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81 relations: Abelian group, Academic Press, Additive category, Adjoint functors, Alexander Grothendieck, Algebraic geometry, Algebraic topology, Axiom, Bilinear map, Biproduct, Category (mathematics), Category of abelian groups, Category theory, Chain complex, Cohomology, Coimage, Cokernel, Commutative algebra, Commutative ring, Comodule, Complete category, Coproduct, David Buchsbaum, Derived functor, Dimension (vector space), Direct product, Direct sum of modules, Enriched category, Epimorphism, Exact category, Exact functor, Exact sequence, Field (mathematics), Filtered category, Finitary, Finite set, Finitely generated abelian group, Finitely generated module, Five lemma, Functor, Functor category, G-module, Grothendieck topology, Grothendieck's Tôhoku paper, Group (mathematics), Homological algebra, Image (category theory), Initial and terminal objects, Kernel (category theory), Lattice (order), ..., Localizing subcategory, Mathematics, Mitchell's embedding theorem, Module (mathematics), Monoidal category, Monomorphism, Morphism, Niels Henrik Abel, Nine lemma, Noetherian ring, Normal morphism, Partially ordered set, Pathological (mathematics), Pre-abelian category, Preadditive category, Product (category theory), Regular category, Ring (mathematics), Sheaf (mathematics), Short five lemma, Snake lemma, Subcategory, Subobject, Subquotient, Tensor product, Topological space, Transactions of the American Mathematical Society, Vector bundle, Vector space, Zero morphism, 0. Expand index (31 more) »
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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Academic Press
Academic Press is an academic book publisher.
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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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Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
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Algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
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Axiom
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
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Bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments.
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Biproduct
In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct.
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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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Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms.
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Category theory
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
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Chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next.
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Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.
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Coimage
In algebra, the coimage of a homomorphism is the quotient of the domain by the kernel.
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Cokernel
In mathematics, the cokernel of a linear mapping of vector spaces f: X → Y is the quotient space Y/im(f) of the codomain of f by the image of f. The dimension of the cokernel is called the corank of f. Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel is a quotient object of the codomain (it maps from the codomain).
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Commutative algebra
Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.
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Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.
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Comodule
In mathematics, a comodule or corepresentation is a concept dual to a module.
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Complete category
In mathematics, a complete category is a category in which all small limits exist.
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Coproduct
In category theory, the coproduct, or categorical sum, is a category-theoretic construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces.
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David Buchsbaum
David Alvin Buchsbaum (born November 6, 1929) is a mathematician at Brandeis University who works on commutative algebra, homological algebra, and representation theory.
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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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Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.
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Direct product
In mathematics, one can often define a direct product of objects already known, giving a new one.
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Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module.
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Enriched category
In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category.
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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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Exact category
In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition of such a sequence.
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Exact functor
In homological algebra, an exact functor is a functor that preserves exact sequences.
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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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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Filtered category
In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category).
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Finitary
In mathematics or logic, a finitary operation is an operation of finite arity, that is an operation that takes a finite number of input values.
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Finite set
In mathematics, a finite set is a set that has a finite number of elements.
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Finitely generated abelian group
In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,..., xs in G such that every x in G can be written in the form with integers n1,..., ns.
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Finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set.
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Five lemma
In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams.
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Functor
In mathematics, a functor is a map between categories.
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Functor category
In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors.
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G-module
In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of ''G''.
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Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space.
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Grothendieck's Tôhoku paper
The article "Sur quelques points d'algèbre homologique" by Alexander Grothendieck, now often referred to as the Tôhoku paper, was published in 1957 in the Tôhoku Mathematical Journal.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting.
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Image (category theory)
In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function.
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Initial and terminal objects
In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists a single morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final.
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Kernel (category theory)
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra.
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Lattice (order)
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.
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Localizing subcategory
In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category.
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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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Mitchell's embedding theorem
Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules.
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Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
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Monoidal category
In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism.
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Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism.
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Morphism
In mathematics, a morphism is a structure-preserving map from one mathematical structure to another one of the same type.
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Niels Henrik Abel
Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields.
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Nine lemma
In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in any abelian category, as well as in the category of groups.
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Noetherian ring
In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; that is, given any chain of left (or right) ideals: there exists an n such that: Noetherian rings are named after Emmy Noether.
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Normal morphism
In category theory and its applications to mathematics, a normal monomorphism or conormal epimorphism is a particularly well-behaved type of morphism.
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Partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.
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Pathological (mathematics)
In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved.
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Pre-abelian category
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.
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Preadditive category
In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups.
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Product (category theory)
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces.
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Regular category
In category theory, a regular category is a category with finite limits and coequalizers of a pair of morphisms called kernel pairs, satisfying certain exactness conditions.
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Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
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Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
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Short five lemma
In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma.
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Snake lemma
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences.
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Subcategory
In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms.
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Subobject
In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category.
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Subquotient
In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject.
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Tensor product
In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.
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Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
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Transactions of the American Mathematical Society
The Transactions of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society.
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Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Zero morphism
In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object.
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0
0 (zero) is both a number and the numerical digit used to represent that number in numerals.
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Abelian Category, Abelian categories.
References
[1] https://en.wikipedia.org/wiki/Abelian_category
