Coalgebraic Update Lenses

O’Connor [6] made the simple but very useful observation with deep consequences that the (very well-behaved) lenses à la Foster et al. [3] are nothing but coalgebras of the array comonads of Power and Shkaravska [7]. The put operation in these lenses is quite rigid in that a whole new view is merged into the source, there is no flexibility for speaking about small changes to the view. We advocate a generalization that is as simple as O’Connor’s, but offers also this flexibility. The idea is to introduce updates (or changes, deltas, edits) that can be composed and applied to views. The generalization derives from the work on directed containers of Ahman et al. [1]. A lens in our generalized sense—an update lens—is parameterized by a fixed set S (of views), a monoid (P, o,⊕) (of updates) and an action ↓ of the monoid on the set (describing the outcome of applying any given update on any given view). These data, sometimes collectively called an act, define a comonad (D, ε, δ) by DX = S × (P → X). We define an update lens to be a coalgebra of this comonad. This is the same as having a set X and maps lkp : X → S and upd : X × P → X satisfying the conditions