Coproduct - Discussion

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 {X_j} will have a coproduct in general, but if it does, then the coproduct is unique in a strong sense: if i_j : X_j → X and k_j : X_j → Y are two coproducts of the family {X_j}, then (by the definition of coproducts) there exists a unique isomorphism f : X → Y such that fi_j = k_j for each j in J.

As with any universal property, the coproduct can be understood as a universal morphism. Let Δ: C → C×C be the diagonal functor which assigns to each object X the ordered pair (X,X) and to each morphism f:X → Y the pair (f,f). Then the coproduct X+Y in C is given by a universal morphism to the functor Δ from the object (X,Y) in C×C.

The coproduct indexed by the empty set (that is, an empty coproduct) is the same as an initial object in C.

If J is a set such that all coproducts for families indexed with J exist, then it is possible to choose the products in a compatible fashion so that the coproduct turns into a functor CJ → C. The coproduct of the family {X_j} is then often denoted by ∐_j X_j, and the maps i_j are known as the natural injections.

Letting Hom_C(U,V) denote the set of all morphisms from U to V in C (that is, a hom-set in C), we have a natural isomorphism

given by the bijection which maps every tuple of morphisms

(a product in Set, the category of sets, which is the Cartesian product, so it is a tuple of morphisms) to the morphism

That this map is a surjection follows from the commutativity of the diagram: any morphism f is the coproduct of the tuple

That it is an injection follows from the universal construction which stipulates the uniqueness of such maps. The naturality of the isomorphism is also a consequence of the diagram. Thus the contravariant hom-functor changes coproducts into products. Stated another way, the hom-functor, viewed as a functor from the opposite category Copp to Set is continuous; it preserves limits (a coproduct in C is a product in Copp).

If J is a finite set, say J = {1,...,n}, then the coproduct of objects X₁,...,X_n is often denoted by X₁⊕...⊕X_n. Suppose all finite coproducts exist in C, coproduct functors have been chosen as above, and 0 denotes the initial object of C corresponding to the empty coproduct. We then have natural isomorphisms

These properties are formally similar to those of a commutative monoid; a category with finite coproducts is an example of a symmetric monoidal category.

If the category has a zero object Z, then we have unique morphism X → Z (since Z is terminal) and thus a morphism X ⊕ Y → Z ⊕ Y. Since Z is also initial, we have a canonical isomorphism Z ⊕ Y ≅ Y as in the preceding paragraph. We thus have morphisms X ⊕ Y → X and X ⊕ Y → Y, by which we infer a canonical morphism X ⊕ Y → X×Y. This may be extended by induction to a canonical morphism from any finite coproduct to the corresponding product. This morphism need not in general be an isomorphism; in Grp it is a proper epimorphism while in Set_* (the category of pointed sets) it is a proper monomorphism. In any preadditive category, this morphism is an isomorphism and the corresponding object is known as the biproduct. A category with all finite biproducts is known as an additive category.

If all families of objects indexed by J have coproducts in C, then the coproduct comprises a functor CJ → C. Note that, like the product, this functor is covariant.