Pre-Abelian category
Encyclopedia
In mathematics
, specifically in category theory
, a pre-abelian category is an additive category
that has all kernels
and cokernels.
Spelled out in more detail, this means that a category C is pre-abelian if:
Note that the zero morphism in item 3 can be identified as the identity element
of the hom-set Hom(A,B), which is an abelian group by item 1; or as the unique morphism A → O → B, where O is a zero object, guaranteed to exist by item 2.
s.
Ab is preadditive because it is a closed monoidal category
, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory
and the cokernel is the quotient map onto the ordinary cokernel from group theory
.
Other common examples:
These will give you an idea of what to think of; for more examples, see abelian category
(every abelian category is pre-abelian).
, and many basic properties of these categories are described under that subject.
This article concerns itself with the properties that exist specifically because of the existence of kernels and cokernels.
Although kernels and cokernels are special kinds of equaliser
s and coequalisers, a pre-abelian category actually has all equalisers and coequalisers.
We simply construct the equaliser of two morphisms f and g as the kernel of their difference g − f; similarly, their coequaliser is the cokernel of their difference.
(The alternative term "difference kernel" for binary equalisers derives from this fact.)
Since pre-abelian categories have all finite products
and coproduct
s (the biproducts) and all binary equalisers and coequalisers (as just described), then by a general theorem of category theory
, they have all finite limits
and colimits.
That is, pre-abelian categories are finitely complete.
The existence of both kernels and cokernels gives a notion of image
and coimage
.
We can define these as
That is, the image is the kernel of the cokernel, and the coimage is the cokernel of the kernel.
Note that this notion of image may not correspond to the usual notion of image, or range, of a function, even assuming that the morphisms in the category are functions
.
For example, in the category of topological abelian groups, the image of a morphism actually corresponds to the inclusion of the closure
of the range of the function.
For this reason, people will often distinguish the meanings of the two terms in this context, using "image" for the abstract categorical concept and "range" for the elementary function-theoretic concept.
In many common situations, such as the category of sets, where images and coimages exist, their objects are isomorphic.
Put more precisely, we have a factorisation of f: A → B as
where the morphism on the left is the coimage, the morphism on the right is the image, and the morphism in the middle (called the parallel of f) is an isomorphism.
In a pre-abelian category, this is not necessarily true.
The factorisation shown above does always exist, but the parallel might not be an isomorphism.
In fact, the parallel of f is an isomorphism for every
morphism f if and only if
the pre-abelian category is an abelian category
.
An example of a non-abelian, pre-abelian category is, once again, the category of topological abelian groups.
As remarked, the image is the inclusion of the closure of the range; however, the coimage is a quotient map onto the range itself.
Thus, the parallel is the inclusion of the range into its closure, which is not an isomorphism unless the range was already closed
.
and colimits exist in a pre-abelian category.
In general category theory
, a functor is called left exact
if it preserves all finite limits and right exact
if it preserves all finite colimits. (A functor is simply exact if it's both left exact and right exact.)
In a pre-abelian category, exact functors can be described in particularly simple terms.
First, recall that an additive functor is a functor F: C → D between preadditive categories that acts as a group homomorphism
on each hom-set.
Then it turns out that a functor between pre-abelian categories is left exact if and only if
it is additive and preserves all kernels, and it's right exact if and only if it's additive and preserves all cokernels.
Note that an exact functor, because it preserves both kernels and cokernels, preserves all images and coimages.
Exact functors are most useful in the study of abelian categories, where they can be applied to exact sequence
s.
The pre-abelian categories most commonly studied are in fact abelian categories; for example, Ab is an abelian category.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically in category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a pre-abelian category is an additive category
Additive category
In mathematics, specifically in category theory, an additive category is a preadditive category C such that all finite collections of objects A1,...,An of C have a biproduct A1 ⊕ ⋯ ⊕ An in C....
that has all kernels
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...
and cokernels.
Spelled out in more detail, this means that a category C is pre-abelian if:
- C is preadditivePreadditive categoryIn mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
, that is enrichedEnriched categoryIn category theory and its applications to mathematics, enriched category is a generalization of category that abstracts the set of morphisms associated with every pair of objects to an opaque object in some fixed monoidal category of "hom-objects" and then defines composition and identity solely...
over the monoidal categoryMonoidal categoryIn mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
of abelian groupAbelian groupIn 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 their order . Abelian groups generalize the arithmetic of addition of integers...
s; - C has all biproductBiproductIn category theory and its applications to mathematics, a biproduct of a finite collection of objects in a category with zero object is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects...
s, which are both finite productsProduct (category theory)In category theory, the product of two 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...
and finite coproductCoproductIn category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
s; - given any morphism f: A → B in C, the equaliserEqualiserIn mathematics, an equaliser, or equalizer, is a set of arguments where two or more functions have equal values.An equaliser is the solution set of an equation.In certain contexts, a difference kernel is the equaliser of exactly two functions....
of f and the zero morphismZero morphismIn category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...
from A to B exists (this is the kernel), as does the coequaliser (this is the cokernel).
Note that the zero morphism in item 3 can be identified as the identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
of the hom-set Hom(A,B), which is an abelian group by item 1; or as the unique morphism A → O → B, where O is a zero object, guaranteed to exist by item 2.
