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Template Games, Simple Games, and Day Convolution Eberhart, Clovis Hirschowitz, Tom Laouar, Alexis Game semantics Day convolution Categorical semantics 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]. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Clovis Eberhart and Tom Hirschowitz and Alexis Laouar 2019 Is Part Of LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.FSCD.2019.16 urn:nbn:de:0030-drops-105237 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.16 eng https://creativecommons.org/licenses/by/3.0/legalcode