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In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects.^[1] The biproduct is a generalization of finite direct sums of modules.

Definition

Let C be a category with zero morphisms. Given a finite (possibly empty) collection of objects A₁, ..., A_n in C, their biproduct is an object ${\textstyle A_{1}\oplus \dots \oplus A_{n}}$ in C together with morphisms

${\textstyle p_{k}\!:A_{1}\oplus \dots \oplus A_{n}\to A_{k}}$ in C (the projection morphisms)
${\textstyle i_{k}\!:A_{k}\to A_{1}\oplus \dots \oplus A_{n}}$ (the embedding morphisms)

satisfying

${\textstyle p_{k}\circ i_{k}=1_{A_{k}}}$ , the identity morphism of $A_{k},$ and
${\textstyle p_{l}\circ i_{k}=0}$ , the zero morphism $A_{k}\to A_{l},$ for $k\neq l,$

and such that

${\textstyle \left(A_{1}\oplus \dots \oplus A_{n},p_{k}\right)}$ is a product for the ${\textstyle A_{k},}$ and
${\textstyle \left(A_{1}\oplus \dots \oplus A_{n},i_{k}\right)}$ is a coproduct for the ${\textstyle A_{k}.}$

If C is preadditive and the first two conditions hold, then each of the last two conditions is equivalent to ${\textstyle i_{1}\circ p_{1}+\dots +i_{n}\circ p_{n}=1_{A_{1}\oplus \dots \oplus A_{n}}}$ when n > 0.^[2] 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. Thus an empty, or nullary, biproduct is always a zero object.

Examples

In the category of abelian groups, biproducts always exist and are given by the direct sum.^[3] The zero object is the trivial group.

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.^[4] 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. This category does not have a zero object.

Block matrix algebra relies upon biproducts in categories of matrices.^[5]

Properties

If the biproduct ${\textstyle A\oplus B}$ 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 ${\textstyle A_{1}\times A_{2}}$ and coproduct ${\textstyle A_{1}\coprod A_{2}}$ both exist for some pair of objects A₁, A₂ then there is a unique morphism ${\textstyle f:A_{1}\coprod A_{2}\to A_{1}\times A_{2}}$ such that

$p_{k}\circ f\circ i_{k}=1_{A_{k}},\ (k=1,2)$
$p_{l}\circ f\circ i_{k}=0$ for ${\textstyle k\neq l.}$ ^{[clarification needed]}

It follows that the biproduct ${\textstyle A_{1}\oplus A_{2}}$ 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 ${\textstyle A_{1}\times A_{2}}$ exists, then there are unique morphisms ${\textstyle i_{k}:A_{k}\to A_{1}\times A_{2}}$ such that

$p_{k}\circ i_{k}=1_{A_{k}},\ (k=1,2)$
$p_{l}\circ i_{k}=0$ for ${\textstyle k\neq l.}$

To see that ${\textstyle A_{1}\times A_{2}}$ is now also a coproduct, and hence a biproduct, suppose we have morphisms ${\textstyle f_{k}:A_{k}\to X,\ k=1,2}$ for some object ${\textstyle X}$ . Define ${\textstyle f:=f_{1}\circ p_{1}+f_{2}\circ p_{2}.}$ Then ${\textstyle f}$ is a morphism from ${\textstyle A_{1}\times A_{2}}$ to ${\textstyle X}$ , and ${\textstyle f\circ i_{k}=f_{k}}$ for ${\textstyle k=1,2}$ .

In this case we always have

${\textstyle i_{1}\circ p_{1}+i_{2}\circ p_{2}=1_{A_{1}\times A_{2}}.}$

An additive category is a preadditive category in which all finite biproducts exist. In particular, biproducts always exist in abelian categories.

References

^ Borceux, 4-5
^ Saunders Mac Lane - Categories for the Working Mathematician, Second Edition, page 194.
^ Borceux, 8
^ Borceux, 7
^ 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, ISSN 0167-6423, doi:10.1016/j.scico.2012.07.012.

