nLab
pairing

Idea

When $a$ and $b$ are elements of sets, the pairing of $a$ and $b$ is the ordered pair $(a,b)$.

It is natural to extend this to generalised elements in any category with binary products.

Definition

Let $X$ and $Y$ be objects of some category $C$, and suppose that the product $X\times Y$ exists in $C$.

Let $G$ be some object of $C$, and let $a:G\to X$ and $b:G\to Y$ be morphisms of $C$. Then, by definition of product, there exists a unique morphism $(a,b):G\to X\times Y$ such that the obvious diagrams commute.

If we think of $a$ and $b$ as $G$-indexed elements of $X$ and $Y$, then $(a,b)$ is a $G$-indexed element of $X\times Y$.

Examples

If $C$ is the category of sets and $G$ is the point, then $a$ and $b$ are simply elements, in the usual sense, of $X$ and $Y$; then $(a,b)$ is an element of $X\times Y$, the usual ordered pair $(a,b)$.

If $Y$ and $G$ are each $X$, with $a$ and $b$ each the identity morphism on $X$, then the pairing $({\mathrm{id}}_X,{\mathrm{id}}_X)$ is the diagonal morphism ${\Delta }_X:X\to X^2$.

Pairings versus products

Since taking the product is a functor (when defined), we can apply it to any two arbitrary morphisms. That is, if $a:G\to X$ and $b:H\to Y$ are morphisms in a category $C$, and if the products $G\times H$ and $X\times Y$ exist, then we have a morphism $a\times b:G\times H\to X\times Y$. This is not the pairing $(a,b)$, whose source is $G$.

A pairing is the composite of a product and a diagonal morphism:

$$G\stackrel{{\Delta }_G}{\to }G\times G\stackrel{a\times b}{\to }X\times Y;$$G \overset{\Delta_G}\to G \times G \overset{a \times b}\to X \times Y ;

conversely, a product is a pairing of two composites:

$$\begin{array}{c}G\times H\to G\stackrel{a}{\to }X,\\ G\times H\to H\stackrel{b}{\to }Y.\end{array}$$\array { G \times H \to G \overset{a}\to X ,\\ G \times H \to H \overset{b}\to Y .}

If $G$ and $H$ are each terminal, however, then $(a,b)$ and $a\times b$ are the same global element of $X\times Y$.