[[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 $\cat C$ the product of objexts $\{ X_{i} \}_{i \in I}$ is an object $X = \prod_{i \in I} X_{i}$ together with morphisms $\pi_{i} \in \cat C(X,X_{i})$ such that for any $Y$ and $f_{i} \in \cat C(Y,X_{i})$,
there exists a unique $f \in \cat C(Y,X)$ so that $f_{i} = \pi_{i}f$.[^loo] #m/def/cat

In a category $\cat C$ the coproduct of objexts $\{ X_{i} \}_{i \in I}$ is an object $X = \coprod_{i \in I} X_{i}$ together with morphisms $\iota_{i} \in \cat C(X_{i}, X)$ such that for any $Y$ and $f_{i} \in \cat C(X_{i}, Y)$,
there exists a unique $f \in \cat C(X,Y)$ so that $f_{i} = f\iota_{i}$.[^loo] #m/def/cat

![[(co)product.svg#invert|c|https://q.uiver.app/#q=WzAsOCxbMCwyLCJYXzEiXSxbNCwyLCJYXzIiXSxbMiwyLCJYXzEgXFx0aW1lcyBYXzIiXSxbMiwwLCJZIl0sWzYsMiwiWF8xIl0sWzEwLDIsIlhfMiJdLFs4LDIsIlhfMSBcXGFtYWxnIFhfMiJdLFs4LDAsIlkiXSxbMiwwLCJcXHBpXzEiXSxbMiwxLCJcXHBpXzIiLDJdLFszLDAsImYiLDJdLFszLDEsImciXSxbMywyLCIoZixnKSIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRvdHRlZCJ9fX1dLFs0LDYsIlxcaW90YV8xIiwyXSxbNSw2LCJcXGlvdGFfMiJdLFs0LDcsImZfMSJdLFs1LDcsImZfMiIsMl0sWzYsNywiZiIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==]]

These are categorical duals; the coproduct is just the product in $\op{\cat C}$.
Each construction, if it exists, is unique up to unique isomorphism.

[^loo]: 2010, [[@looseAlgebraischeTopologie2010|Algebraische Topologie]], Definition 2.2.20, p. 61

> [!check]- Uniqueness up to unique isomorphism
> Uniqueness of the product up to isomorphism is shown by the following commutative diagram:
> <p align="center"><img align="center" 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/></p>
> By flipping the lateral arrows one receives the equivalent argument for the coproduct.
> <span class="QED"/>

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 $\mathscr{D} : \cat J \to \cat C$ be a discrete-shaped diagram (i.e. a diagram in the shape of a [[Discrete category]]) containing a family of objects.
The [[Limits and colimits|limit]] of this diagram is the product, the [[Limits and colimits|colimit]] is the coproduct.

## Examples

- In [[Category of sets|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 meet|join]] and the coproduct corresponds to the [[Join and meet|meet]].
- In [[Category of groups|Grp]] the product is the [[Direct product of groups]] and the coproduct is the [[Free product of groups]].

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