Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Regular Behaviours with Names

On Rational Fixpoints of Endofunctors on Nominal Sets

  • 44 Accesses

  • 3 Citations


Nominal sets provide a framework to study key notions of syntax and semantics such as fresh names, variable binding and α-equivalence on a conveniently abstract categorical level. Coalgebras for endofunctors on nominal sets model, e.g., various forms of automata with names as well as infinite terms with variable binding operators (such as λ-abstraction). Here, we first study the behaviour of orbit-finite coalgebras for functors \(\bar F\) on nominal sets that lift some finitary set functor F. We provide sufficient conditions under which the rational fixpoint of \(\bar F\), i.e. the collection of all behaviours of orbit-finite \(\bar F\)-coalgebras, is the lifting of the rational fixpoint of F. Second, we describe the rational fixpoint of the quotient functors: we introduce the notion of a sub-strength of an endofunctor on nominal sets, and we prove that for a functor G with a sub-strength the rational fixpoint of each quotient of G is a canonical quotient of the rational fixpoint of G. As applications, we obtain a concrete description of the rational fixpoint for functors arising from so-called binding signatures with exponentiation, such as those arising in coalgebraic models of infinitary λ-terms and various flavours of automata.

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


  1. 1.

    Adámek, J.: Introduction to coalgebra. Theory Appl. Categ. 14, 157–199 (2005)

  2. 2.

    Adámek, J., Levy, P., Milius, S., Moss, L., Sousa, L.: On final coalgebras of power-set functors and saturated trees. Appl. Categ. Struct. 23, 609–641 (2015)

  3. 3.

    Adámek, J., Milius, S.: Terminal coalgebras and free iterative theories. Inf. Comput. 204, 1139–1172 (2006)

  4. 4.

    Adámek, J., Milius, S., Velebil, J.: Iterative algebras at work. Math. Struct. Comput. Sci. 16(6), 1085–1131 (2006)

  5. 5.

    Adámek, J., Rosický, J.: Locally presentable and accessible categories. Cambridge University Press (1994)

  6. 6.

    Barr, M.: Terminal coalgebras in well-founded set theory. Theoret. Comput. Sci. 114, 299–315 (1993)

  7. 7.

    Bartels, F.: On Generalised Coinduction and Probabilistic Specification Formats: Distributive Laws in Coalgebraic Modelling. PhD thesis, Vrije Universiteit Amsterdam (2004)

  8. 8.

    Bonsangue, M., Milius, S., Silva, A.: Sound and complete axiomatizations of coalgebraic language equivalence. ACM Trans. Comput. Log. 14(1:7), 52 (2013)

  9. 9.

    Courcelle, B.: Fundamental properties of infinite trees. Theoret. Comput. Sci. 25, 95–169 (1983)

  10. 10.

    Elgot, C.: Monadic computation and iterative algebraic theories. In: Rose, H., Sheperdson, J. (eds.) Monadic Logic Colloquium 1973, vol. 80, pp. 175–230 North Holland (1975)

  11. 11.

    Gabbay, M., Pitts, A.: A new approach to abstract syntax involving binders. In: Logic in Computer Science, LICS 1999, pp. 214–224. IEEE (1999)

  12. 12.

    Gabbay, M., Pitts, A. M.: A new approach to abstract syntax involving binders. In: Logic in Computer Science, LICS 1999, pp. 214–224. IEEE Computer Society Press (1999)

  13. 13.

    Gabriel, P., Ulmer, F.: Lokal präsentierbare Kategorien, vol. 221 of Lect.Notes Math Springer (1971)

  14. 14.

    Gaducci, F., Miculan, M., Montanari, U.: About permutation algebras, (pre)sheaves and named sets. Higher-order Symb Comput. 19, 283–304 (2006)

  15. 15.

    Ginali, S.: Regular trees and the free iterative theory. J. Comput. Syst. Sci. 18, 228–242 (1979)

  16. 16.

    Jacobs, B., Rutten, J.: A tutorial on (co)algebras and (co)induction. EATCS Bulletin 62, 62–222 (1997)

  17. 17.

    Johnstone, P.: Adjoint lifting theorems for categories of algebras. Bull. Lond. Math. Soc. 7, 294–297 (1975)

  18. 18.

    Joyal, A.: Une théorie combinatoire des séries formelles. Adv. Math. 42, 1–82 (1981)

  19. 19.

    Joyal, A.: Foncteurs analytiques et espèces de structures. Lect. Notes Math. 1234, 126–159 (1986)

  20. 20.

    Kock, A.: Strong functors and monoidal monads. Arch. Math. 23, 113–120 (1972)

  21. 21.

    Kozen, D., Mamouras, K., Petrisan, D., Silva, A.: Nominal Kleene coalgebra. In: Automata, Languages, and Programming, ICALP 2015, vol. 9135 of Lect. Notes Comput. Sci., pp. 286–298. Springer (2015)

  22. 22.

    Kurz, A., Petrisan, D., Severi, P., de Vries, F.-J.: Nominal coalgebraic data types with applications to lambda calculus. Log. Meth. Comput. Sci. 9(4) (2013)

  23. 23.

    Kurz, A., Petrisan, D., Velebil, J.: Algebraic theories over nominal sets. CoRR, abs/1006.3027 (2010)

  24. 24.

    Lambek, J.: A fixpoint theorem for complete categories. Math. Z. 103, 151–161 (1968)

  25. 25.

    Makkai, M., Paré, R.: Accessible categories: the foundation of categorical model theory, vol. 104 of Contemporary Math. Am. Math Soc. (1989)

  26. 26.

    Milius, S.: A sound and complete calculus for finite stream circuits. In: Logic in Computer Science, LICS 2010, pp. 449–458. IEEE Computer Society (2010)

  27. 27.

    Milius, S., Wißmann, T.: Finitary corecursion for the infinitary lambda calculus. In: Moss, L., Sobocinski, P. (eds.) Algebra and Coalgebra in Computer Science, CALCO 2015, vol. 35 of LIPIcs, pp. 336–351 Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)

  28. 28.

    Petrişan, D.: Investigations into Algebra and Topology over Nominal Sets. PhD thesis, University of Leicester (2011)

  29. 29.

    Pitts, A.: Nominal logic, a first order theory of names and binding. Inf. Comput. 186, 165–193 (2003)

  30. 30.

    Pitts, A.: Nominal sets: Names and symmetry in computer science. Cambridge university press (2013)

  31. 31.

    Plotkin, G., Turi, D.: Towards a mathematical operational semantics. In: Logic in Computer Science, LICS 1997, pp. 280–291. IEEE (1997)

  32. 32.

    Rutten, J.: Universal coalgebra: a theory of systems. Theoret. Comput. Sci. 249 (1), 3–80 (2000)

  33. 33.

    Rutten, J.: Rational streams coalgebraically. Log. Meth Comput. Sci. 4(3), 9 (2008)

  34. 34.

    Tzevelekos, N.: Full abstraction for nominal general references. In: Logic in Computer Science, LICS 2007, pp. 399–410. IEEE (2007)

  35. 35.

    Worrell, J.: On the final sequence of a finitary set functor. Theoret. Comput. Sci. 338, 184–199 (2005)

Download references

Author information

Correspondence to Lutz Schröder.

Additional information

In fond memory of our colleague and mentor Horst Herrlich

This work forms part of the DFG-funded project COAX (MI 717/5-1 and SCHR 1118/12-1)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Milius, S., Schröder, L. & Wißmann, T. Regular Behaviours with Names. Appl Categor Struct 24, 663–701 (2016). https://doi.org/10.1007/s10485-016-9457-8

Download citation


  • Nominal sets
  • Final coalgebras
  • Rational fixpoints
  • Lifted functors