Skip to main content

Dynamical Systems and Sheaves


A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as ‘machines’ with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special cases include continuous and discrete dynamical systems (e.g. Moore machines). Additionally, morphisms between the different types of systems allow their translation in a common framework. A central goal is to understand the systems that result from arbitrary interconnection of component subsystems, possibly of different types, as well as establish conditions that ensure totality and determinism compositionally. The fundamental categorical tools used here include lax monoidal functors, which provide a language of compositionality, as well as sheaf theory, which flexibly captures the crucial notion of time.

This is a preview of subscription content, access via your institution.


  1. Abramsky, S., Blute, R., Panangaden, P.: Nuclear and trace ideals in tensored \(\ast \)-categories. J. Pure Appl. Algebra, 143(1–3), 3–47 (1999) (special volume on the occasion of the 60th birthday of Professor Michael Barr (Montreal, QC, 1997))

  2. Adámek, J., Borceux, F., Lack, S., Rosický, J.: A classification of accessible categories. J. Pure Appl. Algebra 175(1–3), 7–30 (2002) (special volume celebrating the 70th birthday of Professor Max Kelly)

    Article  MathSciNet  Google Scholar 

  3. Ageron, P.: Effective taxonomies and crossed taxonomies. Cahiers Topologie Géom. Différentielle Catég. 37(2), 82–90 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Barr, M.: Exact categories. In: Exact Categories and Categories of Sheaves, vol. 236, pp. 1–120. Springer, Berlin, Heidelberg (1971).

    Book  MATH  Google Scholar 

  5. Berger, C.: A cellular nerve for higher categories. Adv. Math. 169, 118–175 (2002)

    Article  MathSciNet  Google Scholar 

  6. Berger, C., Melliès, P.-A., Weber, M.: Monads with arities and their associated theories. J. Pure Appl. Algebra 216(8), 2029–2048 (2012)

    Article  MathSciNet  Google Scholar 

  7. Bunge, M., Fiore, M.P.: Unique factorisation lifting functors and categories of linearly-controlled processes. Math. Struct. Comput. Sci. 10(2), 137–163 (2000)

    Article  MathSciNet  Google Scholar 

  8. Day, B., Street, R.: Monoidal bicategories and Hopf algebroids. Adv. Math. 129(1), 99–157 (1997)

    Article  MathSciNet  Google Scholar 

  9. De Paiva, V.C.V.: The Dialectica Categories. Ph.D. thesis, University of Cambridge, UK (1990)

  10. Fiore, M.P.: Fibred models of processes: discrete, continuous, and hybrid systems. In: Proceedings of the International Conference IFIP on Theoretical Computer Science, Exploring New Frontiers of Theoretical Informatics, TCS ’00, pp. 457–473. Springer, Berlin (2000)

    Chapter  Google Scholar 

  11. Fong, B., Spivak, D.: Hypergraph Categories. arXiv:1806.08304 [math.CT] (2018)

  12. Fong, B.: The Algebra of Open and Interconnected Systems. Ph.D. thesis, University of Oxford (2016)

  13. Hedges, J.: Morphisms of open games. Electron. Notes Theor. Comput. Sci. 341, 151–177 (2018)

    Article  MathSciNet  Google Scholar 

  14. Hermida, C.: Representable multicategories. Adv. Math. 151(2), 164–225 (2000)

    Article  MathSciNet  Google Scholar 

  15. Jacobs, B.: Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics, vol. 141. North-Holland Publishing Co., Amsterdam (1999)

    Google Scholar 

  16. Johnstone, P.: A note on discrete Conduché fibrations. Theory Appl. Categ. 5(1), 1–11 (1999)

    MathSciNet  MATH  Google Scholar 

  17. Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium, Volume 43 of Oxford Logic Guides. The Clarendon Press, New York (2002)

    MATH  Google Scholar 

  18. Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102(1), 20–78 (1993)

    Article  MathSciNet  Google Scholar 

  19. Joyal, A., Nielsen, M., Winskel, G.: Bisimulation from open maps. Inf. Computat. 127(2), 164–185 (1996)

    Article  MathSciNet  Google Scholar 

  20. Joyal, A., Street, R., Verity, D.: Traced monoidal categories. Math. Proc. Camb. Philos. Soc. 119(3), 447–468 (1996)

    Article  MathSciNet  Google Scholar 

  21. Katis, P., Sabadini, N., Walters, R.F.C.: On the algebra of systems with feedback and boundary. Rendiconti del Circolo Matematico di Palermo Serie II(63), 123–156 (2000)

