  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.
