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.

中文翻译：

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异步模板游戏和2类的灰色张量积

在最近的关于模板游戏和线性逻辑的探索性工作中，Melli \ es将连续游戏和并发游戏定义为类别，位置作为对象，轨迹作为态射，并由特定的同步模板标记。在本文中，我们将这一概念提出了更高的观点，并主张模板游戏不应仅定义为一维类别，而应将位置，轨迹和改组（或重新安排）的二维类别定义为2个单元。为了达到这个目的，我们认真地考虑了并发异步与2类格雷张量积之间的平行关系。途中的一个技术难题是，配备了灰色张量积的小2类类别的S = 2-Cat类是单曲面的，而不是笛卡尔的。这促使我们扩展了由Melli \ es最初在有限类别S中制定的模板游戏的框架，并将其以Aguiar在量子群上的工作方式进行了升级，使其更趋向于具有coreflexive的单曲面类别S的一般情况。均衡器，由张量积按分量保存。我们以这种方式构造了乘性加法线性逻辑（MALL）的异步模板游戏语义，其中每个公式和每个证明都被解释为带有标记的2类，分别配备了用于异步模板游戏的Gray comonoid和Gray的结构。 bicomodule用于异步策略。s在量子群上的工作，到具有coreflexive均衡器的单曲面类S的更一般情况，并由张量积分量保留。我们以这种方式构造了乘性加法线性逻辑（MALL）的异步模板游戏语义，其中每个公式和每个证明都被解释为带有标记的2类，分别配备了用于异步模板游戏的Gray comonoid和Gray的结构。 bicomodule用于异步策略。s在量子群上的工作，到具有coreflexive均衡器的单曲面类S的更一般情况，并由张量积分量保留。我们以这种方式构造了乘性加法线性逻辑（MALL）的异步模板游戏语义，其中每个公式和每个证明都被解释为带有标记的2类，分别配备了用于异步模板游戏的Gray comonoid和Gray的结构。 bicomodule用于异步策略。