    MATH  Google Scholar 

  22. Kock, J.: Polynomial functors and trees. Int. Math. Res. Not. 2011(3), 609–673 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Lambek, J.: Deductive systems and categories. II. Standard constructions and closed categories. In: Category Theory, Homology Theory and their Applications, I (Battelle Institute Conference, Seattle, Wash., 1968, Vol. 1), pp. 76–122. Springer, Berlin (1969)

    Google Scholar 

  24. Lawvere, F.W.: State categories and response functors. Unpublished manuscript (1986)

  25. Lee, Edward A., Seshia, Sanjit A.: Introduction to Embedded Systems, A Cyber-Physical Systems Approach, 2nd edn. MIT Press, Cambridge (2017)

    MATH  Google Scholar 

  26. Leinster, T.: Nerves of algebras. Talk at CT04, Vancouver (2004)

  27. Leinster, T.: Higher Operads, Higher Categories. Number 298 in London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  28. Markl, M., Merkulov, S., Shadrin, S.: Wheeled props, graph complexes and the master equation. J. Pure Appl. Algebra 213(4), 496–535 (2009)

    Article  MathSciNet  Google Scholar 

  29. Milner, R.: Calculi for interaction. Acta Inform. 33(8), 707–737 (1996)

    Article  MathSciNet  Google Scholar 

  30. Moeller, J., Vasilakopoulou, C.: Monoidal grothendieck construction. arXiv:1809.00727 (2018)

  31. Rupel, D., Spivak, D.I.: The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes CoRR. arxiv:1307.6894 (2013)

  32. Schultz, P., Spivak, D.I.: Temporal type theory: a topos-theoretic approach to systems and behavior. In: Progress in computer science and applied logic, vol. 29, p. 235. Birkhäuser Basel (2019). (2017)

    Book  Google Scholar 

  33. Selinger, P.: First-order axioms for asynchrony. In: International Conference on Concurrency Theory, pp. 376–390. Springer (1997)

  34. Spivak, D.I., Schultz, P., Rupel, D.: String diagrams for traced and compact categories are oriented 1-cobordisms. J Pure Appl Algebra. 221(8), 2064–2110 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  35. Spivak, D.I.: The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits. CoRR, arXiv:1305.0297 (2013)

  36. Spivak, D.I.: The steady states of coupled dynamical systems compose according to matrix arithmetic. arXiv preprint: arXiv:1512.00802 (2015)

  37. Stay, M.: Compact closed bicategories. Theory Appl. Categ. 31, 755–798 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Sztipanovits, J., Ying, S.: Strategic r&d opportunities for 21st century cyber-physical systems. Technical report, National Institute of Standards and Technology (2013)

  39. Vagner, D., Spivak, D.I., Lerman, E.: Algebras of open dynamical systems on the operad of wiring diagrams. Theory Appl. Categ. 30(51), 1793–1822 (2015)

    MathSciNet  MATH  Google Scholar 

  40. Weber, M.: Familial 2-functors and parametric right adjoints. Theory Appl. Categ. 18(22), 665–732 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Willems, J.C., Polderman, J.W.: Introduction to Mathematical Systems Theory: A Behavioral Approach, vol. 26. Springer, Berlin (2013)

    MATH  Google Scholar 

  42. Winskel, G.: Event Structures, pp. 325–392. Springer, Berlin (1987)

    MATH  Google Scholar 

Download references


We greatly appreciate our collaboration with Alberto Speranzon and Srivatsan Varadarajan, who have helped us to understand how the ideas presented here can be applied in practice (specifically for modeling the National Airspace System) and who provided motivating examples with which to test and often augment the theory. We also thank the anonymous reviewers for valuable suggestions; in particular, such a suggestion led to a more abstract formalism of system algebras, explained in Section 2.4.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Christina Vasilakopoulou.

Additional information

Communicated by Richard Garner.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Schultz, Spivak and Vasilakopoulou were supported by AFOSR Grant FA9550–14–1–0031 and NASA Grant NNH13ZEA001N.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Schultz, P., Spivak, D.I. & Vasilakopoulou, C. Dynamical Systems and Sheaves. Appl Categor Struct 28, 1–57 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Dynamical systems
  • Topos theory
  • Sheaf theory
  • Monoidal categories
  • Operads