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.

@article{paperbot2760,
  title = {The Pebble-Relation Comonad in Finite Model Theory},
  author = {Yoàv Montacute and Nihil Shah},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 20, Issue 2},
  year = {2024},
  doi = {10.46298/lmcs-20(2:9)2024}
}