nLab
copairing

Idea

Copairing is dual to pairing.

Definition

Let XX and YY be objects of some category CC, and suppose that the coproduct XYX \sqcup Y exists in CC.

Let TT be some object of CC, and let a:XTa\colon X \to T and b:YTb\colon Y \to T be morphisms of CC. Then, by definition of coproduct, there exists a unique morphism [a,b]:XYT[a,b]\colon X \sqcup Y \to T such that the obvious diagrams commute.

This [a,b][a,b] is the copairing of aa and bb.

Notation

When convenient, it is nice to write it vertically; all of the following are seen:

(ab),[ab],{ab}. \left({a \atop b}\right) ,\quad \left[{a \atop b}\right] ,\quad \left\{{a \atop b}\right\} .

The vertical notation can be combined with pairing to create a matrix calculus for morphisms from a coproduct to a product. This works best when products and coproducts are the same, as described at matrix calculus.

Examples

One often sees a function defined by cases as follows:

f(x)={g(x) ifϕ(x) h(x) ifψ(x). f(x) = \left\{\array{ g(x) & \text {if}\; \phi(x) \\ h(x) & \text {if}\; \psi(x) .}\right.

Such a definition is valid in general iff the domain of ff is the (internal) disjoint union of its subsets {x:ϕ(x)}\{x : \phi(x)\} and {x:ψ(x)}\{x : \psi(x)\}. In that case, let XX be the first subset, let YY be the second, and let TT be the target of ff; let a:XTa\colon X \to T and b:YTb\colon Y \to T be restrictions of gg and hh. Then we have XYX \sqcup Y as the domain of ff, and ff itself is the copairing [a,b][a,b].

Revised on November 15, 2009 19:24:48 by Toby Bartels (173.60.119.197)