![]() As we will see, our work also integrates a number of elements coming from the first approach, and more specifically, the discovery by Melliès that strategies are presented by generators and relations, and for that reason, are prone to factorisation theorems . For that reason, our purpose in this paper is to investigate the connection with the second approach, different in spirit and design, and to define a bicategory of simple games and non-deterministic strategies in the sheaf-theoretic style of Harmer et al. . Interestingly, all the sheaf-theoretic frameworks designed for game semantics today are offsprings of the third approach based on asynchronous games: on the one hand, the notion of concurrent strategy in is a sheaf-theoretic transcription of the notion of receptive ingenuous strategy formulated in on the other hand, the sheaf-theoretic notion of non-deterministic innocent strategy in relies on the diagrammatic and local definition of innocence in alternated asynchronous games . The concurrent and asynchronous approach advocated by Melliès, based on the description of arena games as asynchronous games, and of strategies as causal concurrent structures playing on them, either in an alternated or in a non-alternated way. The combinatorial approach advocated by Hyland, inspired by algebraic topology, and based on the combinatorial description of the structure of pointers in arena games , The logical approach advocated by Girard, and formulated in ludics, polarized linear logic or tensorial logic with its connection to continuations and string diagrams, We recognize three main lines of work here: In this investigation, one benefits from the efforts made in the past twenty-five years to give a clearer mathematical status to the previous generation of game semantics, which was (to a large extent) based on the notion of arena game. For that reason, it is timely to examine more closely the 2-categorical and sheaf-theoretic frameworks available to us in order to formulate these games and strategies in a suitably clean and conceptual way. The games and strategies which determine them are more sophisticated mathematically, and also more difficult to define rigorously, than they were in the deterministic case. A new generation of 2-categorical and sheaf-theoretic game semantics is currently emerging in the field of programming language semantics. ![]()
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