We show that cyclic structures, i.e., finite or possibly infinite structures with backpointers, unwindable into possibly infinite structures, can be elegantly represented as nested datatypes. This representation is free of the various deficiencies characterizing the more naive representation as mixed-variant datatypes. It is inspired by the representation of lambda-terms as a nested datatype via the de Bruijn notation. 1

We have previously demonstrated that dataflow computation is comonadic. Here we argue that attribute evaluation has a lot in common with dataflow computation and admits a similar analysis. We claim that this yields a new, modular way to organize both attribute evaluation programs written directly in a functional language as well as attribute grammar processors.

Abstract. Abbott, Altenkirch, Ghani and others have taught us that many parameterized datatypes (set functors) can be usefully analyzed via container representations in terms of a set of shapes and a set of positions in each shape. This paper builds on the observation that datatypes often carry additional structure that containers alone do not account for. We introduce directed containers to capture the common situation where every position in a datastructure determines another datastructure, informally, the sub-datastructure rooted by that position. Some natural examples are non-empty lists and node-labelled trees, and datastructures with a designated position (zippers). While containers denote set functors via a fully-faithful functor, directed containers interpret fully-faithfully into comonads. But more is true: every comonad whose underlying functor is a container is represented by a directed container. In fact, directed containers are the same as containers that are comonads. We also describe some constructions of directed containers. We have formalized our development in the dependently typed programming language Agda. 1