Editing Coproduct (section)

{{Short description|Category-theoretic construction}}
{{About|coproducts in categories|"coproduct" in the sense of comultiplication|Coalgebra|additional substances that result from the manufacturing of another product|By-product}}

In [[category theory]], the '''coproduct''', or '''categorical sum''', is a construction which includes as examples the [[disjoint union]] of [[Set (mathematics)|sets]] and [[disjoint union (topology)|of topological spaces]], the [[free product]] of [[Group (mathematics)|groups]], and the [[direct sum]] of [[Module (mathematics)|modules]] and [[vector space]]s.  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 (category theory)|dual notion]] to the [[product (category theory)|categorical product]],  which means the definition is the same as the product but with all [[morphism|arrows]] reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products within a given category.