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triangulated category

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Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

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Stable homotopy theory

Contents

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 T X \,;

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^{\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

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 }

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: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}

Properties

Long fiber-cofiber sequences

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.

Lemma

Given a triangulated category, def. 1, and

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

a distinguished triangle, then

gf=0 g\circ f = 0

is the zero morphism.

Proof

Consider the commuting diagram

A id A 0 ΣA id f A f B g B/A h ΣA. \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. 1. Hence by T3 there is an extension to a commuting diagram of the form

A id A 0 ΣA id f Σf A f B g B/A h ΣA. \array{ A &\overset{id}{\longrightarrow}& A &\overset{}{\longrightarrow}& 0 &\overset{}{\longrightarrow}& \Sigma A \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}} && \downarrow && \downarrow^{\mathrlap{\Sigma f}} \\ 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. 1, with shift functor denoted Σ\Sigma and with hom-functor denoted [,] *:Ho op×HoAb[-,-]_\ast \colon Ho^{op}\times Ho \to Ab. Then for XX any object, and for any distinguished triangle

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

the sequences of abelian groups

  1. (long cofiber sequence)

    [ΣA,X] *[h,X] *[B/A,X] *[g,X] *[B,X] *[f,X] *[A,X] * [\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] *[X,f] *[X,B] *[X,g] *[X,B/A] *[X,h] *[X,ΣA] * [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 gf=0g \circ f = 0 by lemma 1, we have an inclusion im([g,X] *)ker([f,X] *)im([g,X]_\ast) \subset ker([f,X]_\ast). Hence it is sufficient to show that if ψ:BX\psi \colon B \to X is in the kernel of [f,X] *[f,X]_\ast in that ψf=0\psi \circ f = 0, then there is ϕ:CX\phi \colon C \to X with ϕg=ψ\phi \circ g = \psi. To that end, consider the commuting diagram

A f B g B/A h ΣA ψ 0 X id X 0, \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 T1T1 in def. 1. Hence by condition T3 there exists ϕ\phi fitting into a commuting diagram of the form

A f B g B/A h ΣA ψ ϕ 0 X id X 0. \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] *[B,X]_\ast. Applying the same reasoning to the distinguished traingle (g,h,Σf)(g,h,-\Sigma f) provided by T2 yields exactness at [C,X] *[C,X]_\ast.

Regarding the second case:

Again, from lemma 1 it is immediate that

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

so that we need to show that for ψ:XB\psi \colon X \to B in the kernel of [X,g] *[X,g]_\ast, hence such that gψ=0g\circ \psi = 0, then there exists ϕ:XA\phi \colon X \to A with fϕ=ψf \circ \phi = \psi.

To that end, consider the commuting diagram

X 0 ΣX id ΣX ψ B g B/A h ΣA Σf ΣB, \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. 1, while the bottom part is a distinguished triangle by applying T2 to the given distinguished triangle. Hence by T3 there exists ϕ˜:ΣXΣA\tilde \phi \colon \Sigma X \to \Sigma A such as to extend to a commuting diagram of the form

X 0 ΣX id ΣX ψ ϕ˜ Σψ B g B/A h ΣA Σf ΣB, \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. 1 that Σ:HoHo\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 ϕ:XA\phi \colon X \to A with ϕ˜=Σϕ\tilde \phi = \Sigma \phi. It being faithful then implies that the whole commuting square on the right is the image under Σ\Sigma of a commuting square

X id X ϕ ψ A f B. \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

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

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

is an equivalence of categories.

In this case (Ho(𝒞),Σ)(Ho(\mathcal{C}),\Sigma) is a triangulated category. (Hovey 99, section 7, for review see also Schwede, section 2).

Examples

References

The concept orignates in the thesis

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)

Discussion of the relation to stable model categories originates in

Other surveys include

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 June 29, 2016 15:15:05 by Urs Schreiber (131.220.184.222)