Borceux, Francis (2008). Handbook of Categorical Algebra 2: Categories and Structures. Cambridge University Press. ISBN 978-0-521-06122-3.^{: Section 1.2}

[1] Borceux, 4-5

[2] Saunders Mac Lane - Categories for the Working Mathematician, Second Edition, page 194.

[3] Borceux, 8

[4] Borceux, 7

[5] 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, ISSN 0167-6423, doi:10.1016/j.scico.2012.07.012.

[1]

[2]

[3]

[4]

[5]

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 {{about|biproducts in mathematics|an incidental product from a process|By-product}}
-In [[category theory]] and its applications to [[mathematics]], a '''biproduct''' of a finite collection of objects in a [[category (mathematics)|category]] with [[zero object]] is both a [[product (category theory)|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 sum of modules|direct sums of modules]].
+In [[category theory]] and its applications to [[mathematics]], a '''biproduct''' of a finite collection of [[Object (category theory)|objects]], in a [[category (mathematics)|category]] with [[zero object]]s, is both a [[product (category theory)|product]] and a [[coproduct]]. In a [[preadditive category]] the notions of product and coproduct coincide for finite collections of objects.<ref>Borceux, 4-5</ref> The biproduct is a generalization of finite [[Direct sum of modules|direct sums of modules]].
 ==Definition==
+Let '''C''' be a [[category (mathematics)|category]] with [[zero morphism|zero morphisms]]. Given a finite (possibly empty) collection of objects ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> in '''C''', their ''biproduct'' is an [[Object (category theory)|object]] <math display="inline">A_1 \oplus \dots \oplus A_n</math> in '''C''' together with [[Morphism|morphisms]]
-Let '''C''' be a [[category (mathematics)|category]] with [[zero object]].
+*<math display="inline">p_k \!: A_1 \oplus \dots \oplus A_n \to A_k</math> in '''C''' (the ''[[Projection (mathematics)|projection]] morphisms'')
-Given objects ''A''<sub>1</sub>,...,''A''<sub>''n''</sub> in '''C''', their ''biproduct'' is an object ''A''<sub>1</sub> ⊕ ··· ⊕ ''A''<sub>''n''</sub> together with morphisms
-*''p''<sub>''k''</sub>: ''A''<sub>1</sub> ⊕ ··· ⊕ ''A''<sub>''n''</sub> → ''A''<sub>''k''</sub> in '''C''' (the ''projection morphisms'')
+*<math display="inline">i_k \!: A_k \to A_1 \oplus \dots \oplus A_n</math> (the ''[[embedding]] morphisms'')
-*''i''<sub>''k''</sub>: ''A''<sub>''k''</sub> → ''A''<sub>1</sub> ⊕ ··· ⊕ ''A''<sub>''n''</sub> (the ''injection morphisms'')
 satisfying
-*''p''<sub>''k''</sub> ∘ ''i''<sub>''k''</sub> = 1<sub>''A''<sub>''k''</sub></sub>, the identity morphism of ''A''<sub>''k''</sub>
+*<math display="inline">p_k \circ i_k = 1_{A_k}</math>, the identity morphism of <math>A_k,</math> and
-*''p''<sub>''l''</sub> ∘ ''i''<sub>''k''</sub> = 0, the [[zero morphism]] ''A''<sub>''k''</sub> → ''A''<sub>''l''</sub>, for ''k'' ≠ ''l''.
+*<math display="inline">p_l \circ i_k = 0</math>, the [[zero morphism]] <math>A_k \to A_l,</math> for <math>k \neq l,</math>
 and such that
-*(''A''<sub>1</sub> ⊕ ··· ⊕ ''A''<sub>''n''</sub>,''p''<sub>''k''</sub>) is a [[product (category theory)|product]] for the ''A''<sub>''k''</sub>
+*<math display="inline">\left( A_1 \oplus \dots \oplus A_n, p_k \right)</math> is a [[product (category theory)|product]] for the <math display="inline">A_k,</math> and
