Preadditive category
Encyclopedia
In mathematics
, specifically in category theory
, a preadditive category is a category
that is enriched
over the monoidal category
of abelian group
s. In other words, the category C is preadditive if every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear
over the integer
s.
A preadditive category is also called an Ab-category, after the notation Ab for the category of abelian groups
. Some authors have used the term additive category for preadditive categories, but Wikipedia follows the current trend of reserving this word for certain special preadditive categories (see special cases below).
. (Note that commutativity
is crucial here; it ensures that the sum of two group homomorphism
s is again a homomorphism. In contrast, the category of all group
s is not closed.) See medial category.
Other common examples:
These will give you an idea of what to think of; for more examples, follow the links to special cases below.
element 0. This is the zero morphism
from A to B. Because composition of morphisms is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition. Extending this analogy, the fact that composition is bilinear in general becomes the distributivity
of multiplication over addition.
Focusing on a single object A in a preadditive category, these facts say that the endomorphism
hom-set Hom(A,A) is a ring, if we define multiplication in the ring to be composition. This ring is the endomorphism ring of A. Conversely, every ring (with identity
) is the endomorphism ring of some object in some preadditive category. Indeed, given a ring R, we can define a preadditive category R to have a single object A, let Hom(A,A) be R, and let composition be ring multiplication. Since R is an Abelian group and multiplication in a ring is bilinear (distributive), this makes R a preadditive category. Category theorists will often think of the ring R and the category R as two different representations of the same thing, so that a particularly perverse
category theorist might define a ring as a preadditive category with exactly one object (in the same way that a monoid
can be viewed as a category with only one object - and forgetting the additive structure of the ring gives us a monoid).
In this way, preadditive categories can be seen as a generalisation of rings. Many concepts from ring theory, such as ideals, Jacobson radical
s, and factor rings can be generalized in a straightforward manner to this setting. When attempting to write down these generalizations, one should think of the morphisms in the preadditive category as the "elements" of the "generalized ring". We won't go into such depth in this article.
F: C → D is additive if it too is enriched over the category Ab. That is, F is additive if and only if
, given any objects A and B of C, the function
F: Hom(A,B) → Hom(F(A),F(B)) is a group homomorphism
. Most functors studied between preadditive categories are additive.
For a simple example, if the rings R and S are represented by the one-object preadditive categories R and S, then a ring homomorphism
from R to S is represented by an additive functor from R to S, and conversely.
If C and D are categories and D is preadditive, then the functor category
Fun(C,D) is also preadditive, because natural transformation
s can be added in a natural way.
If C is preadditive too, then the category Add(C,D) of additive functors and all natural transformations between them is also preadditive.
The latter example leads to a generalization of module
s over rings: If C is a preadditive category, then Mod(C) := Add(C,Ab) is called the module category over C. When C is the one-object preadditive category corresponding to the ring R, this reduces to the ordinary category of (left) R-modules. Again, virtually all concepts from the theory of modules can be generalised to this setting.
in a preadditive category must also be a coproduct
, and conversely. In fact, finite products and coproducts in preadditive categories can be characterised by the following biproduct condition:
This biproduct is often written A1 ⊕ ··· ⊕ An, borrowing the notation for the direct sum
. This is because the biproduct in well known preadditive categories like Ab is the direct sum. However, although infinite direct sums make sense in some categories, like Ab, infinite biproducts do not make sense.
The biproduct condition in the case n = 0 simplifies drastically; B is a nullary biproduct if and only if the identity morphism of B is the zero morphism from B to itself, or equivalently if the hom-set Hom(B,B) is the trivial ring. Note that because a nullary biproduct will be both terminal (a nullary product) and coterminal (a nullary coproduct), it will in fact be a zero object.
Indeed, the term "zero object" originated in the study of preadditive categories like Ab, where the zero object is the zero group
.
A preadditive category in which every biproduct exists (including a zero object) is called additive
. Further facts about biproducts that are mainly useful in the context of additive categories may be found under that subject.
the notion of kernel
and cokernel
make sense. That is, if f: A → B is a
morphism in a preadditive category, then the kernel of f is the
equaliser of f and the zero morphism from A to B, while the cokernel of f is the coequaliser of f and this zero morphism. Unlike with products and coproducts, the kernel and cokernel of f are generally not equal in a preadditive category.
