Documentation

Projection describes a lossless conversion that extracts (projects out) one type from another.

The sole method of this class,

project :: whole -> part

takes a value input :: whole and extracts a value output :: part which was included entirely in the input. project must cover part, that is, every value in part must the output of project for some value in whole. In mathematical terminology, project is surjective:

forall y. exists x. project x = y

We might also say that it projects input :: whole onto its equivalent class; the typeclass gets its name from this terminology.

Projection models the "has-a" relationship used in languages with subtypes (such as in object-oriented programming), but an explicit cast with project is required in Haskell.

Although it is often possible to infer the type parameters of this class, it is advisable to specify one or both of the parameters to project using a type signature or the TypeApplications language extension. Specifying the type parameters will give clearer error messages from the type checker in any case.

Section undoes a Projection.

The method

update :: whole -> part -> whole

is the (right) inverse of project:

project (update x y) = y

In mathematics, section refers generically to the right inverse of a morphism. Keeping with Haskell tradition, we are borrowing this generic term and using it in a very specific way: Section is the right inverse of Projection in particular.

Although it is often possible to infer the type parameters of this class, it is advisable to specify one or both of the parameters to retract using a type signature or the TypeApplications language extension. Specifying the type parameters will give clearer error messages from the type checker in any case.