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Abelian category, the Glossary

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.^[1]

105 relations: AB5 category, Abelian group, Academic Press, Additive category, Adjoint functors, Alexander Grothendieck, Algebraic geometry, Algebraic topology, American Mathematical Society, Axiom, Bilinear map, Biproduct, Category (mathematics), Category of abelian groups, Category theory, Chain complex, Class (set theory), Coherent sheaf, Cohomological dimension, Cohomology, Coimage, Cokernel, Commutative algebra, Commutative ring, Comodule, Complete category, Coproduct, David Buchsbaum, Derived functor, Dimension (vector space), Direct product, Direct sum, 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, Flat module, Flat morphism, Functor, Functor category, G-module, ... Expand index (55 more) »
Additive categories
Niels Henrik Abel

AB5 category

In mathematics, in his "Tôhoku paper" introduced a sequence of axioms of various kinds of categories enriched over the symmetric monoidal category of abelian groups. Abelian category and AB5 category are homological algebra.

See Abelian category and AB5 category

Abelian group

In mathematics, 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. Abelian category and abelian group are Niels Henrik Abel.

See Abelian category and Abelian group

Academic Press

Academic Press (AP) is an academic book publisher founded in 1941.

See Abelian category and Academic Press

Additive category

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

See Abelian category and Additive category

Adjoint functors

In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories.

See Abelian category and Adjoint functors

Alexander Grothendieck

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

See Abelian category and Alexander Grothendieck

Algebraic geometry

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.

See Abelian category and Algebraic geometry

Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.

See Abelian category and Algebraic topology

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.

See Abelian category and American Mathematical Society

Axiom

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

See Abelian category and Axiom

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.

See Abelian category and Bilinear map

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. Abelian category and biproduct are additive categories.

See Abelian category and Biproduct

Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows".

See Abelian category and Category (mathematics)

Category of abelian groups

In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms.

See Abelian category and Category of abelian groups

Category theory

Category theory is a general theory of mathematical structures and their relations.

See Abelian category and Category theory

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. Abelian category and chain complex are homological algebra.

See Abelian category and Chain complex

Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.

See Abelian category and Class (set theory)

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.

See Abelian category and Coherent sheaf

Cohomological dimension

In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. Abelian category and cohomological dimension are homological algebra.

See Abelian category and Cohomological dimension

Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex.

See Abelian category and Cohomology

Coimage

In algebra, the coimage of a homomorphism is the quotient of the domain by the kernel.

See Abelian category and Coimage

Cokernel

The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of.

See Abelian category and Cokernel

Commutative algebra

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.

See Abelian category and Commutative algebra

Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative.

See Abelian category and Commutative ring

Comodule

In mathematics, a comodule or corepresentation is a concept dual to a module.

See Abelian category and Comodule

Complete category

In mathematics, a complete category is a category in which all small limits exist.

See Abelian category and Complete category

Coproduct

In category theory, the coproduct, or categorical sum, is a 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.

See Abelian category and Coproduct

David Buchsbaum

David Alvin Buchsbaum (November 6, 1929 – January 8, 2021) was a mathematician at Brandeis University who worked on commutative algebra, homological algebra, and representation theory.

See Abelian category and David Buchsbaum

Derived functor

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. Abelian category and derived functor are homological algebra.

See Abelian category and Derived functor

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.

See Abelian category and Dimension (vector space)

Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one.

See Abelian category and Direct product

Direct sum

The direct sum is an operation between structures in abstract algebra, a branch of mathematics.

See Abelian category and Direct sum

Direct sum of modules

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module.

See Abelian category and Direct sum of modules

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.

See Abelian category and Enriched category

Epimorphism

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

See Abelian category and Epimorphism

Exact category

In mathematics, specifically in category theory, an exact category is a category equipped with short exact sequences. Abelian category and exact category are additive categories and homological algebra.

See Abelian category and Exact category

Exact functor

In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Abelian category and exact functor are additive categories and homological algebra.

See Abelian category and Exact functor

Exact sequence

An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Abelian category and exact sequence are additive categories and homological algebra.

See Abelian category and Exact sequence

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers.

See Abelian category and Field (mathematics)

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).

See Abelian category and Filtered category

Finitary

In mathematics and logic, an operation is finitary if it has finite arity, i.e. if it has a finite number of input values.

See Abelian category and Finitary

Finite set

In mathematics, particularly set theory, a finite set is a set that has a finite number of elements.

See Abelian category and Finite set

Finitely generated abelian group

In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x.

See Abelian category and Finitely generated abelian group

Finitely generated module

In mathematics, a finitely generated module is a module that has a finite generating set.

See Abelian category and Finitely generated module

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. Abelian category and five lemma are homological algebra.

See Abelian category and Five lemma

Flat module

In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion free modules. Abelian category and flat module are homological algebra.

See Abelian category and Flat module

Flat morphism

In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e., is a flat map for all P in X. A map of rings A\to B is called flat if it is a homomorphism that makes B a flat A-module.

See Abelian category and Flat morphism

Functor

In mathematics, specifically category theory, a functor is a mapping between categories.

See Abelian category and Functor

Functor category

In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in the category).

See Abelian category and Functor category

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''.

See Abelian category and G-module

Graduate Texts in Mathematics

Graduate Texts in Mathematics (GTM) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.

See Abelian category and Graduate Texts in Mathematics

Grothendieck topology

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C that makes the objects of C act like the open sets of a topological space.

See Abelian category and Grothendieck topology

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. Abelian category and Grothendieck's Tôhoku paper are homological algebra.