When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of a kernel
of a homomorphism, if one identifies the ordinary kernel K of f: A → B with its embedding K → A. However, in a general preadditive category there may exist morphisms without kernels and/or cokernels.
There is a convenient relationship between the kernel and cokernel and the Abelian group structure on the hom-sets. Given parallel morphisms f and g, the equaliser of f and g is just the kernel of g − f, if either exists, and the analogous fact is true for coequalisers. The alternative term "difference kernel" for binary equalisers derives from this fact.
A preadditive category in which all biproducts, kernels, and cokernels exist is called pre-Abelian
. Further facts about kernels and cokernels in preadditive categories that are mainly useful in the context of pre-Abelian categories may be found under that subject.
The preadditive 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 preadditive category is a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
that is enriched
Enriched category
In 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 category
Monoidal category
In 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 group
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 their order . Abelian groups generalize the arithmetic of addition of integers...
s. In other words, the category C is preadditive if every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear
Bilinear operator
In mathematics, a bilinear operator is a function combining elements of two vector spaces to yield an element of a third vector space that is linear in each of its arguments. Matrix multiplication is an example.-Definition:...
over the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s.
A preadditive category is also called an Ab-category, after the notation Ab for the category of abelian groups
Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....
. Some authors have used the term additive category for preadditive categories, but Wikipedia follows the current trend of reserving this word for certain special preadditive categories (see special cases below).
Examples
The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal categoryClosed 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...
. (Note that commutativity
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
is crucial here; it ensures that the sum of two 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...
s is again a homomorphism. In contrast, the category of all group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s is not closed.) See medial category.
Other common examples:
- The category of (left) moduleModule (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...
s 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 spacesCategory of vector spacesIn mathematics, especially category theory, the category K-Vect has all vector spaces over a fixed field K as objects and K-linear transformations as morphisms...
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 spaces
- The algebra of matricesMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
over a ring, thought of as a category as described in the article Additive categoryAdditive categoryIn 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....
. - Any ring, thought of as a category with only one object, is a preadditive category. Here composition of morphisms is just ring multiplication and the unique hom-set is the underlying abelian group.
These will give you an idea of what to think of; for more examples, follow the links to special cases below.
Elementary properties
Because every hom-set Hom(A,B) is an abelian group, it has a zero0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
element 0. This is the zero morphism
Zero morphism
In 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. Because composition of morphisms is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition. Extending this analogy, the fact that composition is bilinear in general becomes the distributivity
Distributivity
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
of multiplication over addition.
Focusing on a single object A in a preadditive category, these facts say that the endomorphism
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
hom-set Hom(A,A) is a ring, if we define multiplication in the ring to be composition. This ring is the endomorphism ring of A. Conversely, every ring (with identity
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...
) is the endomorphism ring of some object in some preadditive category. Indeed, given a ring R, we can define a preadditive category R to have a single object A, let Hom(A,A) be R, and let composition be ring multiplication. Since R is an Abelian group and multiplication in a ring is bilinear (distributive), this makes R a preadditive category. Category theorists will often think of the ring R and the category R as two different representations of the same thing, so that a particularly perverse
Abstract nonsense
In mathematics, abstract nonsense, general abstract nonsense, and general nonsense are terms used facetiously by some mathematicians to describe certain kinds of arguments and methods related to category theory. roughly speaking, category theory is the study of the general form of mathematical...
category theorist might define a ring as a preadditive category with exactly one object (in the same way that a monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
can be viewed as a category with only one object - and forgetting the additive structure of the ring gives us a monoid).
In this way, preadditive categories can be seen as a generalisation of rings. Many concepts from ring theory, such as ideals, Jacobson radical
Jacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
s, and factor rings can be generalized in a straightforward manner to this setting. When attempting to write down these generalizations, one should think of the morphisms in the preadditive category as the "elements" of the "generalized ring". We won't go into such depth in this article.
