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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Category theory and its applications to other branches of Mathematics, kernels are a generalization of the kernels of Group homomorphisms and the kernels The motivating prototype example of an abelian category is the category of abelian groups, Ab. In Mathematics, the category Ab has the Abelian groups as objects and Group homomorphisms as Morphisms This is the prototype The theory originated in a tentative attempt to unify several cohomology theories by Alexander Grothendieck. In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany Abelian categories are very stable categories, for example they are regular and they satisfy the Snake lemma. In Category theory, a regular category is a category with finite limits and Coequalizers of kernel pairs satisfying certain exactness conditions In Mathematics, particularly Homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well. In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships These stability properties make them inevitable in homological algebra and beyond; The theory has major applications in algebraic geometry, cohomology and pure category theory. Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets
Contents |
Definitions
A category is abelian if
- it has a zero object,
- it has all pullbacks and pushouts, and
- all monomorphisms and epimorphisms are normal. In Category theory, a branch of Mathematics, a pullback (also called a fibered product or Cartesian square) is the limit of a In Category theory, a branch of Mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square) is the In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which In Category theory and its applications to Mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of Morphism
By a theorem of Peter Freyd, this definition is equivalent to the following "piecemeal" definition:
- A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. Peter J Freyd is an American Mathematician, a professor at the University of Pennsylvania, known for work in Category theory. In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category In Category theory and its applications to Mathematics, an enriched category is a category whose Hom-sets are replaced by objects from some other In Mathematics, a monoidal category (or tensor category) is a category C equipped with a Bifunctor &otimes: C An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the This means that all hom-sets are abelian groups and the composition of morphisms is bilinear. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and
- A preadditive category is additive if every finite set of objects has a biproduct. In Mathematics, specifically in Category theory, an additive category is a Preadditive category C such that any finitely many objects A In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Category theory and its applications to Mathematics, a biproduct is a generalisation of the notion of Direct sum that makes sense in any Preadditive This means that we can form finite direct sums and direct products. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, one can often define a direct product of objectsalready known giving a new one
- An additive category is preabelian if every morphism has both a kernel and a cokernel. In Mathematics, specifically in Category theory, a pre-Abelian category is an Additive category that has all kernels and cokernels In Category theory and its applications to other branches of Mathematics, kernels are a generalization of the kernels of Group homomorphisms and the kernels
- Finally, a preabelian category is abelian if every monomorphism and every epimorphism is normal. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which In Category theory and its applications to Mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of Morphism This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism.
Note that the enriched structure on hom-sets is a consequence of the three axioms of the first definition. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the
The concept of exact sequence arise naturally in this setting, and it turns out that exact functors, i. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Homological algebra, an exact functor is a Functor, from some category to another which preserves Exact sequences Exact functors are very e. the functors preserving exact sequences in various senses, are the relevant functors between Abelian categories. This exactness concept has been axiomatized in the theory of exact categories, forming a very special case of regular categories. 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 In Category theory, a regular category is a category with finite limits and Coequalizers of kernel pairs satisfying certain exactness conditions
Examples
- As mentioned above, the category of all abelian groups is an abelian category. The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups. In Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1
- If R is a ring, then the category of all left (or right) modules over R is an abelian category. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In fact, it can be shown that any small abelian category is equivalent to a full subcategory of such a category of modules (Mitchell's embedding theorem). In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in Mitchell's embedding theorem, also known as the Freyd-Mitchell theorem is a mathematical result about abelian categories; it states that these categories while rather abstractly
- If R is a left-noetherian ring, then the category of finitely generated left modules over R is abelian. In Abstract algebra, a Noetherian ring is a ring that satisfies the Ascending chain condition on ideals. In particular, the category of finitely generated modules over a noetherian commutative ring is abelian; in this way, abelian categories show up in commutative algebra. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings
- As special cases of the two previous examples: the category of vector spaces over a fixed field k is abelian, as is the category of finite-dimensional vector spaces over k. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the dimension of a Vector space V is the cardinality (i
- If X is a topological space, then the category of all (real or complex) vector bundles on X is not usually an abelian category, as there can be monomorphisms that are not kernels. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space
- If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. More generally, the category of sheaves of abelian groups on a Grothendieck site is an abelian category. 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 In this way, abelian categories show up in algebraic topology and algebraic geometry. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with
- If C is a small category and A is an abelian category, then the category of all functors from C to A forms an abelian category (the morphisms of this category are the natural transformations between functors). In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal If C is small and preadditive, then the category of all additive functors from C to A also forms an abelian category. In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category The latter is a generalization of the R-module example, since a ring can be understood as a preadditive category with a single object.
