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Identity-Preserving Lax Extensions and Where to Find Them

Authors Sergey Goncharov , Dirk Hofmann , Pedro Nora , Lutz Schröder , Paul Wild

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Subject Classification

ACM Subject Classification
  • Theory of computation → Modal and temporal logics
  • Theory of computation → Concurrency
Keywords
  • (Bi-)simulations
  • lax extensions
  • modal logics
  • coalgebra

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Abstract

Generic notions of bisimulation for various types of systems (nondeterministic, probabilistic, weighted etc.) rely on identity-preserving (normal) lax extensions of the functor encapsulating the system type, in the paradigm of universal coalgebra. It is known that preservation of weak pullbacks is a sufficient condition for a functor to admit a normal lax extension (the Barr extension, which in fact is then even strict); in the converse direction, nothing is currently known about necessary (weak) pullback preservation conditions for the existence of normal lax extensions. In the present work, we narrow this gap by showing on the one hand that functors admitting a normal lax extension preserve 1/4-iso pullbacks, i.e. pullbacks in which at least one of the projections is an isomorphism. On the other hand, we give sufficient conditions, showing that a functor admits a normal lax extension if it weakly preserves either 1/4-iso pullbacks and 4/4-epi pullbacks (i.e. pullbacks in which all morphisms are epic) or inverse images. We apply these criteria to concrete examples, in particular to functors modelling neighbourhood systems and weighted systems.

Cite As Get BibTex

Sergey Goncharov, Dirk Hofmann, Pedro Nora, Lutz Schröder, and Paul Wild. Identity-Preserving Lax Extensions and Where to Find Them. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.40

Author Details

Sergey Goncharov
  • University of Birmingham, UK
Dirk Hofmann
  • CIDMA, University of Aveiro, Portugal
Pedro Nora
  • Radboud Universiteit, Nijmegen, The Netherlands
Lutz Schröder
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Paul Wild
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

Funding

  • Goncharov, Sergey: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 501369690.
  • Hofmann, Dirk: This work is supported by CIDMA under the FCT (Portuguese Foundation for Science and Technology) Multi-Annual Financing Program for R&D Units.
  • Schröder, Lutz: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 531706730.
  • Wild, Paul: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 434050016.

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