In category theory and its applications to mathematics, a biproduct is a generalisation of the notion of direct sum that makes sense in any preadditive category.


Let C be a preadditive 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 the zero morphism from "A""l" to "A""k" whenever "k" and "l" are distinct.

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.


Biproducts always exist in the category of abelian groups.In that category, the biproduct of several objects is simply their direct sum.The nullary biproduct is the trivial group.Biproducts exist in several other categories with direct sums, such as the category of vector spaces over a given field.But biproducts do not exist in the category of all groups; indeed, this category is not even preadditive.


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 coproducts in the ordinary category-theoretic sense; this is the origin of the term "biproduct".In particular, a nullary biproduct is always a zero 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 in abelian categories.


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Preadditive category — In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups. In other words, the category C is preadditive if every hom set Hom(A,B) in C has the structure of …   Wikipedia

  • Additive category — In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A 1,..., A n of C have a biproduct A 1 ⊕ ⋯ ⊕ A n in C. (Recall that a category C is preadditive if all its… …   Wikipedia

  • Braided Hopf algebra — In mathematics a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter Drinfel d category of a Hopf algebra H . Definition Let H be a Hopf algebra over a field k , and …   Wikipedia

  • Direct sum of modules — For the broader use of the term in mathematics, see Direct sum. 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… …   Wikipedia

  • Pre-Abelian category — In mathematics, specifically in category theory, a pre Abelian category is an additive category that has all kernels and cokernels.Spelled out in more detail, this means that a category C is pre Abelian if: # C is preadditive, that is enriched… …   Wikipedia

  • Zinc nitride — Chembox new Name = Zinc nitride OtherNames = Section1 = Chembox Identifiers CASNo = 1313 49 1 Section2 = Chembox Properties Formula = Zn3N2 MolarMass = 244.15 g/mol Appearance = gray powder Density = 6.22 g/cm³, solid Solubility = insoluble… …   Wikipedia

  • Biwater — Plc is a British water company. It designs and builds Water Treatment Works and Waste Water Treatment Works, mainly in the UK but also around the world.Biwater Plc is split into three arms:* Biwater International Biwater International is… …   Wikipedia

  • Nichols algebra — The Nichols algebra of a braided vector space (with the braiding often induced by a finite group) is a braided Hopf algebra which is denoted by and named after the mathematician Warren Nichols. It takes the role of quantum Borel part of a pointed …   Wikipedia

  • Abelian category — In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of… …   Wikipedia

  • Coproduct — This article is about coproducts in categories. For coproduct in the sense of comultiplication, see Coalgebra. In category theory, the coproduct, or categorical sum, is the category theoretic construction which includes the disjoint union of sets …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”