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.
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language of triangulatedcategories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulatedcategories was to...
the category of chain complexes Kom(A) of A and the derived category D(A) of A when A is abelian; unlike the former it is a triangulatedcategory, and...
required an innovation, the concept of triangulatedcategory, and the construction is based on localization of a category, a generalization of localization...
sequences. An abelian category is a prototypical example of a triangulatedcategory. A derived category is a triangulatedcategory that is not necessary...
paper of Bridgeland, for arbitrary triangulatedcategories. Let D {\displaystyle {\mathcal {D}}} be a triangulatedcategory. A slicing P {\displaystyle {\mathcal...
homotopy category Ho(C) is a triangulatedcategory. A triangulatedcategory T is said to have a dg enhancement C if C is a pretriangulated dg category whose...
G-modules for a given group G. Mathematics portal Triangulatedcategory Mac Lane, Saunders (2013-04-17). Categories for the Working Mathematician. Graduate Texts...
information. Since the (bounded) derived category is triangulated, there is a Grothendieck group for derived categories too. This has applications in representation...
mathematics, a semiorthogonal decomposition is a way to divide a triangulatedcategory into simpler pieces. One way to produce a semiorthogonal decomposition...
which the baker's map is an explicit representation. The notion of triangulatedcategory is a categorified analogue of the shift operator. The shift operator...
using only its triangulated structure. It turns out the correct way of studying derived categories from its objects and triangulated structure is with...
triangulatedcategory constructed from the algebraic geometry of X (the derived category of coherent sheaves on X) and another triangulatedcategory constructed...
involution. If C is a triangulatedcategory, the Karoubi envelope Split(C) can be endowed with the structure of a triangulatedcategory such that the canonical...
sequences as above to be exact triangles, the stable module category becomes a triangulatedcategory. Stable homotopy theory J. F. Carlson, Lisa Townsley, Luis...
algebra. The stable category of a Frobenius category is canonically a triangulatedcategory. Dagger compact category Tannakian category Theorem 2.6 in Happel...
K-groups; Voevodsky, Triangulatedcategories of motives over a field, section 2.2 and Proposition 4.2.9. Voevodsky, Triangulatedcategories of motives over...
stability conditions, a class of stability conditions on elements of a triangulatedcategory. Stability (algebraic geometry) In atmospheric fluid dynamics, atmospheric...
graph Core (group theory), an object in group theory Core of a triangulatedcategory Core, an essential domain of a closed operator; see Unbounded operator...
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