Game comonads offer a categorical view of a number of model-comparison games central to model theory, such as pebble and Ehrenfeucht-Fra\"iss\'e games. Remarkably, the categories of coalgebras for these comonads capture preservation of several fragments of resource-bounded logics, such as (infinitary) first-order logic with n variables or bounded quantifier rank, and corresponding combinatorial parameters such as tree-width and tree-depth. In this way, game comonads provide a new bridge between categorical methods developed for semantics, and the combinatorial and algorithmic methods of resource-sensitive model theory. We give an overview of this framework and outline some of its applications, including the study of homomorphism counting results in finite model theory, and of equi-resource homomorphism preservation theorems in logic using the axiomatic setting of arboreal categories. Finally, we describe some homotopical ideas that arise naturally in the context of game comonads.

翻译：博弈共单子为模型论中一系列核心模型比较博弈（如卵石博弈和Ehrenfeucht-Fraïssé博弈）提供了范畴论视角。值得注意的是，这些共单子的余代数范畴捕捉了若干资源有界逻辑片段（例如具有n个变量或有界量词秩的（无穷）一阶逻辑）的保持性，以及相应的组合参数（如树宽和树深）。通过这种方式，博弈共单子在为语义学发展的范畴论方法与资源敏感模型论的组合及算法方法之间架起了一座新的桥梁。本文概述了这一框架，并勾勒了其部分应用，包括有限模型论中的同态计数结果研究，以及利用树状范畴的公理化设定在逻辑中探讨等资源同态保持定理。最后，我们描述了在博弈共单子语境下自然出现的一些同伦思想。