An invitation to game comonads

Abstract

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.