Template Games, Simple Games, and Day Convolution

Authors Clovis Eberhart, Tom Hirschowitz, Alexis Laouar

Thumbnail PDF


  • Filesize: 0.51 MB
  • 19 pages

Document Identifiers

Author Details

Clovis Eberhart
  • National Institute of Informatics, Tokyo, Japan
Tom Hirschowitz
  • Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000, Chambéry, France
Alexis Laouar
  • Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000, Chambéry, France

Cite AsGet BibTex

Clovis Eberhart, Tom Hirschowitz, and Alexis Laouar. Template Games, Simple Games, and Day Convolution. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Template games [P.-A. Melliès, 2019] unify various approaches to game semantics, by exhibiting them as instances of a double-categorical variant of the slice construction. However, in the particular case of simple games [R. Harmer et al., 2007; C. Jacq and P.-A. Melliès, 2018], template games do not quite yield the standard (bi)category. We refine the construction using factorisation systems, obtaining as an instance a slight generalisation of simple games and strategies. This proves that template games have the descriptive power to capture combinatorial constraints defining well-known classes of games. Another instance is Day’s convolution monoidal structure on the category of presheaves over a strict monoidal category [B. Day, 1970], which answers a question raised in [C. Eberhart, 2018].

Subject Classification

ACM Subject Classification
  • Theory of computation → Denotational semantics
  • Game semantics
  • Day convolution
  • Categorical semantics


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. A. K. Bousfield. Constructions of Factorization Systems in Categories. Journal of Pure and Applied Algebra, 9(2-3):287-329, 1977. Google Scholar
  2. B. Day. On closed categories of functors. In Reports of the Midwest Category Seminar IV, volume 137 of Lecture Notes in Mathematics, pages 1-38. Springer, 1970. Google Scholar
  3. C. Eberhart. Catégories et diagrammes de cordes pour les jeux concurrents. PhD thesis, Université Savoie Mont Blanc, 2018. Google Scholar
  4. C. Eberhart and T. Hirschowitz. What’s in a game?: A theory of game models. In Proc. 33rd Symposium on Logic in Computer Science, pages 374-383. ACM, 2018. URL: http://dx.doi.org/10.1145/3209108.3209114.
  5. C. Ehresmann. Catégories structurées. Annales scientifiques de l'Ecole Normale Supérieure, 80(4):349-426, 1963. Google Scholar
  6. R. Garner. Polycategories. PhD thesis, University of Cambridge, 2006. Google Scholar
  7. R. Garner and M. Shulman. Enriched categories as a free cocompletion. Advances in Mathematics, 289:1-94, 2016. Google Scholar
  8. M. Grandis and R. Paré. Limits in double categories. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 40(3):162-220, 1999. Google Scholar
  9. R. Harmer, J. M. E. Hyland, and P.-A. Melliès. Categorical Combinatorics for Innocent Strategies. In Proc. 22nd Symposium on Logic in Computer Science, pages 379-388. IEEE, 2007. Google Scholar
  10. M. Hirschowitz, A. Hirschowitz, and T. Hirschowitz. A theory for game theories. In Proc. 27th Foundations of Software Technology and Theoretical Computer Science, volume 4855 of Lecture Notes in Computer Science, pages 192-203. Springer, 2007. Google Scholar
  11. J. M. E. Hyland. Game Semantics. In Andrew M. Pitts and Peter Dybjer, editors, Semantics and Logics of Computation, pages 131-184. Cambridge University Press, 1997. Google Scholar
  12. C. Jacq and P.-A. Melliès. Categorical Combinatorics for Non Deterministic Strategies on Simple Games. In Proc. 21st Foundations of Software Science and Computational Structures, volume 10803 of Lecture Notes in Computer Science, pages 39-70. Springer, 2018. URL: http://dx.doi.org/10.1007/978-3-319-89366-2_3.
  13. T. Leinster. Basic Bicategories. arXiv Mathematics e-prints, page math/9810017, October 1998. URL: http://arxiv.org/abs/math/9810017.
  14. P.-A. Melliès. Categorical combinatorics of scheduling and synchronization in game semantics. Proc. ACM Program. Lang., 3(POPL):23:1-23:30, January 2019. URL: http://dx.doi.org/10.1145/3290336.
  15. S. Rideau and G. Winskel. Concurrent Strategies. In Proc. 26th Symposium on Logic in Computer Science, pages 409-418. IEEE, 2011. Google Scholar
  16. R. Street and R. F. C. Walters. The Comprehensive Factorization of a Functor. Bulletin of the American Mathematical Society, 79(5), 1973. Google Scholar
  17. M. Weber. Generic morphisms, parametric representations and weakly cartesian monads. Theory and Applications of Categories, 13:191-234, 2004. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail