nLab
pairing

Pairings

Idea

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

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 XX and YY be objects of some category CC, and suppose that the? product X×YX \times Y exists in CC.

Let GG be some object of CC, and let a:GXa\colon G \to X and b:GYb\colon G \to Y be morphisms of CC. Then, by definition of product, there exists a unique morphism (a,b):GX×Y(a,b)\colon G \to X \times Y such that the obvious diagrams commute.

If we think of aa and bb as GG-indexed elements of XX and YY, then (a,b)(a,b) is a GG-indexed element of X×YX \times Y.

Examples

If CC is the category of sets and GG is the point, then aa and bb are simply elements, in the usual sense, of XX and YY; then (a,b)(a,b) is an element of X×YX \times Y, the usual ordered pair (a,b)(a,b).

If YY and GG are each XX, with aa and bb each the identity morphism on XX, then the pairing (id X,id X)(id_X,id_X) is the diagonal morphism Δ X:XX 2\Delta_X\colon 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:GXa\colon G \to X and b:HYb\colon H \to Y are morphisms in a category CC, and if the products G×HG \times H and X×YX \times Y exist, then we have a morphism a×b:G×HX×Ya \times b\colon G \times H \to X \times Y. This is not the pairing (a,b)(a,b), for which the source is always GG.

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

GΔ GG×Ga×bX×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:

G×HGaX, G×HHbY. \array { G \times H \to G \overset{a}\to X ,\\ G \times H \to H \overset{b}\to Y .}

If GG and HH are each terminal, however, then (a,b)(a,b) and a×ba \times b are the same global element of X×YX \times 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)