Editing Coproduct (section)

== Examples ==
The coproduct in the [[category of sets]] is simply the '''[[disjoint union#Set theory definition|disjoint union]]''' with the maps ''i<sub>j</sub>'' being the [[inclusion map]]s. Unlike [[direct product]]s, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be dramatically different from each other. For example, the coproduct in the [[category of groups]], called the '''[[free product]]''', is quite complicated. On the other hand, in the [[category of abelian groups]] (and equally for [[vector spaces]]), the coproduct, called the '''[[direct sum]]''', consists of the elements of the direct product which have only [[finite set|finitely]] many nonzero terms. (It therefore coincides exactly with the direct product in the case of finitely many factors.)

Given a [[commutative ring]] ''R'', the coproduct in the [[category of commutative algebras|category of commutative ''R''-algebras]] is the [[tensor product of algebras|tensor product]]. In the [[category of rings#R-algebras|category of (noncommutative) ''R''-algebras]], the coproduct is a quotient of the tensor algebra (see ''[[Free product of associative algebras]]'').

In the case of [[topological space]]s, coproducts are disjoint unions with their [[disjoint union (topology)|disjoint union topologies]].  That is, it is a disjoint union of the underlying sets, and the [[open set]]s are sets ''open in each of the spaces'', in a rather evident sense.  In the category of [[pointed space]]s, fundamental in [[homotopy theory]], the coproduct is the [[wedge sum]] (which amounts to joining a collection of spaces with base points at a common base point).

The concept of disjoint union secretly underlies the above examples: the direct sum of abelian groups is the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero), similarly for vector spaces: the space [[linear span|spanned]] by the "almost" disjoint union;  the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute.  This pattern holds for any [[variety (universal algebra)|variety in the sense of universal algebra]].

The coproduct in the category of [[Banach spaces]] with [[short map]]s is the [[Lp space|{{math|''l''<sup>1</sup>}}]] sum, which cannot be so easily conceptualized as an "almost disjoint" sum, but does have a [[unit ball]] almost-disjointly generated by the unit ball is the cofactors.<ref name=Ban1Cat>{{cite web|website=Annoying Precision|title=Banach spaces (and Lawvere metrics, and closed categories)|date=June 23, 2012|author=Qiaochu Yuan|url=https://qchu.wordpress.com/2012/06/23/banach-spaces-and-lawvere-metrics-and-closed-categories/}}</ref>

The coproduct of a [[poset category]] is the [[Join (mathematics)|join operation]].