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triangulated category

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homological algebra

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nonabelian homological algebra

Context

Basic definitions

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diagram chasing

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Idea

Any (infinity,1)-category CC can be flattened, by ignoring higher morphisms, into a 1-category ho(C)ho(C) called its homotopy category. The notion of a triangulated structure is designed to capture the additional structure canonically existing on ho(C)ho(C) when CC 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(𝒜)D(\mathcal{A}) of abelian categories 𝒜\mathcal{A}. These are the homotopy categories of the stable (∞,1)-categories of chain complexes in 𝒜\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 1 below.

Definition

A triangulated category is

TR0: every triangle isomorphic to a distinguished triangle is itself a distinguished triangle;

TR1: the triangle

XId XX0TX X \stackrel{Id_X}{\to} X \to 0 \to T X

is a distinguished triangle;

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

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

TR3: a triangle

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

is a distinguished triangle precisely if

YgZhTXT(f)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

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

and

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

and given morphisms α\alpha and β\beta in

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

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

X f Y g Z h TX α β γ T(α) X f Y g Z h TX \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

XfYhY/XTX YgZkZ/YTY XgfZlZ/XTX \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/XuZ/XvZ/YwT(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

X gf Z k Z/Y k T(Y/X) f g l v T(h) Y Z/X TY h u T(f) Y/X TX \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:CCT : C \to C is often called the suspension functor and denoted ()[1]:XX[1](-)[1] : X \mapsto X[1] (in an algebraic context) or SS or Σ\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)(f,g,h) is a distinguished triangle, then (f,g,h)(f,g,-h) is not generally distinguished, although it is “exact” (induces long exact sequences in homology and cohomology). However, (f,g,h)(f,-g,-h) is always distinguished, since it is isomorphic to (f,g,h)(f,g,h):

X f Y g Z h TX id id 1 id X f Y g Z h TX \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

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.

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.

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 August 25, 2014 03:51:26 by Adeel Khan (132.252.62.217)