-*(''A''<sub>1</sub> ⊕ ··· ⊕ ''A''<sub>''n''</sub>,''i''<sub>''k''</sub>) is a [[coproduct]] for the ''A''<sub>''k''</sub>.
+*<math display="inline">\left( A_1 \oplus \dots \oplus A_n, i_k \right)</math> is a [[coproduct]] for the <math display="inline">A_k.</math>
-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.
+If '''C''' is preadditive and the first two conditions hold, then each of the last two conditions is equivalent to <math display="inline">i_1 \circ p_1 + \dots + i_n\circ p_n  = 1_{A_1 \oplus \dots \oplus A_n}</math> when ''n'' > 0.<ref>Saunders Mac Lane - Categories for the Working Mathematician, Second Edition, page 194.</ref> 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. Thus an empty, or nullary, biproduct is always a [[zero object]].
 ==Examples==
-In the category of [[abelian groups]], biproducts always exist and are given by the [[Direct sum of abelian groups|direct sum]]. Note that the zero object is the [[trivial group]].
+In the category of [[abelian group]]s, biproducts always exist and are given by the [[Direct sum of abelian groups|direct sum]].<ref>Borceux, 8</ref> The zero object is the [[trivial group]].
-Similarly, biproducts exist in the [[category of vector spaces]] over a [[field (algebra)|field]]. The biproduct is again the direct sum, and the zero object is the [[Examples_of_vector_spaces#Trivial_or_zero_vector_space|trivial vector space]].
+Similarly, biproducts exist in the [[category of vector spaces]] over a [[Field (mathematics)|field]]. The biproduct is again the direct sum, and the zero object is the [[Examples_of_vector_spaces#Trivial_or_zero_vector_space|trivial vector space]].
-More generally, biproducts exist in the [[category of modules]] over a ring.
+More generally, biproducts exist in the [[category of modules]] over a [[Ring (mathematics)|ring]].
-On the other hand, biproducts do not exist in the [[category of groups]]. Here, the product is the [[direct product of groups|direct product]], but the coproduct is the [[free product]].
+On the other hand, biproducts do not exist in the [[category of groups]].<ref>Borceux, 7</ref> Here, the product is the [[direct product of groups|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.
+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]]. This category does not have a zero object.
+[[Block matrix]] algebra relies upon biproducts in categories of [[Matrix (mathematics)|matrices]].<ref>H.D. Macedo, J.N. Oliveira, [https://hal.inria.fr/hal-00919866 Typing linear algebra: A biproduct-oriented approach], Science of Computer Programming, Volume 78, Issue 11, 1 November 2013, Pages 2160-2191, {{issn|0167-6423}}, {{doi|10.1016/j.scico.2012.07.012}}.</ref>
 ==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 biproduct <math display="inline">A \oplus B</math> 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 <math display="inline">A_1 \times A_2</math> and coproduct <math display="inline">A_1 \coprod A_2</math> both exist for some pair of objects ''A''<sub>1</sub>, ''A''<sub>2</sub> then there is a unique morphism <math display="inline">f: A_1 \coprod A_2 \to A_1 \times A_2</math> such that
+*<math>p_k \circ f \circ i_k = 1_{A_k},\ (k = 1, 2)</math>
