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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 ...
A DG category C is called pre-triangulated if it has a suspension functor and a class of distinguished triangles compatible with the suspension, such that its 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 homotopy category is ...
If C has products, then given an isomorphism: the mapping :, composed with the canonical map : of symmetry, is a partial 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 functor C → Split(C) becomes a triangulated functor.
In the branch of mathematics called homological algebra, a t-structure is a way to axiomatize the properties of an abelian subcategory of a derived category.A t-structure on consists of two subcategories (,) of a triangulated category or stable infinity category which abstract the idea of complexes whose cohomology vanishes in positive, respectively negative, degrees.
The triangulated subcategory generated by an exceptional object E is equivalent to the derived category () of finite-dimensional k-vector spaces, the simplest triangulated category in this context. (For example, every object of that subcategory is isomorphic to a finite direct sum of shifts of E .)
A Bridgeland stability condition on a triangulated category is a pair (,) consisting of a slicing and a group homomorphism : (), where () is the Grothendieck group of , called a central charge, satisfying
The homotopy category of a stable ∞-category is triangulated. [2] A stable ∞-category admits finite limits and colimits. [3] Examples: the derived category of an abelian category and the ∞-category of spectra are both stable. A stabilization of an ∞-category C having finite limits and base point is a functor from the stable ∞-category ...
One method divides the 3D region of consideration into cubes and determines the intersections of the surface with the edges of the cubes in order to get polygons on the surface, which thereafter have to be triangulated (cutting cube method). [1] [2] The expenditure for managing the data is great. The second and simpler concept is the marching ...