Global Information Lookup Global Information

Triangulated category information


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.

and 22 Related for: Triangulated category information

Request time (Page generated in 0.8167 seconds.)

Triangulated category

Last Update:

language of triangulated categories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulated categories was to...

Word Count : 5798

Homotopy category of chain complexes

Last Update:

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 triangulated category, and...

Word Count : 1051

Derived category

Last Update:

required an innovation, the concept of triangulated category, and the construction is based on localization of a category, a generalization of localization...

Word Count : 4503

Bridgeland stability condition

Last Update:

paper of Bridgeland, for arbitrary triangulated categories. Let D {\displaystyle {\mathcal {D}}} be a triangulated category. A slicing P {\displaystyle {\mathcal...

Word Count : 993

Differential graded category

Last Update:

homotopy category Ho(C) is a triangulated category. A triangulated category T is said to have a dg enhancement C if C is a pretriangulated dg category whose...

Word Count : 825

Glossary of category theory

Last Update:

sequences. An abelian category is a prototypical example of a triangulated category. A derived category is a triangulated category that is not necessary...

Word Count : 10474

Abelian category

Last Update:

G-modules for a given group G. Mathematics portal Triangulated category Mac Lane, Saunders (2013-04-17). Categories for the Working Mathematician. Graduate Texts...

Word Count : 2643

Semiorthogonal decomposition

Last Update:

mathematics, a semiorthogonal decomposition is a way to divide a triangulated category into simpler pieces. One way to produce a semiorthogonal decomposition...

Word Count : 2399

Grothendieck group

Last Update:

information. Since the (bounded) derived category is triangulated, there is a Grothendieck group for derived categories too. This has applications in representation...

Word Count : 4178

Toda bracket

Last Update:

known elements, called Hopf elements. In the case of a general triangulated category the Toda bracket can be defined as follows. Again, suppose that...

Word Count : 1141

Homological mirror symmetry

Last Update:

triangulated category constructed from the algebraic geometry of X (the derived category of coherent sheaves on X) and another triangulated category constructed...

Word Count : 1027

Shift operator

Last Update:

which the baker's map is an explicit representation. The notion of triangulated category is a categorified analogue of the shift operator. The shift operator...

Word Count : 1452

List of homological algebra topics

Last Update:

Filtration (abstract algebra) Spectral sequence Abelian category Triangulated category Derived category Group cohomology Galois cohomology Lie algebra cohomology...

Word Count : 78

Derived noncommutative algebraic geometry

Last Update:

using only its triangulated structure. It turns out the correct way of studying derived categories from its objects and triangulated structure is with...

Word Count : 4702

Outline of category theory

Last Update:

Derived category Triangulated category Model category 2-category Dagger symmetric monoidal category Dagger compact category Strongly ribbon category Closed...

Word Count : 402

Karoubi envelope

Last Update:

involution. If C is a triangulated category, the Karoubi envelope Split(C) can be endowed with the structure of a triangulated category such that the canonical...

Word Count : 833

Motivic cohomology

Last Update:

K-groups; Voevodsky, Triangulated categories of motives over a field, section 2.2 and Proposition 4.2.9. Voevodsky, Triangulated categories of motives over...

Word Count : 2285

Frobenius category

Last Update:

algebra. The stable category of a Frobenius category is canonically a triangulated category. Dagger compact category Tannakian category Theorem 2.6 in Happel...

Word Count : 115

Core

Last Update:

graph Core (group theory), an object in group theory Core of a triangulated category Core, an essential domain of a closed operator; see Unbounded operator...

Word Count : 771

Stable module category

Last Update:

sequences as above to be exact triangles, the stable module category becomes a triangulated category. Stable homotopy theory J. F. Carlson, Lisa Townsley, Luis...

Word Count : 690

Tom Bridgeland

Last Update:

most well-known for defining Bridgeland stability conditions on triangulated categories. Bridgeland was educated at Shelley High School in Huddersfield...

Word Count : 714

Stability

Last Update:

stability conditions, a class of stability conditions on elements of a triangulated category. Stability (algebraic geometry) In atmospheric fluid dynamics, atmospheric...

Word Count : 487

PDF Search Engine © AllGlobal.net