Profunctor Optics, a Categorical Update

Abstract

Profunctor optics are bidirectional data accessors that capture data transformation patterns such as accessing subfields or iterating over containers. They are modular, meaning that we can construct accessors for complex structures by combining simpler ones. Profunctor optics have been studied only using Sets as the enriching category and in the nonmixed case. However, functional programming languages are arguably better described by enriched categories and we have found that some structures in the literature are actually mixed optics. Our work generalizes a classic result by Pastro and Street on Tambara theory and uses it to describe mixed V-enriched profunctor optics and to endow them with V-category structure. We provide some original families of optics and derivations, including an elementary one for traversals that solves an open problem posed by Milewski. Finally, we discuss a Haskell implementation.

Compositional Game Theory, Compositionally

“…Lenses only define a monoidal category when the underlying category is cartesian. In a monoidal category we work with optics [22,5,23,3] instead: Proposition 30. Let C be symmetric monoidal, and A : Prof(C ) a commutative arrow.…”

Escrows are optics

“…commutes, if (⋆⋆) is induced by functoriality of P by the unitor of M. ▶ Remark 6. Tambara modules are the objects of a category Tamb where the hom-set between a Tambara module (P, α) and a Tambara module (Q, β) is the pullback of certain two equalisers (we will not need an explicit description of this hom-set; the reader is invited to consult [4] for the definition). All this will turn out useful in the proof of our Proposition 7.…”

Compositional modelling of network games

“…For the formal definition of an open game, it is convenient to use the notion of lens [18]. (For other references to how lenses are used in functional programming see [19] or [20,8]. )…”