Триангуляцияланған санаттар: математикалық құрылым мен қолданыстары

In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulated categories was to extend properties of sheaves on a space X to complexes of sheaves, viewed as objects of the derived category of sheaves on X. More recently, triangulated categories have become objects of interest in their own right. Many equivalences between triangulated categories of different origins have been proved or conjectured. For example, the homological mirror symmetry conjecture predicts that the derived category of a Calabi–Yau manifold is equivalent to the Fukaya category of its "mirror" symplectic manifold. Shift operator is a decategorified analogue of triangulated category.

Triangulated categories were introduced independently by Dieter Puppe (1962) and Jean Louis Verdier (1963), although Puppe's axioms were less complete (lacking the octahedral axiom (TR 4)). Puppe was motivated by the stable homotopy category. Verdier's key example was the derived category of an abelian category, which he also defined, developing ideas of Alexander Grothendieck. The early applications of derived categories included coherent duality and Verdier duality, which extends Poincaré duality to singular spaces.

Given two exact triangles and a map between the first morphisms in each triangle, there exists a morphism between the third objects in each of the two triangles that makes everything commute. That is, in the following diagram (where the two rows are exact triangles and f and g are morphisms such that gu = u′f), there exists a map h (not necessarily unique) making all the squares commute:

One of the technical complications with triangulated categories is the fact the cone construction is not functorial. For example, given a ring and the partial map of distinguished trianglesin , there are two maps which complete this diagram. This could be the identity map, or the zero mapboth of which are commutative. The fact there exist two maps is a shadow of the fact that a triangulated category is a tool which encodes homotopy limits and colimit. One solution for this problem was proposed by Grothendieck where not only the derived category is considered, but the derived category of diagrams on this category. Such an object is called a Derivator.

Some experts suspectpg 190 (see, for example, ) that triangulated categories are not really the "correct" concept. The essential reason is that the cone of a morphism is unique only up to a non unique isomorphism. In particular, the cone of a morphism does not in general depend functorially on the morphism (note the non uniqueness in axiom (TR 3), for example). This non uniqueness is a potential source of errors. The axioms work adequately in practice, however, and there is a great deal of literature devoted to their study.

One alternative proposal is the theory of derivators proposed in Pursuing stacks by Grothendieck in the 80s By contrast, there is an example of an exact functor from to that does not come from a complex of sheaves on In view of this example, the "right" notion of a morphism between triangulated categories seems to be one that comes from a morphism of underlying dg categories (or stable ∞ categories). Another advantage of stable ∞ categories or dg categories over triangulated categories appears in algebraic K theory. One can define the algebraic K theory of a stable ∞ category or dg category C, giving a sequence of abelian groups for integers i. The group has a simple description in terms of the triangulated category associated to C. But an example shows that the higher K groups of a dg category are not always determined by the associated triangulated category. Thus a triangulated category has a well defined group, but in general not higher K groups. On the other hand, the theory of triangulated categories is simpler than the 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.

For every abelian category A, the derived category D(A) is a triangulated category, containing A as a full subcategory (the complexes concentrated in degree zero). Different abelian categories can have equivalent derived categories, so that it is not always possible to reconstruct A from D(A) as a triangulated category. Alexander Beilinson, Joseph Bernstein and Pierre Deligne described this situation by the notion of a t structure on a triangulated category D. A t structure on D determines an abelian category inside D, and different t structures on D may yield different abelian categories.

Let D be a triangulated category with arbitrary direct sums. A localizing subcategory of D is a strictly full triangulated subcategory that is closed under arbitrary direct sums. To explain the name: if a localizing subcategory S of a compactly generated triangulated category D is generated by a set of objects, then there is a Bousfield localization functor with kernel S. (That is, for every object X in D there is an exact triangle with Y in S and LX in the right orthogonal .) For example, this construction includes the localization of a spectrum at a prime number, or the restriction from a complex of sheaves on a space to an open subset. A parallel notion is more relevant for "small" triangulated categories: a thick subcategory of a triangulated category C is a strictly full triangulated subcategory that is closed under direct summands. (If C is idempotent complete, a subcategory is thick if and only if it is also idempotent complete.) A localizing subcategory is thick. So if S is a localizing subcategory of a triangulated category D, then the intersection of S with the subcategory of compact objects is a thick subcategory of
For example, Devinatz–Hopkins–Smith described all thick subcategories of the triangulated category of finite spectra in terms of Morava K theory. The localizing subcategories of the whole stable homotopy category have not been classified.