Understanding profunctor optics: a representation theorem (extended abstract)

Abstract

Optics, aka functional references, are classes of tools that allow composable access into compound data structures. Usually defined as programming language libraries, they provide combinators to manipulate different shapes of data such as sums, products and collections, that can be composed to operate on larger structures. Together they form a powerful language to describe transformations of data.Among the different approaches to describing optics, one particular type of optics, called profunctor optics, stands out. It describes alternative but equivalent representations of most of the common combinators, and enhances them with elegant composability properties via a higher-order encoding. Notably, it enables easy composition across different optic families.

Profunctor Optics, a Categorical Update

“…1 Double, as used by Pastro and Street [30], should not be confused with double in the sense of double category. 2 A promonoidal category A is a generalization of a monoidal category with functors replaced by profunctors. For instance, a tensor product is a profunctor P : A op ⊗ A ⊗ A → V and the unit is J : A op → V…”

Escrows are optics

“…Optics originated in functional programming as a compositional solution to the problem of accessing fields of nested data structures; currently, optics find application in diverse branches of applied category theory like categorical probability theory [19,20] the theory of open games [5,6] (see also the connection with lenses, particular optics: [8,7]; see [21] for a categorical overview and generalisation) functional programming [15,14,2,3]; see [1] for a string-diagrammatic calculus (which is not the one we employ here; instead, we use [18]).…”

“…So, both Asmodeus and Belial have no economic incentive in not upholding their side of the trade and may do so only if motivated by irrational motives such as pure trolling. 2 The basic structure of an escrow is depicted in the figure above. This paper aims to describe escrows formally, with a particular focus on their compositional nature.…”

Profunctor optics and traversals

“…They provide a powerful abstraction of many common data transformations. This abstraction is compositional thanks to a representation in terms of profunctors endowed with an algebraic structure called Tambara module [Mil17].There exists a general definition of optic [Boi17,Ril18] in terms of coends that, after some elementary application of the Yoneda lemma, particularizes in each one of the basic optics. Traversals used to be the exception; we show an elementary derivation of traversals and discuss some other new derivations for optics.…”

“…There exists a general definition of optic [Boi17,Ril18] in terms of coends that, after some elementary application of the Yoneda lemma, particularizes in each one of the basic optics. Traversals used to be the exception; we show an elementary derivation of traversals and discuss some other new derivations for optics.…”