  The pebbling comonad, introduced by Abramsky, Dawar and Wang, provides a
categorical interpretation for the k-pebble games from finite model theory. The
coKleisli category of the pebbling comonad specifies equivalences under
different fragments and extensions of infinitary k-variable logic. Moreover,
the coalgebras over this pebbling comonad characterise treewidth and correspond
to tree decompositions. In this paper we introduce the pebble-relation comonad,
which characterises pathwidth and whose coalgebras correspond to path
decompositions. We further show that the existence of a coKleisli morphism in
this comonad is equivalent to truth preservation in the restricted conjunction
fragment of k-variable infinitary logic. We do this using Dalmau's
pebble-relation game and an equivalent all-in-one pebble game. We then provide
a similar treatment to the corresponding coKleisli isomorphisms via a bijective
version of the all-in-one pebble game. Finally, we show as a consequence a new
Lov\'asz-type theorem relating pathwidth to the restricted conjunction fragment
of k-variable infinitary logic with counting quantifiers.
