  In previous work, Abramsky, Dawar and Wang (LiCS 2017) and Abramsky and Shah
(CSL 2018) have shown how a range of model comparison games which play a
central role in finite model theory, including Ehrenfeucht-Fraisse, pebbling,
and bisimulation games, can be captured in terms of resource-indexed comonads
on the category of relational structures. Moreover, the coalgebras for these
comonads capture important combinatorial parameters such as tree-width and
tree-depth.
  The present paper extends this analysis to quantifier-guarded fragments of
first-order logic. We give a systematic account, covering atomic, loose and
clique guards. In each case, we show that coKleisli morphisms capture winning
strategies for Duplicator in the existential guarded bisimulation game, while
back-and-forth bisimulation, and hence equivalence in the full guarded
fragment, is captured by spans of open morphisms. We study the coalgebras for
these comonads, and show that they correspond to guarded tree decompositions.
We relate these constructions to a syntax-free setting, with a comonad on the
category of hypergraphs.
