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.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aczel, P.: Non-Well-Founded Sets. CSLI Lecture Notes, vol. 14. CSLI Publications, Stanford (1988)
Aczel, P., Adámek, J., Milius, S., Velebil, J.: Infinite trees and completely iterative Theories: a coalgebraic view. Theoretical Computer Science 300, 1–45 (2003)
Aczel, P., Adámek, J., Velebil, J.: A coalgebraic view of infinite trees and iteration. Electronic Notes in Theoretical Computer Science 44(1) (2001)
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)
Adámek, J., Milius, S., Velebil, J.: On coalgebra based on classes. Theoretical Computer Science 316(1-3), 3–23 (2004)
Adámek, J., Milius, S., Velebil, J.: Elgot algebras (2005) (preprint)
Adámek, J., Trnková, V.: Automata and Algebras in Categories. Kluwer Academic Publishers Group, Dordrecht (1990)
Barwise, J., Moss, L., Circles, V.: CSLI Lecture Notes, vol. 60. CSLI Publications, Stanford (1996)
Cancila, D.: Ph.D. Dissertation, University of Udine Computer Science Department (2003)
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)
Devlin, K.: The Joy of Sets, 2nd edn. Springer, Heidelberg (1993)
Freyd, P.: Real coalgebra, post on categories mailing list(December 22, 1999), available via http://www.mta.ca/~cat-dist
Levy, A.: Basic Set Theory. Springer, Heidelberg (1979)
Milius, S.: Completely iterative algebras and completely iterative monads. Inform. and Comput. 196, 1–41 (2005)
Milius, S.: Ph.D. Dissertation, Institute of Theoretical Computer Science, Technical University of Braunschweig (2005)
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)
Moss, L.S.: Coalgebraic logic. Annals of Pure and Applied Logic 96(1-3), 277–317 (1999)
Moss, L.S.: Parametric corecursion. Theoretical Computer Science 260(1-2), 139–163 (2001)
Moss, L.S., Danner, N.: On the foundations of corecursion. Logic Journal of the IGPL 5(2), 231–257 (1997)
Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Theoretical Computer Science 249(1), 3–80 (2000)
Turi, D.: Functorial Operational Semantics and its Denotational Dual Ph.D. thesis, CWI, Amsterdam (1996)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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)