Elementary properties
Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. In Category theory, a zero morphism is a special kind of "trivial" Morphism. This can be defined as the zero element of the hom-set Hom(A,B), since this is an abelian group. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and Alternatively, it can be defined as the unique composition A → 0 → B, where 0 is the zero object of the abelian category.
In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. This epimorphism is called the coimage of f, while the monomorphism is called the image of f. In Mathematics, particularly in algebra, the coimage of a Homomorphism f: A → B Given a category C and a Morphism fX\rightarrow Y in C, the image of f is a Monomorphism hI\rightarrow
Subobjects and quotient objects are well-behaved in abelian categories. In Category theory, there is a general definition of subobject extending the idea of Subset and Subgroup. In Category theory, there is a general definition of subobject extending the idea of Subset and Subgroup. Mathematicians (and those in related sciences very frequently speak of whether a mathematical object &mdash a Number, a function, a set, a space For example, the poset of subobjects of any given object A is a bounded lattice. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements'
Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product of a finitely generated abelian group G and any object A of A. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector The abelian category is also a comodule; Hom(G,A) can be interpreted as an object of A. In Mathematics, a comodule is a concept Dual to a module. The definition of a comodule over a Coalgebra is formed by dualizing the definition If A is complete, then we can remove the requirement that G be finitely generated; most generally, we can form finitary enriched limits in A. In Mathematics, a complete category is a category in which all small limits exist In Mathematics or Logic, a finitary operation is one like those of Arithmetic, that takes a finite number of input values to produce an output
Related concepts
Abelian categories are the most general setting for homological algebra. Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequences, and derived functors. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Mathematics, certain Functors may be derived to obtain other functors closely related to the original ones Important theorems that apply in all abelian categories include the five lemma (and the short five lemma as a special case), as well as the snake lemma (and the nine lemma as a special case). In Mathematics, especially Homological algebra and other applications of Abelian category theory the five lemma is an important and widely used lemma In Mathematics, especially Homological algebra and other applications of Abelian category theory the short five lemma is a special case of the Five In Mathematics, particularly Homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the In Mathematics, the nine lemma is a statement about Commutative diagrams and Exact sequences valid in any Abelian category, as well as in the category
History
Abelian categories were introduced by Alexander Grothendieck in his famous Tôhoku paper in the middle of the 1950s in order to unify various cohomology theories. Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany Year 1950 ( MCML) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar. In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain At the time, there was a cohomology theory for sheaves, and a cohomology theory for groups. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element The two were defined completely differently, but they had formally almost identical properties. In fact, much of category theory was developed as a language to study these similarities. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Grothendieck managed to unify the two theories: they both arise as derived functors on abelian categories; on the one hand the abelian category of sheaves of abelian groups on a topological space, on the other hand the abelian category of G-modules for a given group G. In Mathematics, certain Functors may be derived to obtain other functors closely related to the original ones
References
- P. Freyd. Abelian Categories, Harper and Row, New York, 1964. Available online.
- Alexandre Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Mathematics Journal, 1957
- Barry Mitchell: Theory of Categories, New York, Academic Press, 1965.
- N. Popescu: Abelian categories with applications to rings and modules, Academic Press, London, 1973.
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