In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, a preadditive category is
another name for an Ab-category, i.e., a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
that is
enriched over the
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: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object of ...
, Ab.
That is, an Ab-category C is a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
such that
every
hom-set
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
Hom(''A'',''B'') in C has the structure of an abelian group, and composition of morphisms is
bilinear, in the sense that composition of morphisms distributes over the group operation.
In formulas:
and
where + is the group operation.
Some authors have used the term ''additive category'' for preadditive categories, but here we follow the current trend of reserving this term for certain special preadditive categories (see below).
Examples
The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a
closed monoidal category. Note that
commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
is crucial here; it ensures that the sum of two
group homomorphisms is again a homomorphism. In contrast, the category of all
groups is not closed. See
Medial category.
Other common examples:
* The category of (left)
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s over a
ring ''R'', in particular:
** the
category of vector spaces over a
field ''K''.
* The algebra of
matrices over a ring, thought of as a category as described in the article
Additive category
In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts.
Definition
A category C is preadditive if all its hom-sets are abelian groups and composition of ...
.
* 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 below.
Elementary properties
Because every hom-set Hom(''A'',''B'') is an abelian group, it has a
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...
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
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmeti ...
of multiplication over addition.
Focusing on a single object ''A'' in a preadditive category, these facts say that the
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
hom-set Hom(''A'',''A'') is a
ring, if we define multiplication in the ring to be composition. This ring is the
endomorphism ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a ...
of ''A''. Conversely, every ring (with
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
) 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
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
object (in the same way that a
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
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
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
s,
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition y ...
s, and
factor ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. I ...
s 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".
Additive functors
If
and
are preadditive categories, then a
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
is additive if it too is
enriched over the category
. That is,
is additive
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
, given any objects
and
of
, the
function is a
group homomorphism. Most functors studied between preadditive categories are additive.
For a simple example, if the rings
and
are represented by the one-object preadditive categories
and
, then a
ring homomorphism from
to
is represented by an additive functor from
to
, and conversely.
If
and
are categories and
is preadditive, then the
functor category is also preadditive, because
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 (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s can be added in a natural way.
If
is preadditive too, then the category
of additive functors and all natural transformations between them is also preadditive.
The latter example leads to a generalization of
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s over rings: If
is a preadditive category, then
is called the module category over
. When
is the one-object preadditive category corresponding to the ring
, this reduces to the ordinary category of
(left) -modules. Again, virtually all concepts from the theory of modules can be generalised to this setting.
-linear categories
More generally, one can consider a category enriched over the monoidal category of
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, called an -linear category. In other words, each
hom-set
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
in has the structure of an -module, and composition of morphisms is -bilinear.
When considering functors between two -linear categories, one often restricts to those that are -linear, so those that induce -linear maps on each hom-set.
Biproducts
Any
finite product 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'':
:The object ''B'' is a biproduct of the objects ''A''
1, ..., ''A
n''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
there are ''projection morphisms'' ''p''
''j'': ''B'' → ''A''
''j'' and ''injection morphisms'' ''i''
''j'': ''A''
''j'' → ''B'', such that (''i''
1∘''p''
1) + ··· + (''i
n''∘''p
n'') is the identity morphism of ''B'', ''p
j''∘''i
j'' is the
identity morphism of
Aj, and ''p''
''j''∘''i
k'' is the zero morphism from ''A''
''k'' to ''A
j'' whenever ''j'' and ''k'' are
distinct.
This biproduct is often written ''A''
1 ⊕ ··· ⊕ ''A
n'', borrowing the notation for the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
. 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 (see {{section link, Category of abelian groups#Properties).
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
In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for ...
. Note that because a nullary biproduct will be both
terminal
Terminal may refer to:
Computing Hardware
* Terminal (electronics), a device for joining electrical circuits together
* Terminal (telecommunication), a device communicating over a line
* Computer terminal, a set of primary input and output devi ...
(a nullary product) and
initial
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph tha ...
(a nullary coproduct), it will in fact be a
zero object
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 .
The dual notion is that of a terminal object (also called terminal element): ...
.
Indeed, the term "zero object" originated in the study of preadditive categories like Ab, where the zero object is the
zero group
In mathematics, a trivial group or zero 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 and so it is usuall ...
.
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.
Kernels and cokernels
Because the hom-sets in a preadditive category have zero morphisms,
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.
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 category
In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts.
Definition
A category C is preadditive if all its hom-sets are abelian groups and composition of ...
'' is a preadditive category with all finite biproducts.
* A ''
pre-abelian category'' is an additive category with all kernels and cokernels.
* An ''
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 prototypical example of an abelian category is the category of ...
'' is a pre-abelian category such that every
monomorphism and
epimorphism is
normal.
The preadditive categories most commonly studied are in fact abelian categories; for example, Ab is an abelian category.
References
*
Nicolae Popescu
Nicolae Popescu (; 22 September 1937 – 29 July 2010) was a Romanian mathematician and professor at the University of Bucharest. He also held a research position at the Institute of Mathematics of the Romanian Academy, and was elected correspo ...
; 1973;
Abelian Categories with Applications to Rings and Modules; Academic Press, Inc.; out of print
*
Charles Weibel; 1994;
An introduction to homological algebras; Cambridge Univ. Press
Additive categories