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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.
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
Category theory is a field of mathematics which deals in an abstract way with mathematical structures and relationships between them. Arising as an abstraction of homological algebra , which itself was affectionately called " abstract nonsense ", category theory is sometimes called " generalized abstract nonsense ".
From Wikipedia, the free encyclopedia. Redirect page. Redirect to: Triangulated category