Distributive Laws of Directed Containers

Containers [1] are an elegant representation of a wide class of datatypes in terms of shapes and positions in shapes. In our FoSSaCS 2012 work [2], we introduced directed containers as a special case to account for the common situation where every position in a shape determines another shape, informally the subshape rooted by that position; some examples being the datatypes of nonempty lists and trees and the corresponding zipper datatypes. While containers interpret into set functors via a fully faithful monoidal functor, directed containers interpret into comonads. Further, it is also true that every comonad whose underlying functor is a container is represented by a directed container. In this paper, we develop a characterization of distributive laws between such comonads. A container S C P is given by a set S (of shapes) and a shape-indexed family P : S → Set (of positions). A morphism between containers S C P and S′ C P ′ is a pair t C q of maps t : S → S′ and q : Π{s : S}. P ′ (t s) → P s. (We use Agda’s syntax of braces for implicit arguments.) Containers form a category Cont carrying a monoidal structure defined by Id = 1 C λ ∗ . 1 and (S0 C P0) · (S1 C P1) = Σs : S0. P0 s → S1 C λ (s, v). Σp0 : P0 s. P1 (v p0) together suitable unital and associativity laws. The interpretation of a container S C P is the set functor given by JSCP KcX = Σs : S. P s→ X, JSCP Kc f (s, v) = (s, f ◦v). The interpretation of a container map tCq is the natural transformation JtCqKc(s, v) = (t s, v ◦q {s}). J−Kc is a fully faithful monoidal functor from Cont to [Set,Set]. A directed container is a container S C P together with three operations