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 ...
and its applications to
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 ...
, a biproduct of a finite collection of
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...
, in 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) ...
with
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): ...
s, is both a
product and a
coproduct. In a
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.
That is, an Ab-category C is a category such that
every hom- ...
the notions of product and coproduct coincide for finite collections of objects. The biproduct is a generalization of finite
direct sums of modules.
Definition
Let C be 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) ...
with
zero morphisms. Given a finite (possibly empty) collection of objects ''A''
1, ..., ''A''
''n'' in C, their ''biproduct'' is an
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
in C together with
morphisms
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 C (the ''
projection morphisms'')
*
(the ''
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
morphisms'')
satisfying
*
, the identity morphism of
and
*
, the
zero morphism for
and such that
*
is a
product for the
and
*
is a
coproduct for the
If C is preadditive and the first two conditions hold, then each of the last two conditions is equivalent to
when ''n'' > 0. An empty, or
nullary, product is always a
terminal 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): ...
in the category, and the empty coproduct is always an
initial 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): ...
in the category. Thus an empty, or nullary, biproduct is always 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): ...
.
Examples
In the category of
abelian groups, biproducts always exist and are given by 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 ...
. The zero object is the
trivial 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 ...
.
Similarly, biproducts exist in the
category of vector spaces over a
field. The biproduct is again the direct sum, and the zero object is the
trivial vector space.
More generally, biproducts exist in the
category of modules
In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ri ...
over a
ring.
On the other hand, biproducts do not exist in the
category of groups. Here, the product is the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
, but the coproduct is the
free product.
Also, biproducts do not exist in the
category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
. For, the product is given by the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
, whereas the coproduct is given by the
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
. This category does not have a zero object.
Block matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original mat ...
algebra relies upon biproducts in categories of
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
.
[H.D. Macedo, J.N. Oliveira]
Typing linear algebra: A biproduct-oriented approach
Science of Computer Programming, Volume 78, Issue 11, 1 November 2013, Pages 2160-2191, , .
Properties
If the biproduct
exists for all pairs of objects ''A'' and ''B'' in the category C, and C has a zero object, then all finite biproducts exist, making C both a
Cartesian monoidal category and a co-Cartesian monoidal category.
If the product
and coproduct
both exist for some pair of objects ''A''
1, ''A''
2 then there is a unique morphism
such that
*
*
for
It follows that the biproduct
exists if and only if ''f'' is an
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. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
.
If C is a
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.
That is, an Ab-category C is a category such that
every hom- ...
, then every finite product is a biproduct, and every finite coproduct is a biproduct. For example, if
exists, then there are unique morphisms
such that
*
*
for
To see that
is now also a coproduct, and hence a biproduct, suppose we have morphisms
for some object
. Define
Then
is a morphism from
to
, and
for
.
In this case we always have
*
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
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.
That is, an Ab-category C is a category such that
every hom- ...
in which all finite biproducts exist. In particular, biproducts always exist in
abelian categories
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 abe ...
.
References
*{{rp, at=Section 1.2
Additive categories
Limits (category theory)