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.

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 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

If
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.
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 triangles
in, there are two maps which complete this diagram. This could be the identity map, or the zero map
both of which are commutative. The fact there exist two maps is a shadow of the fact 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.

Examples

Are there better axioms?

Some experts suspect 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.
One alternative proposal is the theory of derivators initiated by Grothendieck in 1991. Another is that 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.

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. 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 sends a complex X to the object in A. That is, an exact triangle in D determines a long exact sequence in A:
using that.

Exact functors and equivalences

An exact 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 , 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 identity functors.

Compactly generated triangulated categories

Let D be a triangulated category such that direct sums indexed by an arbitrary set exist in D. An object X in D is called compact if the functor commutes with direct sums. Explicitly, this means that for every family of objects in D indexed by a set S, the natural homomorphism of abelian groups is an isomorphism. This is different from the general notion of a compact object in category theory, which involves all colimits rather than only coproducts.
For example, a compact object in the stable homotopy category is a finite spectrum. A compact object in the derived category of a ring, or in the quasi-coherent derived category of a scheme, is a perfect complex. In the case of a smooth projective variety X over a field, the category Perf of perfect complexes can also be viewed as the bounded derived category of coherent sheaves,.
A triangulated category D is compactly generated if
Many naturally occurring "large" triangulated categories are compactly generated:
Amnon Neeman generalized the Brown representability theorem to any compactly generated triangulated category, as follows. Let D be a compactly generated triangulated category, a cohomological functor which takes coproducts to products. Then H is representable. For another version, let D be a compactly generated triangulated category, T any triangulated category. If an exact functor sends coproducts to coproducts, then F has a right adjoint.
The Brown representability theorem can be used to define various functors between triangulated categories. In particular, Neeman used it to simplify and generalize the construction of the exceptional inverse image functor for a morphism f of schemes, the central feature of coherent duality theory.

t-structures

For every abelian category A, the derived category D is a triangulated category, containing A as a full subcategory. Different abelian categories can have equivalent derived categories, so that it is not always possible to reconstruct A from D 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.

Localizing and thick subcategories

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. 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. 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.