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triangulated category

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homological algebra

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nonabelian homological algebra

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diagram chasing

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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(𝒜)D(\mathcal{A}) of abelian categories 𝒜\mathcal{A}. These are the homotopy categories of the (∞,1)-categories of chain complexes in 𝒜\mathcal{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

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.

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 27, 2014 02:13:12 by Urs Schreiber (82.136.246.44)