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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 …