# Extensive Definition

In category
theory, the coproduct, or categorical sum, is the
category-theoretic construction which subsumes 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 "most general" 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 innocuous-looking change
in the name and notation, coproducts can be and typically are
dramatically different from products.

## Definition

The formal definition is as follows: Let C be a category and let be an indexed family of objects in C. The coproduct of the set is an object X together with a collection of morphisms ij : Xj → X (called canonical injections although they need not be injections or even monic) 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 following diagram commutes (for each j):The coproduct of the family is often denoted

- X = \coprod_X_j

- X = \bigoplus_ X_j.

Sometimes the morphism f may be denoted

- f=\coprod_ f_j: \coprod_ X_j \to Y

If the family of objects consists of only two
members the product is usually written X1 ∐ X2 or X1 ⊕ X2 or
sometimes simply X1 + X2, and the diagram takes the form:

The unique arrow f making this diagram commute is
then correspondingly denoted f1 ∐ f2 or f1 ⊕ f2 or f1 + f2 or [f1,
f2].

## Examples

The coproduct in the category of sets is simply the disjoint union with the maps ij being the inclusion maps. 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). 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 with their disjoint
union topologies. That is it is a disjoint union of the
underlying sets, and the open sets are
sets open in each of the spaces, in a rather evident sense. 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.

## 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 will have a coproduct in
general, but if it does, then the coproduct is unique in a strong
sense: if ij : Xj → X and kj : Xj → Y are two coproducts of the
family , then (by the definition of coproducts) there exists a
unique isomorphism f
: X → Y such that ij = kj f 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 is then often denoted by ∐j Xj, and the
maps ij are known as the natural injections.

Letting HomC(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

- \operatorname_C\left(\coprod_X_j,Y\right) \cong \prod_\operatorname_C(X_j,Y)

- (f_j)_ \in \prod_\operatorname(X_j,Y)

- \coprod_ f_j \in \operatorname\left(\coprod_X_j,Y\right).

- (f\circ i_j)_.

If J is a finite set,
say J = , then the coproduct of objects X1,...,Xn is often denoted
by X1⊕...⊕Xn. 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

- X\oplus (Y \oplus Z)\cong (X\oplus Y)\oplus Z\cong X\oplus Y\oplus Z
- X\oplus 0 \cong 0\oplus X \simeq X
- X\oplus Y \cong Y\oplus X

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.

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
respects the injections.

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.

coproducts in German:
Koprodukt