Skip to main content
SpringerLink
Log in

Uniform Functors on Sets

  • Chapter
Algebra, Meaning, and Computation

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4060))

Abstract

This paper studies uniformity conditions for endofunctors on sets following Aczel [1], Turi [21], and others. The “usual” functors on sets are uniform in our sense, and assuming the Anti-Foundation Axiom AFA, a uniform functor H has the property that its greatest fixed point H * is a final coalgebra whose structure is the identity map. We propose a notion of uniformity whose definition involves notions from recent work in coalgebraic recursion theory: completely iterative monads and completely iterative algebras (cias). Among our new results is one which states that for a uniform H, the entire set-theoretic universe V is a cia: the structure is the inclusion of HV into the universe V itself.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

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.

References

  1. Aczel, P.: Non-Well-Founded Sets. CSLI Lecture Notes, vol. 14. CSLI Publications, Stanford (1988)

    MATH  Google Scholar 

  2. Aczel, P., Adámek, J., Milius, S., Velebil, J.: Infinite trees and completely iterative Theories: a coalgebraic view. Theoretical Computer Science 300, 1–45 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aczel, P., Adámek, J., Velebil, J.: A coalgebraic view of infinite trees and iteration. Electronic Notes in Theoretical Computer Science 44(1) (2001)

    Google Scholar 

  4. Aczel, P., Mendler, N.: A final coalgebra theorem. In: Pitt, D.H., et al. (eds.) Category Theory and Computer Science, pp. 357–365. Springer, Heidelberg (1989)

    Chapter  Google Scholar 

  5. Adámek, J., Milius, S., Velebil, J.: On coalgebra based on classes. Theoretical Computer Science 316(1-3), 3–23 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Adámek, J., Milius, S., Velebil, J.: Elgot algebras (2005) (preprint)

    Google Scholar 

  7. Adámek, J., Trnková, V.: Automata and Algebras in Categories. Kluwer Academic Publishers Group, Dordrecht (1990)

    MATH  Google Scholar 

  8. Barwise, J., Moss, L., Circles, V.: CSLI Lecture Notes, vol. 60. CSLI Publications, Stanford (1996)

    Google Scholar 

  9. Cancila, D.: Ph.D. Dissertation, University of Udine Computer Science Department (2003)

    Google Scholar 

  10. Cancila, D., Honsell, F., Lenisa, M.: Properties of set functors. In: Honsell, F., et al. (eds.) Proceedings of COMETA 2003. ENTCS, vol. 104, pp. 61–80 (2004)

    Google Scholar 

  11. Devlin, K.: The Joy of Sets, 2nd edn. Springer, Heidelberg (1993)

    MATH  Google Scholar 

  12. Freyd, P.: Real coalgebra, post on categories mailing list(December 22, 1999), available via http://www.mta.ca/~cat-dist

  13. Levy, A.: Basic Set Theory. Springer, Heidelberg (1979)

    MATH  Google Scholar 

  14. Milius, S.: Completely iterative algebras and completely iterative monads. Inform. and Comput. 196, 1–41 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Milius, S.: Ph.D. Dissertation, Institute of Theoretical Computer Science, Technical University of Braunschweig (2005)

    Google Scholar 

  16. Milius, S., Moss, L.S.: The category theoretic solution of recursive program schemes. In: Fiadeiro, J.L., Harman, N.A., Roggenbach, M., Rutten, J. (eds.) CALCO 2005. LNCS, vol. 3629, pp. 293–312. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  17. Moss, L.S.: Coalgebraic logic. Annals of Pure and Applied Logic 96(1-3), 277–317 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Moss, L.S.: Parametric corecursion. Theoretical Computer Science 260(1-2), 139–163 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Moss, L.S., Danner, N.: On the foundations of corecursion. Logic Journal of the IGPL 5(2), 231–257 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Theoretical Computer Science 249(1), 3–80 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Turi, D.: Functorial Operational Semantics and its Denotational Dual Ph.D. thesis, CWI, Amsterdam (1996)

    Google Scholar 

  22. Turi, D., Rutten, J.J.M.M.: On the foundations of final semantics: non-standard sets, metric spaces, partial orders. Mathematical Structures in Computer Science 8(5), 481–540 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Indiana University, Bloomington, IN, 47405, USA

    Lawrence S. Moss

Authors
  1. Lawrence S. Moss
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology (JAIST),  

    Kokichi Futatsugi

  2. LIX, Projet INRIA TypiCal, École Polytechnique and CNRS, 91400, Palaiseau, France

    Jean-Pierre Jouannaud

  3. CS Dept., University of Illinois at Urbana-Champaign, USA

    José Meseguer

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Moss, L.S. (2006). Uniform Functors on Sets. In: Futatsugi, K., Jouannaud, JP., Meseguer, J. (eds) Algebra, Meaning, and Computation. Lecture Notes in Computer Science, vol 4060. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11780274_22

Download citation

  • DOI: https://doi.org/10.1007/11780274_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35462-8

  • Online ISBN: 978-3-540-35464-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation