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Concept
Coproduct
Summary
In , the coproduct, or categorical sum, is a construction which includes as examples 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 "least specific" object to which each object in the family admits a morphism. It is the category-theoretic to the , which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.
Let be a and let and be objects of An object is called the coproduct of and written or or sometimes simply if there exist morphisms and satisfying the following universal property: for any object and any morphisms and there exists a unique morphism such that and That is, the following diagram commutes:
The unique arrow making this diagram commute may be denoted or The morphisms and are called , although they need not be injections or even monic.
The definition of a coproduct can be extended to an arbitrary family of objects indexed by a set The coproduct of the family is an object together with a collection of morphisms such that, for any object and any collection of morphisms there exists a unique morphism such that That is, the following diagram commutes for each :
The coproduct of the family is often denoted or
Sometimes the morphism may be denoted to indicate its dependence on the individual s.
The coproduct in the 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 , called the free product, is quite complicated.
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Related concepts (32)
Product (category theory)
In , the product of two (or more) in a is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the whose s are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds, with compactly generated spaces as objects and continuous maps as morphisms or with the .
Initial and terminal objects
In , a branch of mathematics, an initial object of a C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists exactly one morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object.