Universal construction

Products and coproducts

Products and coproducts are tuples of objects and morphisms within a category which, if they exist, are unique up to isomorphism The categorical product and coproduct generalise the cartesian product and Disjoint union in Category of sets respectively.

In a category the product of objects is an object together with morphisms such that for any and , there exists a unique so that .¹ cat

In a category the coproduct of objexts is an object together with morphisms such that for any and , there exists a unique so that .¹ cat

These are categorical duals; the coproduct is just the product in . Each construction, if it exists, is unique up to unique isomorphism.

Uniqueness up to unique isomorphism

Uniqueness of the product up to isomorphism is shown by the following commutative diagram:

By flipping the lateral arrows one receives the equivalent argument for the coproduct.

The product and coproduct may be generalized to the Fibre product and coproduct. A category with finitary products is a special kind of monoidal category called a Cartesian category, whereas one with finitary coproducts is a Cocartesian category.

Limits and colimits

Let be a discrete-shaped diagram (i.e. a diagram in the shape of a Discrete category) containing a family of objects. The limit of this diagram is the product, the colimit is the coproduct.

Examples

• In Set the product is the Cartesian product and the coproduct is the Disjoint union
• In a Posetal category viewed as a poset, the product corresponds to the join and the coproduct corresponds to the meet.
• In Grp the product is the Direct product of groups and the coproduct is the Free product of groups.

---

tidy | sembr | en

Footnotes

1. 2010, Algebraische Topologie, Definition 2.2.20, p. 61 ↩ ↩²