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Constructing Strictly Positive Types Michael Abbott a , Thorsten Altenkirch b , Neil Ghani a
 

Summary: Containers
Constructing Strictly Positive Types
Michael Abbott a , Thorsten Altenkirch b , Neil Ghani a
a Department of Mathematics and Computer Science, University of Leicester
b School of Computer Science and Information Technology, Nottingham University
Abstract
We introduce container functors as a representation of data types providing a new
conceptual analysis of data structures and polymorphic functions. Our development
exploits Type Theory as a convenient way to define constructions within locally cartesian
closed categories. We show that container morphisms can be full and faithfully interpreted
as polymorphic functions (i.e. natural transformations) and that in the presence of W­types
all strictly positive types (including nested inductive and coinductive types) give rise to
containers.
Key words: Type Theory, Category Theory, Container Functors, W­Types, Induction,
Coinduction, Initial Algebras, Final Coalgebras.
1 Introduction
One of the strengths of modern functional programming languages like Haskell or
CAML is that they support recursive datatypes such as lists and various forms of
trees. When reasoning about functional programs we commonly restrict our view
to total functions and view types as sets. David Turner called this elementary strong

  

Source: Abbott, Michael - Department of Computer Science, University of Leicester

 

Collections: Computer Technologies and Information Sciences