Composition and Recursion for Causal Structures

Authors Henning Basold , Tanjona Ralaivaosaona

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Henning Basold
  • LIACS, Leiden University, The Netherlands
Tanjona Ralaivaosaona
  • LIACS, Leiden University, The Netherlands

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Henning Basold and Tanjona Ralaivaosaona. Composition and Recursion for Causal Structures. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Causality appears in various contexts as a property where present behaviour can only depend on past events, but not on future events. In this paper, we compare three different notions of causality that capture the idea of causality in the form of restrictions on morphisms between coinductively defined structures, such as final coalgebras and chains, in fairly general categories. We then focus on one presentation and show that it gives rise to a traced symmetric monoidal category of causal morphisms. This shows that causal morphisms are closed under sequential and parallel composition and, crucially, under recursion.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Causal morphisms
  • Final Coalgebras
  • Final Chains
  • Metric Maps
  • Guarded Recursion
  • Traced Symmetric Monoidal Category


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  1. J Adámek, S Milius, and LS Moss. Initial algebras, terminal coalgebras, and the theory of fixed points of functors, 2019. Google Scholar
  2. John C. Baez, Brandon Coya, and Franciscus Rebro. Props in Network Theory, June 2018. URL:
  3. Henning Basold. Coinduction in Flow: The Later Modality in Fibrations. In Markus Roggenbach and Ana Sokolova, editors, 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019), volume 139 of LIPIcs, pages 8:1-8:22, Dagstuhl, Germany, 2019. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL:
  4. Lars Birkedal, Rasmus Ejlers Møgelberg, Jan Schwinghammer, and Kristian Støvring. First steps in synthetic guarded domain theory: Step-indexing in the topos of trees. LMCS, 8(4), 2012. URL:
  5. Jasmin Christian Blanchette, Aymeric Bouzy, Andreas Lochbihler, Andrei Popescu, and Dmitriy Traytel. Friends with Benefits. In Hongseok Yang, editor, Programming Languages and Systems, Lecture Notes in Computer Science, pages 111-140, Berlin, Heidelberg, 2017. Springer. URL:
  6. Francis Borceux. Handbook of Categorical Algebra: Volume 2: Categories and Structures, volume 2 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge, 1994. URL:
  7. Helle Hvid Hansen, Clemens Kupke, and Jan Rutten. Stream differential equations: specification formats and solution methods. arXiv preprint arXiv:1609.08367, 2016. Google Scholar
  8. Bart Jacobs. Introduction to coalgebra, volume 59. Cambridge University Press, 2017. Google Scholar
  9. André Joyal, Ross Street, and Dominic Verity. Traced monoidal categories. In Mathematical proceedings of the cambridge philosophical society, volume 119, pages 447-468. Cambridge University Press, 1996. Google Scholar
  10. Max Kelly. Basic Concepts of Enriched Category Theory. Number 64 in Lecture Notes in Mathematics. Cambridge University Press, reprints in theory and applications of categories, no. 10 (2005) edition, 1982. Google Scholar
  11. Saunders Mac Lane. Categories for the Working Mathematician. Number 5 in Graduate Texts in Mathematics. Springer, 2 edition, 1998. Google Scholar
  12. Saunders MacLane. Categorical algebra. Bulletin of the American Mathematical Society, 71(1):40-106, 1965. Google Scholar
  13. Luís Monteiro. Observation systems. Electronic Notes in Theoretical Computer Science, 33:261-275, 2000. Google Scholar
  14. Jurriaan Rot and Damien Pous. Companions, causality and codensity. Logical Methods in Computer Science, 15, 2019. Google Scholar
  15. Jan JMM Rutten. A tutorial on coinductive stream calculus and signal flow graphs. Theoretical Computer Science, 343(3):443-481, 2005. Google Scholar
  16. David Sprunger and Bart Jacobs. The differential calculus of causal functions. arXiv preprint arXiv:1904.10611, 2019. Google Scholar