theory of stable ∞-categories or dg-categories, and in many applications the triangulated structure is sufficient. An example is the proof of the Bloch–Kato conjecture, where many computations were done at the level of triangulated categories, and the additional structure of ∞-categories or dg-categories was not required. Cohomology in triangulated categories Triangulated categories admit a notion of cohomology, and every triangulated category has a large supply of cohomological functors. A cohomological functor F from a triangulated category D to an abelian category A is a functor such that for every exact triangle the sequence in A is exact. Since an exact triangle determines an infinite sequence of exact triangles in both directions, a cohomological functor F actually gives a long exact sequence in the abelian category A: A key example is: for each object B in a triangulated category D, the functors and are cohomological, with values in the category of abelian groups. (To be precise, the latter is a contravariant functor, which can be considered as a functor on the opposite category of D.) That is, an exact triangle determines two long exact sequences of abelian groups: and For particular triangulated categories, these exact sequences yield many of the important exact sequences in sheaf cohomology, group cohomology, and other areas of mathematics. One may also use the notation for integers i, generalizing the Ext functor in an abelian category. In this notation, the first exact sequence above would be written: For an abelian category A, another basic example of a cohomological functor on the derived category D(A) sends a complex X to the object in A. That is, an exact triangle in D(A) determines a long exact sequence in A: using that . Exact functors and equivalences An exact functor (also called triangulated functor) from a triangulated category D to a triangulated category E is an additive functor which, loosely speaking, commutes with translation and sends exact triangles to exact triangles. In more detail, an exact functor comes with a natural isomorphism (where the first denotes the translation functor of D and the second denotes the translation functor of E), such that whenever is an exact triangle in D, is an exact triangle in E. An equivalence of triangulated categories is an exact functor that is also an equivalence of categories. In this case, there is an exact functor such that FG and GF are naturally isomorphic to the respective