The free product is actually the coproduct in the category of groups.

The coproduct construction given above is actually a special case of a colimit in category theory.

There the coproduct is given by a more general free product of algebras).

Dually, the coproduct of an empty family is an initial object.

When used in the construction of the limit, the result is the product; for the colimit, one gets the coproduct.

It's quite possible for the coproduct of nontrivial rings to be trivial.

This serves to define the coproduct in , when it exists.

The coproduct is given by the disjoint union of topological spaces.

The wedge sum can be understood as the coproduct in the category of pointed spaces.

Instead, the coproduct is the tensor product of rings.

Remarks opposite to the above apply: the pushout is a coproduct with additional structure.

Any finite product in a preadditive category must also be a coproduct, and conversely.

The tensor algebra is not a bialgebra with this coproduct.

However, the following more complicated coproduct does yield a bialgebra:

Note that in the category of commutative rings, the direct sum is not the coproduct.

When the sets are pairwise disjoint, the usual union is another realization of the coproduct.

In this case, the direct sum is indeed the coproduct in the category of abelian groups.

There is no coproduct in Met.

For crystal bases, the coproduct , given by , is adopted.

In this case, we obtain the coproduct in the appropriate category of all objects carrying the additional structure.

Here, the product is the direct product, but the coproduct is the free product.

An initial object in a category (an empty coproduct, and so an identity under coproducts)

The coproduct of a family of rings exists and is given by a construction analogous to the free product of groups.

Some metal centers are oxidized by thiols, the coproduct being hydrogen gas:

The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.