Game comonads, introduced by Abramsky, Dawar and Wang and developed by Abramsky and Shah, give an interesting categorical semantics to some Spoiler-Duplicator games that are common in finite model theory. In particular they expose connections between one-sided and two-sided games, and parameters such as treewidth and treedepth and corresponding notions of decomposition. In the present paper, we expand the realm of game comonads to logics with generalised quantifiers. In particular, we introduce a comonad graded by two parameter $n \leq k$ such that isomorphisms in the resulting Kleisli category are exactly Duplicator winning strategies in Hella's $n$-bijection game with $k$ pebbles. We define a one-sided version of this game which allows us to provide a categorical semantics for a number of logics with generalised quantifiers. We also give a novel notion of tree decomposition that emerges from the construction.

中文翻译：

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游戏共生体和广义量词

由 Abramsky、Dawar 和 Wang 引入并由 Abramsky 和 ​​Shah 开发的博弈共生体为有限模型理论中常见的一些 Spoiler-Duplicator 博弈提供了有趣的分类语义。特别是它们揭示了单边和双边博弈之间的联系，以及树宽和树深等参数以及相应的分解概念。在本文中，我们将博弈共生的领域扩展到具有广义量词的逻辑。特别地，我们引入了一个由两个参数 $n \leq k$ 分级的共生体，这样得到的 Kleisli 类别中的同构就是 Hella 的 $n$-双射游戏中的复制器获胜策略，其中包含 $k$ 鹅卵石。我们定义了这个游戏的单方面版本，它允许我们为具有广义量词的许多逻辑提供分类语义。