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 𝖢 the product of objects {𝑋𝑖}𝑖∈𝐼 is an object 𝑋 =∏𝑖∈𝐼𝑋𝑖 together with morphisms 𝜋𝑖 ∈𝖢(𝑋,𝑋𝑖) such that for any 𝑌 and 𝑓𝑖 ∈𝖢(𝑌,𝑋𝑖), there exists a unique 𝑓 ∈𝖢(𝑌,𝑋) so that 𝑓𝑖 =𝜋𝑖𝑓.¹ cat

In a category 𝖢 the coproduct of objexts {𝑋𝑖}𝑖∈𝐼 is an object 𝑋 =∐𝑖∈𝐼𝑋𝑖 together with morphisms 𝜄𝑖 ∈𝖢(𝑋𝑖,𝑋) such that for any 𝑌 and 𝑓𝑖 ∈𝖢(𝑋𝑖,𝑌), there exists a unique 𝑓 ∈𝖢(𝑋,𝑌) so that 𝑓𝑖 =𝑓𝜄𝑖.¹ cat

These are categorical duals; the coproduct is just the product in 𝖢𝐨𝐩. 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 →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.

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Footnotes

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