Uniform Functors on Sets

• Lawrence S. Moss
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

Keywords

Natural Transformation Foundation Axiom Equation Morphism Standard Functor Constant Functor
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.

References

1. 1.
Aczel, P.: Non-Well-Founded Sets. CSLI Lecture Notes, vol. 14. CSLI Publications, Stanford (1988)
2. 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)
3. 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. 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)
5. 5.
Adámek, J., Milius, S., Velebil, J.: On coalgebra based on classes. Theoretical Computer Science 316(1-3), 3–23 (2004)
6. 6.
7. 7.
Adámek, J., Trnková, V.: Automata and Algebras in Categories. Kluwer Academic Publishers Group, Dordrecht (1990)
8. 8.
Barwise, J., Moss, L., Circles, V.: CSLI Lecture Notes, vol. 60. CSLI Publications, Stanford (1996)Google Scholar
9. 9.
Cancila, D.: Ph.D. Dissertation, University of Udine Computer Science Department (2003)Google Scholar
10. 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. 11.
Devlin, K.: The Joy of Sets, 2nd edn. Springer, Heidelberg (1993)
12. 12.
Freyd, P.: Real coalgebra, post on categories mailing list(December 22, 1999), available via http://www.mta.ca/~cat-dist
13. 13.
Levy, A.: Basic Set Theory. Springer, Heidelberg (1979)
14. 14.
Milius, S.: Completely iterative algebras and completely iterative monads. Inform. and Comput. 196, 1–41 (2005)
15. 15.
Milius, S.: Ph.D. Dissertation, Institute of Theoretical Computer Science, Technical University of Braunschweig (2005)Google Scholar
16. 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)
17. 17.
Moss, L.S.: Coalgebraic logic. Annals of Pure and Applied Logic 96(1-3), 277–317 (1999)
18. 18.
Moss, L.S.: Parametric corecursion. Theoretical Computer Science 260(1-2), 139–163 (2001)
19. 19.
Moss, L.S., Danner, N.: On the foundations of corecursion. Logic Journal of the IGPL 5(2), 231–257 (1997)
20. 20.
Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Theoretical Computer Science 249(1), 3–80 (2000)
21. 21.
Turi, D.: Functorial Operational Semantics and its Denotational Dual Ph.D. thesis, CWI, Amsterdam (1996)Google Scholar
22. 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)