Editing Biproduct

{{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 [[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]] 

*<math display="inline">p_k \!: A_1 \oplus \dots \oplus A_n \to A_k</math> in '''C''' (the ''[[Projection (mathematics)|projection]] morphisms'')
*<math display="inline">i_k \!: A_k \to A_1 \oplus \dots \oplus A_n</math> (the ''[[embedding]] morphisms'')
satisfying
*<math display="inline">p_k \circ i_k = 1_{A_k}</math>, the identity morphism of <math>A_k,</math> and
*<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
*<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
*<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>

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 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 (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 (mathematics)|ring]].

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]]. 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 <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>
*<math>p_l \circ i_k = 0 </math> for <math display="inline">k \neq l.</math>

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>.

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>

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]]
[[Category:Limits (category theory)]]