Skip to main content
Springer Nature Link
Log in

Derivatives of Containers

  • Conference paper
  • First Online:
Typed Lambda Calculi and Applications (TLCA 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2701))

Included in the following conference series:

  • 436 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  • M. Abbott, T. Altenkirch, and N. Ghani. Categories of containers. In Proceedings of Foundations of Software Science and Computation Structures, 2003.

    Google Scholar 

  • J. Adámek and J. Rosický. Locally Presentable and Accessible Categories. Number 189 in LMS Lecture Note Series. CUP, 1994.

    Google Scholar 

  • P. Dybjer. Representing inductively defined sets by wellorderings in Martin-Löf’s type theory. Theoretical Computer Science, 170(1–2):245–276, 1996.

    MathSciNet  Google Scholar 

  • T. Ehrhard and L. Regnier. The differential lambda-calculus. Available electronically, 2001. URL http://iml.univ-mrs.fr/ftp/regnier/Articles.html.

  • R. Hasegawa. Two applications of analytic functors. Theoretical Comput. Sci., 272(1–2):112–175, 2002.

    MathSciNet  Google Scholar 

  • M. Hofmann. On the interpretation of type theory in locally cartesian closed catetories. In Computer Science Logic, volume 933 of LNCS, 1994.

    Google Scholar 

  • G. Huet. The zipper. Journal of Functional Programming, 7(5):549–554, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  • G. Huet. Linear contexts and the sharing functor: Techniques for symbolic computation. In Workshop on Thirty Five Years of Automath, 2003.

    Google Scholar 

  • A. Joyal. Foncteurs analytiques et espéces de structures. In Combinatoire énumérative, number 1234 in LNM, pages 126–159. 1986.

    Google Scholar 

  • G. Leibniz. Nova methodus pro maximis et minimis, itemque tangentibus, qua nec irrationals quantitates moratur. Acta eruditorum, 1684.

    Google Scholar 

  • C. McBride. The derivative of a regular type is its type of one-hole contexts. Available electronically, 2001. URL http://www.dur.ac.uk/c.t.mcbride/.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Leicester, Leicester

    Michael Abbott & Neil Ghani

  2. School of Computer Science and Information Technology, Nottingham University, Nottingham

    Thorsten Altenkirch

  3. Department of Computer Science, University of Durham, Durham

    Conor McBride

Authors
  1. Michael Abbott
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thorsten Altenkirch
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Neil Ghani
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Conor McBride
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Universität München, Oettingenstr. 67, 80538, München, Germany

    Martin Hofmann

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Abbott, M., Altenkirch, T., Ghani, N., McBride, C. (2003). Derivatives of Containers. In: Hofmann, M. (eds) Typed Lambda Calculi and Applications. TLCA 2003. Lecture Notes in Computer Science, vol 2701. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44904-3_2

Download citation

  • DOI: https://doi.org/10.1007/3-540-44904-3_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40332-6

  • Online ISBN: 978-3-540-44904-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics