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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 ...
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 ...
In general, there need not exist dg enhancements of triangulated categories or functors between them, for example stable homotopy category can be shown not to arise from a dg category in this way. However, various positive results do exist, for example the derived category D(A) of a Grothendieck abelian category A admits a unique dg enhancement.
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
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 category is compactly generated if any object can be expressed as a filtered colimit of compact objects in . For example, any vector space V is the filtered colimit of its finite-dimensional (i.e., compact) subspaces. Hence the category of vector spaces (over a fixed field) is compactly generated.