Editing Biproduct (section)

==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]].