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Summary: Derivatives of Containers
Michael Abbott1, Thorsten Altenkirch2, Neil Ghani1, and Conor McBride3
1 Department of Mathematics and Computer Science, University of Leicester
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2 School of Computer Science and Information Technology, Nottingham University
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3 Department of Computer Science, University of Durham
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Abstract. We are investigating McBride's idea that the type of one-hole contexts
are the formal derivative of a functor from a categorical perspective. Exploiting
our recent work on containers we are able to characterise derivatives by a
universal property and show that the laws of calculus including a rule for initial
algebras as presented by McBride hold -- hence the differentiable containers
include those generated by polynomials and least fixpoints. Finally, we discuss
abstract containers (i.e. quotients of containers) -- this includes a container which
plays the role of ex in calculus by being its own derivative.
1 Introduction
In his classic functional pearl Huet (1997) shows how to represent a tree with one of
its subtrees `in focus' by a pair of the subtree and the one-hole context (or `zipper')
in which it sits. The unpublished article McBride (2001) gives a `generic program' for
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