Simple game semantics and Day convolution

TL;DR

This paper refines a categorical construction for simple game semantics using factorisation systems, aligning it with standard models and connecting it to Day convolution on presheaf categories.

Contribution

It introduces a refined categorical framework for simple game semantics that matches standard models and relates to Day convolution structures.

Findings

Refined the categorical construction for simple game semantics.

Showed the new construction yields standard categories of games and strategies.

Connected the construction to Day's convolution on presheaves.

Abstract

Game semantics has provided adequate models for a variety of programming languages, in which types are interpreted as two-player games and programs as strategies. Melli\`es (2018) suggested that such categories of games and strategies may be obtained as instances of a simple abstract construction on weak double categories. However, in the particular case of simple games, his construction slightly differs from the standard category. We refine the abstract construction using factorisation systems, and show that the new construction yields the standard category of simple games and strategies. Another perhaps surprising instance is Day's convolution monoidal structure on the category of presheaves over a strict monoidal category.