  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.