Additive functors
If C and D are preadditive categories, then a functorFunctor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
F: C → D is additive if it too is enriched over the category Ab. That is, F is additive 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....
, given any objects A and B of C, the function
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...
F: Hom(A,B) → Hom(F(A),F(B)) is 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...
. Most functors studied between preadditive categories are additive.
For a simple example, if the rings R and S are represented by the one-object preadditive categories R and S, then a ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
from R to S is represented by an additive functor from R to S, and conversely.
If C and D are categories and D is preadditive, then the functor category
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...
Fun(C,D) is also preadditive, because natural transformation
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 of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
s can be added in a natural way.
If C is preadditive too, then the category Add(C,D) of additive functors and all natural transformations between them is also preadditive.
The latter example leads to a generalization of module
Module (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...
s over rings: If C is a preadditive category, then Mod(C) := Add(C,Ab) is called the module category over C. When C is the one-object preadditive category corresponding to the ring R, this reduces to the ordinary category of (left) R-modules. Again, virtually all concepts from the theory of modules can be generalised to this setting.
Biproducts
Any finite productProduct (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...
in a preadditive category must also be a 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...
, and conversely. In fact, finite products and coproducts in preadditive categories can be characterised by the following biproduct condition:
- The object B is a biproduct of the objects A1,...,An if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
there are projection morphisms pj: B → Aj and injection morphisms ij: Aj → B, such that (i1 o p1) + ··· + (in o pn) is the identity morphism of B, pj o ij is the identity morphism of Aj, and pj o ik is the zero morphism from Ak to Aj whenever j and k are distinct.
This biproduct is often written A1 ⊕ ··· ⊕ An, borrowing the notation for the direct sum
Direct sum
In mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...
. This is because the biproduct in well known preadditive categories like Ab is the direct sum. However, although infinite direct sums make sense in some categories, like Ab, infinite biproducts do not make sense.
The biproduct condition in the case n = 0 simplifies drastically; B is a nullary biproduct if and only if the identity morphism of B is the zero morphism from B to itself, or equivalently if the hom-set Hom(B,B) is the trivial ring. Note that because a nullary biproduct will be both terminal (a nullary product) and coterminal (a nullary coproduct), it will in fact be a zero object.
Indeed, the term "zero object" originated in the study of preadditive categories like Ab, where the zero object is the zero group
Trivial group
In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...
.
A preadditive category in which every biproduct exists (including a zero object) is called additive
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....
. Further facts about biproducts that are mainly useful in the context of additive categories may be found under that subject.
Kernels and cokernels
Because the hom-sets in a preadditive category have zero morphisms,the notion of kernel
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 cokernel
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....
make sense. That is, if f: A → B is a
morphism in a preadditive category, then the kernel of f is the
equaliser of f and the zero morphism from A to B, while the cokernel of f is the coequaliser of f and this zero morphism. Unlike with products and coproducts, the kernel and cokernel of f are generally not equal in a preadditive category.
When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of a kernel
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...
of a homomorphism, if one identifies the ordinary kernel K of f: A → B with its embedding K → A. However, in a general preadditive category there may exist morphisms without kernels and/or cokernels.
There is a convenient relationship between the kernel and cokernel and the Abelian group structure on the hom-sets. Given parallel morphisms f and g, the equaliser of f and g is just the kernel of g − f, if either exists, and the analogous fact is true for coequalisers. The alternative term "difference kernel" for binary equalisers derives from this fact.
A preadditive category in which all biproducts, kernels, and cokernels exist is called pre-Abelian
Pre-Abelian category
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:...
. Further facts about kernels and cokernels in preadditive categories that are mainly useful in the context of pre-Abelian categories may be found under that subject.
Special cases
Most of these special cases of preadditive categories have all been mentioned above, but they're gathered here for reference.- A ring is a preadditive category with exactly one object.
- An additive categoryAdditive categoryIn 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....
is a preadditive category with all finite biproducts. - A pre-Abelian categoryPre-Abelian categoryIn 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:...
is an additive category with all kernels and cokernels. - 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 preadditive categories most commonly studied are in fact Abelian categories; for example, Ab is an Abelian category.