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Triangulated CategoryIn mathematics, a triangulated category is a category satisfying some axioms that are based on the properties of a derived category. Some examples are the category of complexes over an abelian category, and the derived category of an abelian category. Introduction The notion of a derived category was introduced in his thesis by Verdier, based on some 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. 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 Tn. A triangle is a set of 3 objects X, Y, and Z, together with morphisms from X to Y, Y to Z and Z to X1. A triangulated category is an additive category D with a translation functor and a class of distringuished 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, somethimes 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".
Comments on the axioms The axioms above seem rather artificial. It is strongly suspected that triangulated categories are not really the "correct" concept. However they do seem to work adequately in practice, and no one has been able to find any really convincing replacements for them. The last axiom is called the octahedral axiom because drawing all the objects and morphisms gives the skeleton of an octahedron, 4 of whose faces are distinguished triangles. There seems to be no really satisfactory way to draw everything in 2 dimensions. 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 source of many errors. In particular the mapping cone of a morphism does not in general depend functorially on the morphism. Deligne has found further axioms that could be added, which are generalizations of (and even more complicated than) the octahedral axiom. 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. References Part of Verdier's thesis is reprinted in SGA 4 1/2 ISBN 038708066X. Two textbooks that discuss triangulated categories are: Methods of Homological Algebra by Sergei I. Gelfand, Yuri I. Manin ISBN 3540435832 Homological Algebra by S. I. Gelfand, Yu. I. Manin ISBN 3540653783 Another standard reference is Faisceaux pervers, Beilinson, Bernstein, and Deligne. Astrisque 100.
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