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When Is a Container a Comonad?
"... 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 addi ..."
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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 subdatastructure rooted by that position. Some natural examples are nonempty lists and nodelabelled trees, and datastructures with a designated position (zippers). While containers denote set functors via a fullyfaithful functor, directed containers interpret fullyfaithfully 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
A Categorical Treatment of Ornaments
"... Abstract—Ornaments aim at taming the multiplication of specialpurpose datatypes in dependently typed programming languages. In type theory, purpose is logic. By presenting datatypes as the combination of a structure and a logic, ornaments relate these specialpurpose datatypes through their common ..."
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Abstract—Ornaments aim at taming the multiplication of specialpurpose datatypes in dependently typed programming languages. In type theory, purpose is logic. By presenting datatypes as the combination of a structure and a logic, ornaments relate these specialpurpose datatypes through their common structure. In the original presentation, the concept of ornament was introduced concretely for an example universe of inductive families in type theory, but it was clear that the notion was more general. This paper digs out the abstract notion of ornaments in the form of a categorical model. As a necessary first step, we abstract the universe of datatypes using the theory of polynomial functors. We are then able to characterise ornaments as cartesian morphisms between polynomial functors. We thus gain access to powerful mathematical tools that shall help us understand and develop ornaments. We shall also illustrate the adequacy of our model. Firstly, we rephrase the standard ornamental constructions into our framework. Thanks to its conciseness, we gain a deeper understanding of the structures at play. Secondly, we develop new ornamental constructions, by translating categorical structures into type theoretic artefacts.
A LINEAR CATEGORY OF POLYNOMIAL FUNCTORS (EXTENSIONAL PART)
, 2014
"... Abstract. We construct a symmetric monoidal closed category of polynomial endofunctors (as objects) and simulation cells (as morphisms). This structure is defined using universal properties without reference to representing polynomial diagrams and is reminiscent of Day’s convolution on presheaves. W ..."
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Abstract. We construct a symmetric monoidal closed category of polynomial endofunctors (as objects) and simulation cells (as morphisms). This structure is defined using universal properties without reference to representing polynomial diagrams and is reminiscent of Day’s convolution on presheaves. We then make this category into a model for intuitionistic linear logic by defining an additive and exponential structure.
Logical Methods in Computer Science
, 2012
"... Vol. 10(3:14)2014, pp. 1–48 www.lmcsonline.org ..."
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POLYNOMIALS IN CATEGORIES WITH PULLBACKS
"... Abstract. The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed category E, is generalised for E just having pullbacks. The 2categorical analogue of the theory of polynomials and polynomial functors is given, and its relationship with Street’s theory of fibration ..."
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Abstract. The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed category E, is generalised for E just having pullbacks. The 2categorical analogue of the theory of polynomials and polynomial functors is given, and its relationship with Street’s theory of fibrations within 2categories is explored. Johnstone’s notion of “bagdomain data ” is adapted to the present framework to make it easier to completely exhibit examples of polynomial monads. 1.