h}{\to }& TX\\ {}^{\mathrm{id}}↓& & {}^{\mathrm{id}}↓& & {}^{-1}↓& & {↓}^{\mathrm{id}}\\ X& \stackrel{f}{\to }& Y& \stackrel{-g}{\to }& Z& \stackrel{-h}{\to }& TX\end{array}$\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 $K\left(𝒜\right)$ 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 $X\stackrel{f}{\to }Y\to \mathrm{Cone}\left(f\right)\to TX$ in ${\mathrm{Ch}}_{•}\left(𝒜\right)$.

• 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 $N↪C$, 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 $\left(\infty ,1\right)$-category, and usually of a stable model category.

## References

The original reference is the thesis of Verdier:

• Verdier, Jean-Louis, Des Catégories Dérivées des Catégories Abéliennes, Astérisque (Paris: Société Mathématique de France) 239. Available in electronic format courtesy of Georges Maltsiniotis.

A comprehensive monograph is

• Amnon Neeman, Triangulated Categories , Princeton University Press (2001)

and a survey is in section 10 of

section 3 of

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

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

There was also some discussion at the nForum.

Revised on May 9, 2013 19:25:49 by Zoran Škoda (161.53.130.104)