Examples
The original example of an additive category is the category Ab of abelian groupAbelian 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 their order . Abelian groups generalize the arithmetic of addition of integers...
s.
Ab is preadditive because it is a closed monoidal category
Closed monoidal category
In mathematics, especially in category theory, aclosed monoidal category is a context where we can take tensor products of objects and also form 'mapping objects'. A classic example is the category of sets, Set, where the tensor product of sets A and B is the usual cartesian product A \times B, and...
, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
and the cokernel is the quotient map onto the ordinary cokernel from group theory
Cokernel
In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
.
Other common examples:
- The category of (left) modulesModule (mathematics)In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over a ringRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R, in particular:- the category of vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s over a fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K.
- the category of vector space
- The category of (HausdorffHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
) abelianAbelian groupIn 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 their order . Abelian groups generalize the arithmetic of addition of integers...
topological groupTopological groupIn mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
s.
These will give you an idea of what to think of; for more examples, see abelian category
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. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
(every abelian category is pre-abelian).
Elementary properties
Every pre-abelian category is of course an additive categoryAdditive category
In mathematics, specifically in category theory, an additive category is a preadditive category C such that all finite collections of objects A1,...,An of C have a biproduct A1 ⊕ ⋯ ⊕ An in C....
, and many basic properties of these categories are described under that subject.
This article concerns itself with the properties that exist specifically because of the existence of kernels and cokernels.
Although kernels and cokernels are special kinds of equaliser
Equaliser
In mathematics, an equaliser, or equalizer, is a set of arguments where two or more functions have equal values.An equaliser is the solution set of an equation.In certain contexts, a difference kernel is the equaliser of exactly two functions....
s and coequalisers, a pre-abelian category actually has all equalisers and coequalisers.
We simply construct the equaliser of two morphisms f and g as the kernel of their difference g − f; similarly, their coequaliser is the cokernel of their difference.
(The alternative term "difference kernel" for binary equalisers derives from this fact.)
Since pre-abelian categories have all finite products
Product (category theory)
In category theory, the product of two 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...
and coproduct
Coproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
s (the biproducts) and all binary equalisers and coequalisers (as just described), then by a general theorem of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, they have all finite limits
Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
and colimits.
That is, pre-abelian categories are finitely complete.
The existence of both kernels and cokernels gives a notion of image
Image (category theory)
Given a category C and a morphismf\colon X\to Y in C, the image of f is a monomorphism h\colon I\to Y satisfying the following universal property:#There exists a morphism g\colon X\to I such that f = hg....
and coimage
Coimage
In algebra, the coimage of a homomorphismis the quotientof domain and kernel.The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies....
.
We can define these as
- im f := ker coker f;
- coim f := coker ker f.
That is, the image is the kernel of the cokernel, and the coimage is the cokernel of the kernel.
Note that this notion of image may not correspond to the usual notion of image, or range, of a function, even assuming that the morphisms in the category are functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
.
For example, in the category of topological abelian groups, the image of a morphism actually corresponds to the inclusion of the closure
Closure (topology)
In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
of the range of the function.
For this reason, people will often distinguish the meanings of the two terms in this context, using "image" for the abstract categorical concept and "range" for the elementary function-theoretic concept.
In many common situations, such as the category of sets, where images and coimages exist, their objects are isomorphic.
Put more precisely, we have a factorisation of f: A → B as
- A → C → I → B,
where the morphism on the left is the coimage, the morphism on the right is the image, and the morphism in the middle (called the parallel of f) is an isomorphism.
In a pre-abelian category, this is not necessarily true.
The factorisation shown above does always exist, but the parallel might not be an isomorphism.
In fact, the parallel of f is an isomorphism for every
Universal quantification
In predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....
morphism f if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
the pre-abelian category is an abelian category
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. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
.
An example of a non-abelian, pre-abelian category is, once again, the category of topological abelian groups.
As remarked, the image is the inclusion of the closure of the range; however, the coimage is a quotient map onto the range itself.
Thus, the parallel is the inclusion of the range into its closure, which is not an isomorphism unless the range was already closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
.
Exact functors
Recall that all finite limitsLimit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
and colimits exist in a pre-abelian category.
In general category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a functor is called left exact
Exact functor
In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...
if it preserves all finite limits and right exact
Exact functor
In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...
if it preserves all finite colimits. (A functor is simply exact if it's both left exact and right exact.)
In a pre-abelian category, exact functors can be described in particularly simple terms.
First, recall that an additive functor is a functor F: C → D between preadditive categories that acts as a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
on each hom-set.
Then it turns out that a functor between pre-abelian categories is left exact if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is additive and preserves all kernels, and it's right exact if and only if it's additive and preserves all cokernels.
Note that an exact functor, because it preserves both kernels and cokernels, preserves all images and coimages.
Exact functors are most useful in the study of abelian categories, where they can be applied to exact sequence
Exact sequence
An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
s.
Special cases
- An abelian categoryAbelian categoryIn 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. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
is a pre-abelian category such that every monomorphismMonomorphismIn the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
and epimorphismEpimorphismIn category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
is normal.
The pre-abelian categories most commonly studied are in fact abelian categories; for example, Ab is an abelian category.