Biproduct
Encyclopedia
In 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. The biproduct is a generalization of finite direct sums of modules
.
with zero object.
Given objects A1,...,An in C, their biproduct is an object A1 ⊕ ··· ⊕ An together with morphisms
satisfying
and such that
An empty, or nullary, product is always a terminal object in the category, and the empty coproduct is always an initial object
in the category. Since our category C has a zero object, the empty biproduct exists and is isomorphic to the zero object.
.
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 over a ring.
On the other hand, biproducts do not exist in the category of groups
. Here, the product is the direct product
, but the coproduct is the free product
.
Also, biproducts do not exist in the category of sets
. For, the product is given by the Cartesian product
, whereas the coproduct is given by the disjoint union
. Note also that this category does not have a zero object.
If the product A1 × A2 and coproduct A1 ∐ A2 both exist for some pair of objects Ai, then there is a unique morphism f: A1 ∐ A2 → A1 × A2 such that
It follows that the biproduct A1 ⊕ A2 exists if and only if f is an isomorphism.
If C is a preadditive category
, then every finite product is a biproduct, and every finite coproduct is a biproduct. For example, if A1 × A2 exists, then there are unique morphisms ik: Ak → A1 × A2 such that
To see that A1 × A2 is now also a coproduct, and hence a biproduct, suppose we have morphisms fk: Ak → X for some object X. Define f := f1 ∘ p1 + f2 ∘ p2. Then f: A1 × A2 → X is a morphism and f ∘ ik = fk.
Note also that in this case we always have
An additive category
is a preadditive category in which all finite biproduct exist. In particular, biproducts always exist in abelian categories.
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...
and its applications to mathematics
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...
, a biproduct of a finite collection of objects in 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...
with zero object is both a product
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 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...
. In a preadditive category
Preadditive category
In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
the notions of product and coproduct coincide for finite collections of objects. The biproduct is a generalization of finite direct sums of modules
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
.
Definition
Let C be a categoryCategory (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...
with zero object.
Given objects A1,...,An in C, their biproduct is an object A1 ⊕ ··· ⊕ An together with morphisms
- pk: A1 ⊕ ··· ⊕ An → Ak in C (the projection morphisms)
- ik: Ak → A1 ⊕ ··· ⊕ An (the injection morphisms)
satisfying
- pk ∘ ik = 1Ak, the identity morphism of Ak
- pl ∘ ik = 0, the zero morphism Ak → Al, for k ≠ l.
and such that
- (A1 ⊕ ··· ⊕ An,pk) is a 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...
for the Ak - (A1 ⊕ ··· ⊕ An,ik) is a 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...
for the Ak.
An empty, or nullary, product is always a terminal object in the category, and the empty coproduct is always an initial object
Initial object
In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
in the category. Since our category C has a zero object, the empty biproduct exists and is isomorphic to the zero object.
Examples
In the category of abelian groups, biproducts always exist and are given by the direct sum. Note that the zero object is the trivial groupTrivial 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...
.
Similarly, biproducts exist in the category of vector spaces
Category of vector spaces
In 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 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 over a ring.
On the other hand, biproducts do not exist in the category of groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
. Here, the product is the direct product
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
, but the coproduct is the free product
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
.
Also, biproducts do not exist in the category of sets
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 all functions from A to B...
. For, the product is given by the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
, whereas the coproduct is given by the disjoint union
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
. Note also that this category does not have a zero object.
Properties
If the biproduct A ⊕ B exists for all pairs of objects A and B in the category C, then all finite biproducts exist.If the product A1 × A2 and coproduct A1 ∐ A2 both exist for some pair of objects Ai, then there is a unique morphism f: A1 ∐ A2 → A1 × A2 such that
- pk ∘ f ∘ ik = 1Ak
- pl ∘ f ∘ ik = 0 for k ≠ l.
It follows that the biproduct A1 ⊕ A2 exists if and only if f is an isomorphism.
If C is a preadditive category
Preadditive category
In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
, then every finite product is a biproduct, and every finite coproduct is a biproduct. For example, if A1 × A2 exists, then there are unique morphisms ik: Ak → A1 × A2 such that
- pk ∘ ik = 1Ak
- pl ∘ ik = 0 for k ≠ l.
To see that A1 × A2 is now also a coproduct, and hence a biproduct, suppose we have morphisms fk: Ak → X for some object X. Define f := f1 ∘ p1 + f2 ∘ p2. Then f: A1 × A2 → X is a morphism and f ∘ ik = fk.
Note also that in this case we always have
- i1 ∘ p1 + i2 ∘ p2 = 1A1 × A2.
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....
is a preadditive category in which all finite biproduct exist. In particular, biproducts always exist in abelian categories.