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Concept
Biproduct
Résumé
In and its applications to mathematics, a biproduct of a finite collection of , in a with zero objects, is both a and a coproduct. In a the notions of product and coproduct coincide for finite collections of objects. The biproduct is a generalization of finite direct sums of modules.
Let C be a with zero morphisms. Given a finite (possibly empty) collection of objects A1, ..., An in C, their biproduct is an in C together with morphisms
in C (the projection morphisms)
(the embedding morphisms)
satisfying
the identity morphism of and
the zero morphism for
and such that
is a for the and
is a coproduct for the
If C is preadditive and the first two conditions hold, then each of the last two conditions is equivalent to when n > 0. 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.
In the category of abelian groups, biproducts always exist and are given by the direct sum. The zero object is the trivial group.
Similarly, biproducts exist in the 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 over a ring.
On the other hand, biproducts do not exist in the . Here, the product is the direct product, but the coproduct is the free product.
Also, biproducts do not exist in the . 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.
If the biproduct 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 and a co-Cartesian monoidal category.
If the product and coproduct both exist for some pair of objects A1, A2 then there is a unique morphism such that
for
It follows that the biproduct exists if and only if f is an isomorphism.
Source officielle
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