  Pebble games are a powerful tool in the study of finite model theory,
constraint satisfaction and database theory. Monads and comonads are basic
notions of category theory which are widely used in semantics of computation
and in modern functional programming. We show that existential k-pebble games
have a natural comonadic formulation. Winning strategies for Duplicator in the
k-pebble game for structures A and B are equivalent to morphisms from A to B in
the coKleisli category for this comonad. This leads on to comonadic
characterisations of a number of central concepts in Finite Model Theory: -
Isomorphism in the co-Kleisli category characterises elementary equivalence in
the k-variable logic with counting quantifiers. - Symmetric games corresponding
to equivalence in full k-variable logic are also characterized. - The treewidth
of a structure A is characterised in terms of its coalgebra number: the least k
for which there is a coalgebra structure on A for the k-pebbling comonad. -
Co-Kleisli morphisms are used to characterize strong consistency, and to give
an account of a Cai-F\"urer-Immerman construction. - The k-pebbling comonad is
also used to give semantics to a novel modal operator. These results lay the
basis for some new and promising connections between two areas within logic in
computer science which have largely been disjoint: (1) finite and algorithmic
model theory, and (2) semantics and categorical structures of computation.
