Coproduct

In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.

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Other articles related to "coproducts, coproduct":

Category Of Rings - Properties - Limits and Colimits
... creates (and preserves) limits and filtered colimits, but does not preserve either coproducts or coequalizers ... The coproduct of a family of rings exists and is given by a construction analogous to the free product of groups ... It's quite possible for the coproduct of nontrivial rings to be trivial ...
Exterior Algebra - Duality - Bialgebra Structure
... forms defined above is dual to a coproduct defined on Λ(V), giving the structure of a coalgebra ... The coproduct is a linear function Δ Λ(V) → Λ(V) ⊗ Λ(V) given on decomposable elements by For example, This extends by linearity to an operation defined on the whole exterior algebra ... In terms of the coproduct, the exterior product on the dual space is just the graded dual of the coproduct where the tensor product on the right-hand ...
Tensor Product Of Algebras
... of A and B to A ⊗R B given by These maps make the tensor product a coproduct in the category of commutative R-algebras ... (The tensor product is not the coproduct in the category of all R-algebras ... There the coproduct is given by a more general free product of algebras) ...
Braided Hopf Algebra - Radford's Biproduct
... The algebra structure of is given by where, (Sweedler notation) is the coproduct of, and is the left action of H on R ... Further, the coproduct of is determined by the formula Here denotes the coproduct of r in R, and is the left coaction of H on ...
Coproduct - Discussion
... The coproduct construction given above is actually a special case of a colimit in category theory ... The coproduct in a category C can be defined as the colimit of any functor from a discrete category J into C ... Not every family {Xj} will have a coproduct in general, but if it does, then the coproduct is unique in a strong sense if ij Xj → X and kj Xj → Y are two ...