Coproduct (category theory)

In category theory, the coproduct, or categorical sum, is a category-theoretic construction which includes as examples 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.

Let C be a category and let X₁ and X₂ be objects in that category. An object is called the coproduct of these two objects, written X₁ ∐ X₂ or X₁ ⊕ X₂ or sometimes simply X₁ + X₂, if there exist morphisms i₁ : X₁ → X₁ ∐ X₂ and i₂ : X₂ → X₁ ∐ X₂ satisfying a universal property: for any object Y and morphisms f₁ : X₁ → Y and f₂ : X₂ → Y, there exists a unique morphism f : X₁ ∐ X₂ → Y such that f₁ = f ∘ i₁ and f₂ = f ∘ i₂. That is, the following diagram commutes:

The unique arrow f making this diagram commute may be denoted f₁ ∐ f₂ or f₁ ⊕ f₂ or f₁ + f₂ or [f₁, f₂]. The morphisms i₁ and i₂ are called canonical injections, although they need not be injections nor even monic.

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