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We compare two approaches to concurrent game semantics, one by Tsukada and Ong for a simply-typed λ-calculus and the other by the authors and collaborators for CCS and the π-calculus. The two approaches are obviously related, as they both define strategies as sheaves for the Grothendieck topology induced by embedding ‘views’ into ‘plays’. However, despite this superficial similarity, the notions of views and plays differ significantly: the former is based on standard justified sequences, the latter uses string diagrams. In this paper, we relate both approaches at the level of plays. Specifically, we design a notion of play (resp. view) for the simply-typed λ-calculus, based on string diagrams as in our previous work, into which we fully embed Tsukada and Ong's plays (resp. views). We further provide a categorical explanation of why both notions yield essentially the same model, thus demonstrating that the difference is a matter of presentation. In passing, we introduce an abstract framework for producing sheaf models based on string diagrams, which unifies our present and previous models.

Justified Sequences in String Diagrams: a Comparison Between Two Approaches to Concurrent Game Semantics