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 $𝖲 𝖾 𝗍$ respectively.

In a category $𝖢$ the product of objects ${𝑋 𝑖} 𝑖 \in 𝐼$ is an object $𝑋 = \prod 𝑖 \in 𝐼 𝑋 𝑖$ together with morphisms $𝜋 𝑖 \in 𝖢 (𝑋, 𝑋 𝑖)$ such that for any $𝑌$ and $𝑓 𝑖 \in 𝖢 (𝑌, 𝑋 𝑖)$ , there exists a unique $𝑓 \in 𝖢 (𝑌, 𝑋)$ so that $𝑓 𝑖 = 𝜋 𝑖 𝑓$ .¹ #m/def/cat

In a category $𝖢$ the coproduct of objexts ${𝑋 𝑖} 𝑖 \in 𝐼$ is an object $𝑋 = ∐ 𝑖 \in 𝐼 𝑋 𝑖$ together with morphisms $𝜄 𝑖 \in 𝖢 (𝑋 𝑖, 𝑋)$ such that for any $𝑌$ and $𝑓 𝑖 \in 𝖢 (𝑋 𝑖, 𝑌)$ , there exists a unique $𝑓 \in 𝖢 (𝑋, 𝑌)$ so that $𝑓 𝑖 = 𝑓 𝜄 𝑖$ .¹ #m/def/cat

These are categorical duals; the coproduct is just the product in $𝖢 𝐨 𝐩$ . 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 $𝒟 : 𝖩 \to 𝖢$ 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.

#state/tidy | #SemBr | #lang/en

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

Products and coproducts

Limits and colimits

Examples

Footnotes