Triangulated category
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.
History
Triangulated categories were introduced independently by Dieter Puppe and Jean-Louis Verdier, although Puppe's axioms were less complete. 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.Definition
A shift or translation functor on a category D is an additive automorphism from D to D. It is common to write for integers n.A triangle consists of three objects X, Y, and Z, together with morphisms, and. Triangles are generally written in the unravelled form:
or
for short.
A triangulated category is an additive category D with a translation functor and a class of triangles, called exact triangles, satisfying the following properties,, and.
TR 1
- For every object X, the following triangle is exact:
- For every morphism, there is an object Z fitting into an exact triangle
- Every triangle isomorphic to an exact triangle is exact. This means that if
TR 2
is an exact triangle, then so are the two rotated triangles
and
In view of the last triangle, the object Z is called a fiber of the morphism.
The second rotated triangle has a more complex form when and are not isomorphisms but only mutually inverse equivalences of categories, since is a morphism from to, and to obtain a morphism to one must compose with the natural transformation. This leads to complex questions about possible axioms one has to impose on the natural transformations making and into a pair of inverse equivalences. Due to this issue, the assumption that and are mutually inverse isomorphisms is the usual choice in the definition of a triangulated category.
TR 3
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, there exists a map h making all the squares commute:TR 4: The octahedral axiom
Let and be morphisms, and consider the composed morphism. Form exact triangles for each of these three morphisms according to TR 1. The octahedral axiom states that the three mapping cones can be made into the vertices of an exact triangle so that "everything commutes."More formally, given exact triangles
there exists an exact triangle
such that
This axiom is called the "octahedral axiom" because drawing all the objects and morphisms gives the skeleton of an octahedron, four of whose faces are exact triangles. The presentation here is Verdier's own, and appears, complete with octahedral diagram, in. In the following diagram, u and v are the given morphisms, and the primed letters are the cones of various maps. Various arrows have been marked with to indicate that they are of "degree 1"; e.g. the map from Z′ to X is in fact from Z′ to X. The octahedral axiom then asserts the existence of maps f and g forming an exact triangle, and so that f and g form commutative triangles in the other faces that contain them:
Two different pictures appear in . The first presents the upper and lower pyramids of the above octahedron and asserts that given a lower pyramid, one can fill in an upper pyramid so that the two paths from Y to Y′, and from Y′ to Y, are equal. The triangles marked + are commutative and those marked "d" are exact:
The second diagram is a more innovative presentation. Exact triangles are presented linearly, and the diagram emphasizes the fact that the four triangles in the "octahedron" are connected by a series of maps of triangles, where three triangles are given and the existence of the fourth is claimed. One passes between the first two by "pivoting" about X, to the third by pivoting about Z, and to the fourth by pivoting about X′. All enclosures in this diagram are commutative but the other commutative square, expressing the equality of the two paths from Y′ to Y, is not evident. All the arrows pointing "off the edge" are degree 1:
This last diagram also illustrates a useful intuitive interpretation of the octahedral axiom. In triangulated categories, triangles play the role of exact sequences, and so it is suggestive to think of these objects as "quotients", and. In those terms, the existence of the last triangle expresses on the one hand
Putting these together, the octahedral axiom asserts the "third isomorphism theorem":
If the triangulated category is the derived category D of an abelian category A, and X, Y, Z are objects of A viewed as complexes concentrated in degree 0, and the maps and are monomorphisms in A, then the cones of these morphisms in D are actually isomorphic to the quotients above in A.
Finally, formulates the octahedral axiom using a two-dimensional commutative diagram with 4 rows and 4 columns. also give generalizations of the octahedral axiom.
Properties
Here are some simple consequences of the axioms for a triangulated category D.- Given an exact triangle
- Given a morphism, TR 1 guarantees the existence of a cone Z completing an exact triangle. Any two cones of u are isomorphic, but the isomorphism is not always uniquely determined.
- Every monomorphism in D is the inclusion of a direct summand,, and every epimorphism is a projection. A related point is that one should not talk about "injectivity" or "surjectivity" for morphisms in a triangulated category. Every morphism that is not an isomorphism has a nonzero "cokernel" Z and also a nonzero "kernel", namely Z.
Non-functoriality of the cone construction
Examples
Are there better axioms?
Some experts suspectpg 190 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. 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.Derivators
One alternative proposal is the theory of derivators proposed in Pursuing stacks by Grothendieck in the 80spg 191, and later developed in the 90s in his manuscript on the topic. Essentially, these are a system of homotopy categories given by the diagram categories for a category with a class of weak equivalences. These categories are then related by the morphisms of diagrams. This formalism has the advantage of being able to recover the homotopy limits and colimits, which replaces the cone construction.Stable ∞-categories
Another alternative built is the theory of stable ∞-categories. The homotopy category of a stable ∞-category is canonically triangulated, and moreover mapping cones become essentially unique. Moreover, a stable ∞-category naturally encodes a whole hierarchy of compatibilities for its homotopy category, at the bottom of which sits the octahedral axiom. Thus, it is strictly stronger to give the data of a stable ∞-category than to give the data of a triangulation of its homotopy category. Nearly all triangulated categories that arise in practice come from stable ∞-categories. A similar enrichment of triangulated categories is the notion of a dg-category.In some ways, stable ∞-categories or dg-categories work better than triangulated categories. One example is the notion of an exact functor between triangulated categories, discussed below. For a smooth projective variety X over a field k, the bounded derived category of coherent sheaves comes from a dg-category in a natural way. For varieties X and Y, every functor from the dg-category of X to that of Y comes from a complex of sheaves on by the Fourier–Mukai transform. 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.
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.