+*<math>p_l \circ f \circ i_k = 0 </math> for <math display="inline">k \neq l.</math>{{clarify|reason=Surely we need to require that the category has zero morphisms, or at least a zero object, since otherwise this equation doesn't make sense.|date=April 2020}}
+It follows that the biproduct <math display="inline">A_1 \oplus A_2</math> 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 <math display="inline">A_1 \times A_2</math> exists, then there are unique morphisms <math display="inline">i_k: A_k \to A_1 \times A_2</math> such that
+*<math>p_k \circ i_k = 1_{A_k},\ (k = 1, 2)</math>
-If the product ''A''<sub>1</sub> × ''A''<sub>2</sub> and coproduct ''A''<sub>1</sub> ∐ ''A''<sub>2</sub> both exist for some pair of objects ''A''<sub>i</sub>, then there is a unique morphism ''f'': ''A''<sub>1</sub> ∐ ''A''<sub>2</sub> → ''A''<sub>1</sub> × ''A''<sub>2</sub> such that
+*<math>p_l \circ i_k = 0 </math> for <math display="inline">k \neq l.</math>
-*''p''<sub>''k''</sub> ∘ ''f'' ∘ ''i''<sub>k</sub> = 1<sub>''A''<sub>k</sub></sub>
-*''p''<sub>''l''</sub> ∘ ''f'' ∘ ''i''<sub>k</sub> = 0 for ''k'' ≠ ''l''.
-It follows that the biproduct ''A''<sub>1</sub> ⊕ ''A''<sub>2</sub> exists if and only if ''f'' is an isomorphism.
+To see that <math display="inline">A_1 \times A_2</math> is now also a coproduct, and hence a biproduct, suppose we have morphisms <math display="inline">f_k: A_k \to X,\ k=1,2</math> for some object <math display="inline">X</math>. Define <math display="inline">f := f_1 \circ p_1 + f_2 \circ p_2.</math> Then <math display="inline">f</math> is a morphism from <math display="inline">A_1 \times A_2</math> to <math display="inline">X</math>, and <math display="inline">f \circ i_k = f_k</math> for <math display="inline">k = 1, 2</math>.
-If '''C''' is a [[preadditive category]], then every finite product is a biproduct, and every finite coproduct is a biproduct. For example, if ''A''<sub>1</sub> × ''A''<sub>2</sub> exists, then there are unique morphisms ''i''<sub>k</sub>: ''A''<sub>k</sub> → ''A''<sub>1</sub> × ''A''<sub>2</sub> such that
-*''p''<sub>k</sub> ∘ ''i''<sub>k</sub> = 1<sub>''A''<sub>k</sub></sub>
-*''p''<sub>l</sub> ∘ ''i''<sub>k</sub> = 0 for ''k'' ≠ ''l''.
-To see that ''A''<sub>1</sub> × ''A''<sub>2</sub> is now also a coproduct, and hence a biproduct, suppose we have morphisms ''f''<sub>k</sub>: ''A''<sub>k</sub> → ''X'' for some object ''X''. Define ''f'' := ''f''<sub>1</sub> ∘ ''p''<sub>1</sub> + ''f''<sub>2</sub> ∘ ''p''<sub>2</sub>. Then ''f'': ''A''<sub>1</sub> × ''A''<sub>2</sub> → ''X'' is a morphism and ''f'' ∘ ''i''<sub>k</sub> = ''f''<sub>k</sub>.
-Note also that in this case we always have
+In this case we always have
+*<math display="inline">i_1 \circ p_1 + i_2 \circ p_2 = 1_{A_1 \times A_2}.</math>
-*''i''<sub>1</sub> ∘ ''p''<sub>1</sub> + ''i''<sub>2</sub> ∘ ''p''<sub>2</sub> = 1<sub>''A''<sub>1</sub> × ''A''<sub>2</sub></sub>.
-An [[additive category]] is a preadditive category in which all finite biproduct exist. In particular, biproducts always exist in [[abelian categories]].
+An [[additive category]] is a [[preadditive category]] in which all finite biproducts exist. In particular, biproducts always exist in [[abelian categories]].
 ==References==
 {{reflist}}
+*{{cite book|last1=Borceux|first1=Francis|title=Handbook of Categorical Algebra 2: Categories and Structures|date=2008|publisher=[[Cambridge University Press]]|isbn=978-0-521-06122-3}}{{rp|at=Section 1.2}}
 [[Category:Additive categories]]