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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 ...
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.
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 ...
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.
From Wikipedia, the free encyclopedia. Redirect page. Redirect to: Triangulated category
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 .)
For a triangulated category C which admits all coproducts, Neeman (2001) defines an object to be compact if (,):, (,) commutes with coproducts. The relation of this notion and the above is as follows: suppose C arises as the homotopy category of a stable ∞-category admitting all filtered colimits.
It is convention to assume the category is essentially small, so that the collection of all stability conditions on forms a set (). In good circumstances, for example when D = D b Coh ( X ) {\displaystyle {\mathcal {D}}={\mathcal {D}}^{b}\operatorname {Coh} (X)} is the derived category of coherent sheaves on a complex manifold X ...