vitrea

Pure profunctor lenses and optics via the profunctor representation theorem. You can read more about this representation in the following articles.

• “Profunctor optics: the categorical view”, Milewski (2017).
• “What you needa know about Yoneda”, Boisseau, Gibbons (2018).
• “Categories of optics”, Riley (2018).
• “Profunctor optics: a categorical update”, Clarke, Elkins, Gibbons, Loregian, Milewski, Pillmore, Román (2019).

All the optics on this library are particular cases of a single unified definition. All the combinators on this library are instantiations of the profunctor optic to a particular Tambara module.

The usual notions of lawfulness (as in “Categories of optics”) apply to non-mixed optics (lenses, prisms, traversals…). However, there is not a consensus on what the natural notion of laws for a mixed optic should be, apart from specific cases like monadic lenses.

Examples

The library follows, where possible, the style of combinators of the lens library. Example code from “Profunctor optics: a categorical update” works directly under the library.

Example 1. (Lenses and prisms) The composition of a prism (address) and a lens (street) is used to parse a string and then access and modify one of its subfields.

let home = "221b Baker St, London, UK"
address :: Prism Address String
street :: Lens String Address

  >>> home?. address.street
  Just "221b Baker St"
  >>> home & address.street. ̃ "4 Marylebone Rd"
  "4 Marylebone Rd, London, UK"

Example 2. (Monadic lenses) A polymorphic family of type-changing monadic lenses for a logging monad is used to track each time a data holder (Box) is accessed.

box :: (Show b) => MonadicLens IO a b (Box a) (Box b)

  >>> return (Box 42)
        >>= mupdate box "hello"
        >>= mupdate box "world"
  [box]: contents changed to "hello".
  [box]: contents changed to "world".
  Box{"world"}

Example 3. (Algebraic lenses) A classifying lens (measure) is used both for accessing a the measurements of a point in the iris dataset and to classify new measurements into a species (Versicolor).

let iris =
[ Iris Setosa 4.9 3.0 1.4, 0.2
, Iris Setosa 4.7 3.2 1.3 0.2
, ...
, Iris Virginica 5.9 3.0 5.1 1.8 ]
measure :: AlgLens [] Measurements Flower

  >>> iris!!4 ˆ. measure
  (5.0, 3.6, 1.4, 0.2)
  >>> iris & measure .? Measurements (4.8, 3.2, 3.5, 2.1)
  Iris Versicolor (4.8, 3.1, 1.5, 0.1)

Example 4. (Traversals) The composition of a traversal (each) with a prism (address) and a lens (city) is used to parse a collection of strings and modify one of their subfields.

let mail =
[ "43 Adlington Rd, Wilmslow, United Kingdom"
, "26 Westcott Rd, Princeton, USA"
, "St James's Square, London, United Kingdom"
]
each :: Traversal String [String]
address :: Prism Address String
city :: Lens String Address

  >>> mail & each.address.city % ̃ uppercase
  [ "43 Adlington Rd, WILMSLOW, United Kingdom"
  , "26 Westcott Rd, PRINCETON, USA"
  , "St James's Square, LONDON, United Kingdom"
  ]

Example 5. (Kaleidoscopes) Following the previous Example 3, a kaleidoscope (aggregate) is composed with an algebraic lens to create a new point on the dataset by aggregating measurements with some function (mean, maximum) and then classifying it.

measure :: AlgebraicLens [] Measurements Flower
aggregate :: Kaleidoscope Float Measurements

  >>> iris & measure.aggregate >- mean
  Iris Versicolor; Sepal (5.843, 3.054); Petal (3.758, 1.198)

About

This is a complete rewriting from the first prototype of vitrea, maintained by @emilypi and @mroman42.