- Biproduct
In

category theory and its applications tomathematics , a**biproduct**is a generalisation of the notion ofdirect sum that makes sense in anypreadditive category .**Definition**Let

**C**be apreadditive category .In particular, morphisms in**C**can be added.Given objects "A"

_{1},...,"A"_{"n"}in**C**, suppose that we have:

*an object "A"_{1}⊕ ··· ⊕ "A"_{"n"}in**C**(the "biproduct");

*morphisms "p"_{"k"}: "A"_{1}⊕ ··· ⊕ "A"_{"n"}→ "A"_{"k"}in**C**(the "projection morphisms"); and

* morphisms "i"_{"k"}: "A"_{"k"}→ "A"_{1}⊕ ··· ⊕ "A"_{"n"}(the "injection morphisms").Additionally, suppose that:

*("i"_{1}_{°}"p"_{1}) + ··· + ("i"_{"n"}_{°}"p"_{"n"}) equals the identity morphism of "A"_{1}⊕ ··· ⊕ "A"_{"n"};

*"p"_{"k"}_{°}"i"_{"k"}equals the identity morphism of "A"_{"k"}; and

*"p"_{"k"}_{°}"i"_{"l"}is thezero morphism from "A"_{"l"}to "A"_{"k"}whenever "k" and "l" aredistinct .Then "A"

_{1}⊕ ··· ⊕ "A"_{"n"}is a biproduct of "A"_{1},...,"A"_{"n"}.Note that if we take "n" = 0 in the above definition, then only the first condition applies, and we have for the "

nullary biproduct" an object "O" such that the identity morphism on "O" is equal to the zero morphism from "O" to itself.**Examples**Biproducts always exist in the category of

abelian group s.In that category, the biproduct of several objects is simply theirdirect sum .The nullary biproduct is thetrivial group .Biproducts exist in several other categories with direct sums, such as the category ofvector space s over a given field.But biproducts do not exist in the category of all groups; indeed, this category is not even preadditive.**Properties**If a nullary biproduct exists and all binary biproducts "A"

_{1}⊕ "A"_{2}exist, then all biproducts whatsoever must also exist.Biproducts in preadditive categories are always both products and

coproduct s in the ordinary category-theoretic sense; this is the origin of the term "biproduct".In particular, a nullary biproduct is always azero object .Conversely, any finitary product or coproduct in a preadditive category must be a biproduct.An "

additive category " is a preadditive category in which every biproduct exists.In particular, biproducts always exist inabelian categories .**References**

*Wikimedia Foundation.
2010.*