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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 [],
This is a list of homological algebra topics, by Wikipedia page. Basic techniques ... Triangulated category; Derived category; Applications. Group cohomology;
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.
A triangulated torus Another triangulation of the torus A triangulated dolphin shape. In mathematics, triangulation describes the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism. A space that admits such a homeomorphism is called a triangulable space.
In category theory, a branch of mathematics, a stable ∞-category is an ∞-category such that [1] (i) It has a zero object. (ii) Every morphism in it admits a fiber and cofiber. (iii) A triangle in it is a fiber sequence if and only if it is a cofiber sequence. The homotopy category of a stable ∞-category is triangulated. [2]
Here is a full list of the ceremony's 94 award categories. The 2025 Grammy Awards are scheduled for Feb. 2 in Los Angeles. Here is a full list of the ceremony's 94 award categories.
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