Coproduct
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In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. Basically, this means the definition is the same as the product but with all arrows reversed. Despite this innocuous-looking change in the name and notation, coproducts can be dramatically different from products.
The formal definition is as follows: Let C be a category and let {Xj | j ∈ J} be a 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 injections) 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 O ij. That is, the follow diagram commutes (for each j):
Coproduct-01.png
Image:Coproduct-01.png
The coproduct of the family {Xj} is often denoted
- <math> X = \coprod_{j\in J}X_j<math>
Coproducts are actually special cases of colimits in category theory. The coproduct can be defined as the colimit of a discrete subcategory in C. It follows that if coproducts exists in a given category (they need not) they are unique up to a unique isomorphism that respect the injections.
The coproduct indexed by the empty set (that is, an empty coproduct) is the same as an initial object in C.
The coproduct in the category of sets is simply the disjoint union with the maps ij being the inclusions. Unlike direct products, 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 finitely many nonzero terms (this therefore coincides exactly with the direct product, in the case of finitely many factors—so much for "dramatically different"). As a consequence, since most introductory linear algebra courses deal with only finite-dimensional vector spaces, nobody really hears much about direct sums until later on.
In the case of topological spaces coproducts are disjoint unions on the underlying sets, and the open sets are sets open in each of the spaces, in a rather evident sense (see disjoint union (topology)). In the category of pointed spaces, 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).
Despite all this dissimilarity, there is still, at the heart of the whole thing, a disjoint union: 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 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.