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Triangulated categories admit a notion of cohomology, and every triangulated category has a large supply of cohomological functors. A cohomological functor F from a triangulated category D to an abelian category A is a functor such that for every exact triangle [],
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 ...
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.
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.
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.
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
From Wikipedia, the free encyclopedia. Redirect page. Redirect to: Triangulated category