Let be some object of , and let and be morphisms of . Then, by definition of coproduct, there exists a unique morphism such that the obvious diagrams commute.
This is the copairing of and .
When convenient, it is nice to write it vertically; all of the following are seen:
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.
One often sees a function defined by cases as follows:
Such a definition is valid in general iff the domain of is the (internal) disjoint union of its subsets and . In that case, let be the first subset, let be the second, and let be the target of ; let and be restrictions of and . Then we have as the domain of , and itself is the copairing .
Revised on November 15, 2009 19:24:48
by Toby Bartels