When Is a Container a Comonad?

When Is a Container a Comonad?

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by Danel Ahman , James Chapman , Tarmo Uustalu

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Datum	Value	Source
TITLE	When Is a Container a Comonad?	SVM HeaderParse 0.2
AUTHOR NAME	Danel Ahman	SVM HeaderParse 0.2
AUTHOR AFFIL	1; Computer Laboratory, University of Cambridge,	SVM HeaderParse 0.2
AUTHOR ADDR	15 J. J. Thomson Avenue, Cambridge CB3 0FD, United Kingdom	SVM HeaderParse 0.2
AUTHOR NAME	James Chapman	SVM HeaderParse 0.2
AUTHOR AFFIL	2; Institute of Cybernetics, Tallinn University of Technology,	SVM HeaderParse 0.2
AUTHOR ADDR	Akadeemia tee 21, 12618 Tallinn, Estonia	SVM HeaderParse 0.2
AUTHOR NAME	Tarmo Uustalu	SVM HeaderParse 0.2
AUTHOR AFFIL	2; Institute of Cybernetics, Tallinn University of Technology,	SVM HeaderParse 0.2
AUTHOR ADDR	Akadeemia tee 21, 12618 Tallinn, Estonia	SVM HeaderParse 0.2
ABSTRACT	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	SVM HeaderParse 0.2
CITATIONS	22 found	ParsCit 1.0