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The coproduct in the category of sets is simply the disjoint union with the maps i j 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 ...
In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, in the ...
In category theory the disjoint union is defined as a coproduct in the category of sets. As such, the disjoint union is defined up to an isomorphism, and the above definition is just one realization of the coproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct.
For any objects ,, of a category with finite products and coproducts, there is a canonical morphism + (+), where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct X × Y + X × Z {\displaystyle X\times Y+X\times Z} guarantees the existence of unique arrows filling out the following diagram ...
The coproduct in Ab is given by the direct sum; since Ab has cokernels, it follows that Ab is also cocomplete. We have a forgetful functor Ab → Set which assigns to each abelian group the underlying set, and to each group homomorphism the underlying function. This functor is faithful, and therefore Ab is a concrete category.
The free product is the coproduct in the category of groups. That is, the free product plays the same role in group theory that disjoint union plays in set theory, or that the direct sum plays in module theory. Even if the groups are commutative, their free product is not, unless one of the two groups is the trivial group.
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In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct.