Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology.
A shift or translation functor on a category D is an additive automorphism (or for some authors, an auto-equivalence)
This leads to complex questions about possible axioms one has to impose on the natural transformations making
are mutually inverse isomorphisms is the usual choice in the definition of a triangulated category.
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: Let
The octahedral axiom states (roughly) that the three mapping cones can be made into the vertices of an exact triangle so that "everything commutes."
The presentation here is Verdier's own, and appears, complete with octahedral diagram, in (Hartshorne 1966).
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 (Beilinson, Bernstein & Deligne 1982) (Gelfand and Manin (2006) also present the first one).
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",
Finally, Neeman (2001) formulates the octahedral axiom using a two-dimensional commutative diagram with 4 rows and 4 columns.
Beilinson, Bernstein, and Deligne (1982) also give generalizations of the octahedral axiom.
One of the technical complications with triangulated categories is the fact the cone construction is not functorial.
Some experts suspect[11]pg 190 (see, for example, (Gelfand & Manin 2006, Introduction, Chapter IV)) that triangulated categories are not really the "correct" concept.
The axioms work adequately in practice, however, and there is a great deal of literature devoted to their study.
This formalism has the advantage of being able to recover the homotopy limits and colimits, which replaces the cone construction.
The homotopy category of a stable ∞-category is canonically triangulated, and moreover mapping cones become essentially unique (in a precise homotopical sense).
Moreover, a stable ∞-category naturally encodes a whole hierarchy of compatibilities for its homotopy category, at the bottom of which sits the octahedral axiom.
For a smooth projective variety X over a field k, the bounded derived category of coherent sheaves
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
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.
One may also use the notation for integers i, generalizing the Ext functor in an abelian category.
such that FG and GF are naturally isomorphic to the respective identity functors.
Let D be a triangulated category such that direct sums indexed by an arbitrary set (not necessarily finite) exist in D. An object X in D is called compact if the functor
This is different from the general notion of a compact object in category theory, which involves all colimits rather than only coproducts.
In particular, Neeman used it to simplify and generalize the construction of the exceptional inverse image functor
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.
For example, Devinatz–Hopkins–Smith described all thick subcategories of the triangulated category of finite spectra in terms of Morava K-theory.
Some textbook introductions to triangulated categories are: A concise summary with applications is: Some more advanced references are: