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Regular Behaviours with Names

On Rational Fixpoints of Endofunctors on Nominal Sets

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Abstract

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.

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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)

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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

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Keywords

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