Pairings

Idea

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

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

For products of higher arity, one can say tripling, quadrupling, etc, or just tupling.

Definition

Let $X$ and $Y$ be objects of some category $C$, and suppose that the product $X×Y$ 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 $\left(a,b\right):G\to X×Y$ such that the obvious diagrams commute.

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

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 $\left(a,b\right)$ is an element of $X×Y$, the usual ordered pair $\left(a,b\right)$.

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

Pairings versus products

Since taking products (when these always exist) is a functor, we can apply this operation to any two morphisms. That is, if $a:G\to X$ and $b:H\to Y$ are morphisms in a category $C$, and if the products $G×H$ and $X×Y$ exist, then we have a morphism $a×b:G×H\to X×Y$. This is not the pairing $\left(a,b\right)$, for which the source is always $G$.

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

$G\stackrel{{\Delta }_{G}}{\to }G×G\stackrel{a×b}{\to }X×Y;$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:

$\begin{array}{c}G×H\to G\stackrel{a}{\to }X,\\ G×H\to H\stackrel{b}{\to }Y.\end{array}$\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 $\left(a,b\right)$ and $a×b$ are the same global element of $X×Y$. Thus, both product morphisms and pairings are generalisations of ordered pairs in Set.

Revised on November 1, 2011 07:19:32 by Toby Bartels (71.31.218.235)