See Abelian category and Grothendieck's Tôhoku paper

Group (mathematics)

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

See Abelian category and Group (mathematics)

Homological algebra

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting.

See Abelian category and Homological algebra

Image (category theory)

In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function.

See Abelian category and Image (category theory)

Initial and terminal objects

In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in, there exists precisely one morphism.

See Abelian category and Initial and terminal objects

Isomorphism

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.

See Abelian category and Isomorphism

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.

See Abelian category and Kernel (category theory)

Lattice (order)

A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.

See Abelian category and Lattice (order)

Lie group

In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

See Abelian category and Lie group

Local ring

In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

See Abelian category and Local ring

Localizing subcategory

In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category. Abelian category and localizing subcategory are homological algebra.

See Abelian category and Localizing subcategory

Maschke's theorem

In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces.

See Abelian category and Maschke's theorem

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Abelian category and Mathematics

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. Abelian category and Mitchell's embedding theorem are additive categories.

See Abelian category and Mitchell's embedding theorem

Module (mathematics)

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring.

See Abelian category and Module (mathematics)

Monoidal category

In mathematics, a monoidal category (or tensor category) is a category \mathbf 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.

See Abelian category and Monoidal category

Monomorphism

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

See Abelian category and Monomorphism

Morphism

In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces.

See Abelian category and Morphism

Nine lemma

right In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category. Abelian category and nine lemma are homological algebra.

See Abelian category and Nine lemma

Noetherian ring

In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively.

See Abelian category and Noetherian ring

Noetherian scheme

In algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatornameSpec A_i, where each A_i is a Noetherian ring.

See Abelian category and Noetherian scheme

Normal morphism

In category theory and its applications to mathematics, a normal monomorphism or conormal epimorphism is a particularly well-behaved type of morphism.

See Abelian category and Normal morphism

Partially ordered set

In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other.

See Abelian category and Partially ordered set

Pathological (mathematics)

In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological.

See Abelian category and Pathological (mathematics)

Pre-abelian category

In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Abelian category and pre-abelian category are additive categories.

See Abelian category and Pre-abelian category

Preadditive category

In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. Abelian category and preadditive category are additive categories.

See Abelian category and Preadditive category

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 or rings, and the product of topological spaces.

See Abelian category and Product (category theory)

Pullback (category theory)

In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain.

See Abelian category and Pullback (category theory)

Pushout (category theory)

In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms f: Z → X and g: Z → Y with a common domain.

See Abelian category and Pushout (category theory)

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.

See Abelian category and Regular category

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

See Abelian category and Representation theory

Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.

See Abelian category and Ring (mathematics)

Samuel Eilenberg

Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra.

See Abelian category and Samuel Eilenberg

Saunders Mac Lane

Saunders Mac Lane (August 4, 1909 – April 14, 2005), born Leslie Saunders MacLane, was an American mathematician who co-founded category theory with Samuel Eilenberg.

See Abelian category and Saunders Mac Lane

Scheme (mathematics)

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

See Abelian category and Scheme (mathematics)

Semisimple module

In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts.

See Abelian category and Semisimple module

Sheaf (mathematics)

In mathematics, a sheaf (sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them.

See Abelian category and Sheaf (mathematics)

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. Abelian category and short five lemma are homological algebra.

See Abelian category and Short five lemma

Snake lemma

The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. Abelian category and snake lemma are homological algebra.

See Abelian category and Snake lemma

Subcategory

In mathematics, specifically category theory, 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.

See Abelian category and Subcategory

Subobject

In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category.

See Abelian category and Subobject

Subquotient

In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject.

See Abelian category and Subquotient

Tensor product

In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted.

See Abelian category and Tensor product

Tohoku Mathematical Journal

The Tohoku Mathematical Journal is a mathematical research journal published by Tohoku University in Japan.

See Abelian category and Tohoku Mathematical Journal

Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.

See Abelian category and Topological space

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.

See Abelian category and Transactions of the American Mathematical Society

Triangulated category

In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Abelian category and triangulated category are homological algebra.

See Abelian category and Triangulated category

Unipotent

In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n. In particular, a square matrix M is a unipotent matrix if and only if its characteristic polynomial P(t) is a power of t − 1.

See Abelian category and Unipotent

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.

See Abelian category and Vector bundle

References

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

Also known as Abelian categories.

, Graduate Texts in Mathematics, Grothendieck topology, Grothendieck's Tôhoku paper, Group (mathematics), Homological algebra, Image (category theory), Initial and terminal objects, Isomorphism, Kernel (category theory), Lattice (order), Lie group, Local ring, Localizing subcategory, Maschke's theorem, Mathematics, Mitchell's embedding theorem, Module (mathematics), Monoidal category, Monomorphism, Morphism, Nine lemma, Noetherian ring, Noetherian scheme, Normal morphism, Partially ordered set, Pathological (mathematics), Pre-abelian category, Preadditive category, Product (category theory), Pullback (category theory), Pushout (category theory), Regular category, Representation theory, Ring (mathematics), Samuel Eilenberg, Saunders Mac Lane, Scheme (mathematics), Semisimple module, Sheaf (mathematics), Short five lemma, Snake lemma, Subcategory, Subobject, Subquotient, Tensor product, Tohoku Mathematical Journal, Topological space, Transactions of the American Mathematical Society, Triangulated category, Unipotent, Vector bundle, Vector space, Zariski tangent space, Zero morphism, 0.

Abelian category, the Glossary

Table of Contents

See also

Additive categories

Niels Henrik Abel

References