一句话：引入 pebble-relation 余单子，用余Kleisli 映射刻画 k 变量无穷逻辑受限合取片段，余代数对应 pathwidth 与路径分解。
Abramsky–Dawar–Wang 的 pebbling 余单子为有限模型论中的 k-pebble 游戏提供范畴解释：其余Kleisli 范畴刻画无穷 k 变量逻辑不同片段的等价，其余代数表征 treewidth 并对应树分解。本文引入 pebble-relation 余单子，表征 pathwidth，并使其余代数对应路径分解。作者证明该余单子的余Kleisli 态射等价于 k 变量无穷逻辑受限合取片段的真值保持，证明使用 Dalmau 的 pebble-relation 游戏及等价的 all-in-one pebble 游戏；又用带隐藏 pebble 放置的双射版 all-in-one 游戏处理余Kleisli 同构。推论给出联系 pathwidth 与带计数量词的该逻辑片段的新 Lovász 型定理。
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 with a hidden pebble placement. Finally, we show as a consequence a new Lovász-type theorem relating pathwidth to the restricted conjunction fragment of k-variable infinitary logic with counting quantifiers.