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Triangulated category

Index Triangulated category

In mathematics, a triangulated category is a category together with the additional structure of a "translation functor" and a class of "distinguished triangles". [1]

45 relations: Abelian category, Abelian group, Additive category, Adjoint functors, Alexander Grothendieck, American Mathematical Society, Brown's representability theorem, Cardinal number, Category (mathematics), Coherent sheaf, Commutative diagram, Derivator, Derived category, Derived functor, Epimorphism, Equivalence of categories, Exact sequence, Exceptional inverse image functor, Ext functor, Frobenius algebra, Functor, Glossary of algebraic geometry, Homotopy, Homotopy category of chain complexes, J. Peter May, Localization of a category, Mapping cone (homological algebra), Mathematics, Modular representation theory, Monomorphism, Natural transformation, Noetherian scheme, Norm residue isomorphism theorem, Octahedron, Opposite category, Quasi-category, Quasi-compact morphism, Quasi-isomorphism, Scheme (mathematics), Section (category theory), Spectrum (topology), Springer Science+Business Media, Stable ∞-category, Stable module category, T-structure.

Abelian category

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.

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Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

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Additive category

In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts.

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Adjoint functors

In mathematics, specifically category theory, adjunction is a possible relationship between two functors.

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Alexander Grothendieck

Alexander Grothendieck (28 March 1928 – 13 November 2014) was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry.

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American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

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Brown's representability theorem

In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor.

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Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.

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Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.

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Coherent sheaf

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space.

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Commutative diagram

The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result.

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Derivator

In mathematics, derivators are a proposed new framework for homological algebra and various generalisations.

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Derived category

In mathematics, the derived category D(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes.

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Derived functor

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones.

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Epimorphism

In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f: X → Y that is right-cancellative in the sense that, for all morphisms, Epimorphisms are categorical analogues of surjective functions (and in the category of sets the concept corresponds to the surjective functions), but it may not exactly coincide in all contexts; for example, the inclusion \mathbb\to\mathbb is a ring-epimorphism.

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Equivalence of categories

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same".

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Exact sequence

An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry.

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Exceptional inverse image functor

In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves.

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Ext functor

In mathematics, the Ext functors of homological algebra are derived functors of Hom functors.

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Frobenius algebra

In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories.

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Functor

In mathematics, a functor is a map between categories.

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Glossary of algebraic geometry

This is a glossary of algebraic geometry.

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Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

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Homotopy category of chain complexes

In homological algebra in mathematics, the homotopy category K(A) of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences.

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J. Peter May

Jon Peter May (born September 16, 1939 in New York) is an American mathematician, working in the fields of algebraic topology, category theory, homotopy theory, and the foundational aspects of spectra.

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Localization of a category

In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms.

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Mapping cone (homological algebra)

In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Modular representation theory

Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field K of positive characteristic.

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Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism.

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Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved.

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Noetherian scheme

In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings.

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Norm residue isomorphism theorem

In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor ''K''-theory and Galois cohomology.

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Octahedron

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.

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Opposite category

In category theory, a branch of mathematics, the opposite category or dual category Cop of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism.

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Quasi-category

In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category.

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Quasi-compact morphism

In algebraic geometry, a morphism f: X \to Y between schemes is said to be quasi-compact if Y can be covered by open affine subschemes V_i such that the pre-images f^(V_i) are quasi-compact (as topological space).

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Quasi-isomorphism

In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism A → B of chain complexes (respectively, cochain complexes) such that the induced morphisms of homology groups (respectively, of cohomology groups) are isomorphisms for all n. In the theory of model categories, quasi-isomorphisms are sometimes used as the class of weak equivalences when the objects of the category are chain or cochain complexes.

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Scheme (mathematics)

In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.

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Section (category theory)

In category theory, a branch of mathematics, a section is a right inverse of some morphism.

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Spectrum (topology)

In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory.

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Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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Stable ∞-category

In category theory, a branch of mathematics, a stable ∞-category is an ∞-category such that.

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Stable module category

In representation theory, the stable module category is a category in which projectives are "factored out.".

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T-structure

In mathematics, more specifically in homological algebra, a t-structure is an additional piece of structure that can be put on a triangulated category or a stable infinity category that axiomatizes the properties of complexes whose positive or negative cohomology vanishes.

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Redirects here:

Distinguished triangle, T-category, Triangle category, Triangulated categories, Triangulated functor.

References

[1] https://en.wikipedia.org/wiki/Triangulated_category

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