Asynchronous Template Games and the Gray Tensor Product of 2-Categories

In his recent and exploratory work on template games and linear logic, Melli\`es defines sequential and concurrent games as categories with positions as objects and trajectories as morphisms, labelled by a specific synchronization template. In the present paper, we bring the idea one dimension higher and advocate that template games should not be just defined as 1-dimensional categories but as 2-dimensional categories of positions, trajectories and reshufflings (or reschedulings) as 2-cells. In order to achieve the purpose, we take seriously the parallel between asynchrony in concurrency and the Gray tensor product of 2-categories. One technical difficulty on the way is that the category S=2-Cat of small 2-categories equipped with the Gray tensor product is monoidal, and not cartesian. This prompts us to extend the framework of template games originally formulated by Melli\`es in a category S with finite limits, and to upgrade it in the style of Aguiar's work on quantum groups to the more general situation of a monoidal category S with coreflexive equalizers, preserved by the tensor product componentwise. We construct in this way an asynchronous template game semantics of multiplicative additive linear logic (MALL) where every formula and every proof is interpreted as a labelled 2-category equipped, respectively, with the structure of Gray comonoid for asynchronous template games, and of Gray bicomodule for asynchronous strategies.