Optics are bidirectional data accessors that capture data transformation patterns such as accessing subfields or iterating over containers. Profunctor optics are a particular choice of representation supporting modularity, meaning that we can construct accessors for complex structures by combining simpler ones. Profunctor optics have previously been studied only in an unenriched and non-mixed setting, in which both directions of access are modelled in the same category. 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, with access directions modelled in different categories. 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. Finally, we discuss a Haskell implementation.

Bryce Clarke

Derek Elkins

Jeremy Gibbons

Fosco Loregian

Bartosz Milewski

Emily Pillmore

Mario Román

Compositionality

Profunctor Optics, a Categorical Update

Profunctor 透镜：一个范畴论的更新

Profunctor Optics, a Categorical Update

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