Triangulated category

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A triangulated category is a mathematical category satisfying some axioms that are based on the properties of a derived category. Some examples are the homotopy category of spectra, and the derived category of an abelian category.

1 History
2 Definition
3 Comments on the axioms
4 Examples
5 t-structure and cores
6 References

[edit] History

The notion of a derived category was introduced in his thesis by Verdier, based on the ideas of Grothendieck. He also defined the notion of a triangulated category, by noting that a derived category had some special "triangles" and writing down axioms for the basic properties of these triangles.

[edit] Definition

A translation functor on a category D is an automorphism T from D to D. The image of X under Tⁿ is usually written as X[n].

A triangle (X, Y, Z, u, v, w) is a set of 3 objects X, Y, and Z, together with morphisms u from X to Y, v from Y to Z and w from Z to X[1]. If (X, Y, Z, u, v, w) is a triangle then the rotated triangle is (Z[−1],X, Y, −w[−1], u, v).

A triangulated category is an additive category D with a translation functor and a class of distinguished triangles, satisfying the following properties.

Any triangle isomorphic to a distinguished triangle is distinguished.
The rotation of a distinguished triangle is distinguished.
Any morphism can be completed to a distinguished triangle. (The third object in the triangle is called a mapping cone of the morphism.)
The identity morphism of an object can be completed to a distinguished triangle with the third object 0.
Given a map between two morphisms, there is a morphism between their mapping cones that makes "everything commute".

So far all the axioms are reasonably natural and obvious. The final axiom, sometimes called the octahedral axiom, is notorious for being incomprehensible.

Suppose we have morphisms from X to Y and Y to Z, so that we also have a composed morphism from X to Z. Form distinguished triangles for each of these three morphisms. The octahedral axiom states (roughly) that the three mapping cones can be made into the vertices of a distinguished triangle so that "everything commutes".

[edit] Comments on the axioms

The axioms above have seemed rather artificial. It is strongly suspected by experts that triangulated categories are not really the "correct" concept. They do however seem to work adequately in practice; and there is no current and convincing replacement.

The last axiom is called the octahedral axiom, because drawing all the objects and morphisms gives the skeleton of an octahedron, four of whose faces are distinguished triangles. There seems to be no really satisfactory way to draw everything in two dimensions (see the book of Kashiwara and Schapira for details), although Neeman's book (see reference below) gives a way of expressing the octahedral axiom using a two dimensional commutative diagram with 4 rows and 4 columns.

The axioms above are not independent. In particular, the axiom implying the existence of a morphism between mapping cones can be deduced from the others.

The mapping cone of a morphism is unique up to a non-unique isomorphism. This non-uniqueness is a potential source of errors. In particular the mapping cone of a morphism does not in general depend functorially on the morphism.

Pierre Deligne has found further axioms that could be added, which are generalizations of (and even more complicated than) the octahedral axiom.

[edit] Examples

If A is an abelian category, then the category Kom(A) has as objects all complexes of objects of A, and as morphisms the homotopy classes of morphisms of complexes. Then Kom(A) is a triangulated category, where the distinguished triangles consist of triangles isomorphic to a morphism with its mapping cone (in the sense of chain complexes). Variations: use complexes that are bounded on the left, or on the right, or on both sides.

A localization of a triangulated category is also triangulated. In particular the derived category of A, which is a localization of Kom(A), is triangulated.

[edit] t-structure and cores

In the derived category D of an Abelian category A, there are natural subcategories $D^{\le n}$ and $D^{\ge m}$ , consisting of complexes whose cohomology vanishes in degrees larger then n or smaller than m. These have the following properties:

$D^{\le n}=D^{\le 0}[-n]$ , $D^{\ge n}=D^{\ge 0}[-n]$
$Hom(D^{\le 0},D^{\ge 1})=0$
$D^{\le 0}\subset D^{\le 1}$ , $D^{\ge 1}\subset D^{\ge 0}$
Every object Y can be embedded in a distinguished triangle

(X, Y, Z, u, v, w) with $X\in D^{\le 0}$ and $Z\in D^{\ge 1}$ .

A t-structure on a triangulated category consists of full subcategories $D^{\le n}$ and $D^{\ge m}$ satisfying the conditions above. The letter t stands for "truncation".

The core of a t-structure is the category $D^{\le 0}\cap D^{\ge 0}$ . It is an abelian category. (A triangulated category is additive but is not usually abelian).

The core of a t-structure of the derived category of A can be thought of as a sort of twisted version of A, which sometimes has better properties. For example, the category of perverse sheaves is the core of a certain (quite complicated) t-structure on the derived category of the category of sheaves. Over a space with singularities, the category of perverse sheaves is similar to the category of sheaves but behaves better.

[edit] References

Part of Verdier's thesis is reprinted in

SGA 4 1/2 ISBN 0-387-08066-X.

Two textbooks that discuss triangulated categories are:

Methods of Homological Algebra by Sergei I. Gelfand, Yuri I. Manin ISBN 3-540-43583-2
Homological Algebra by S. I. Gelfand, Yu. I. Manin ISBN 3-540-65378-3
Triangulated Categories (AM-148)(Annals of Mathematics Studies) Amnon Neeman ISBN 0-691-08686-9

Another standard reference is:

Faisceaux pervers, Beilinson, Bernstein, and Deligne. Astérisque 100.
Sheaves on Manifolds (1990) M. Kashiwara and P. Schapira (concise introduction, and applications)

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Category: Homological algebra