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Concept
Product (category theory)
Summary
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.
Fix a category Let and be objects of A product of and is an object typically denoted equipped with a pair of morphisms satisfying the following universal property:
For every object and every pair of morphisms there exists a unique morphism such that the following diagram commutes:
Whether a product exists may depend on or on and If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of the product. This has the following meaning: let be another cartesian product, there exists a unique isomorphism such that and .
The morphisms and are called the canonical projections or projection morphisms; the letter (pronounced pi) alliterates with projection. Given and the unique morphism is called the product of morphisms and and is denoted
Instead of two objects, we can start with an arbitrary family of objects indexed by a set
Given a family of objects, a product of the family is an object equipped with morphisms satisfying the following universal property:
For every object and every -indexed family of morphisms there exists a unique morphism such that the following diagrams commute for all
The product is denoted If then it is denoted and the product of morphisms is denoted
Alternatively, the product may be defined through equations. So, for example, for the binary product:
Existence of is guaranteed by existence of the operation
Commutativity of the diagrams above is guaranteed by the equality: for all and all
Uniqueness of is guaranteed by the equality: for all
The product is a special case of a . This may be seen by using a (a family of objects without any morphisms, other than their identity morphisms) as the required for the definition of the limit.
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Ontological neighbourhood
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Related concepts (24)
Coproduct
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.
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.
Complete category
In mathematics, a complete category is a in which all small s exist. That is, a category C is complete if every F : J → C (where J is ) has a limit in C. , a cocomplete category is one in which all small colimits exist. A bicomplete category is a category which is both complete and cocomplete. The existence of all limits (even when J is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a : for any two objects there can be at most one morphism from one object to the other.