Definition
The formal definition is as follows: Let C be a category and let {Xj : j ∈ J} be an indexed family of objects in C. The coproduct of the set {Xj} is an object X together with a collection of morphisms ij : Xj → X (called canonical injections although they need not be injections or even monic) which satisfy a universal property: for any object Y and any collection of morphisms fj : Xj → Y, there exists a unique morphism f from X to Y such that fj = f ∘ ij. That is, the following diagram commutes (for each j):
The coproduct of the family {Xj} is often denoted
or
Sometimes the morphism f may be denoted
to indicate its dependence on the individual fj.
If the family of objects consists of only two members the coproduct is usually written X1 ∐ X2 or X1 ⊕ X2 or sometimes simply X1 + X2, and the diagram takes the form:
The unique arrow f making this diagram commute is then correspondingly denoted f1 ∐ f2 or f1 ⊕ f2 or f1 + f2 or .
Read more about this topic: Coproduct
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