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.

翻译：由Abramsky、Dawar和Wang引入的卵石余单子，为有限模型论中的k-卵石游戏提供了范畴论解释。该卵石余单子的余Kleisli范畴刻画了无穷k-变量逻辑不同片段及扩展下的等价关系。此外，该卵石余单子上的余代数刻画了树宽，并与树分解相对应。本文引入卵石关系余单子，该结构刻画了路径宽度，其上的余代数对应于路径分解。我们进一步证明，在该余单子中存在余Kleisli态射等价于在k-变量无穷逻辑的受限合取片段中真值保持性成立。这一结论通过Dalmau的卵石关系游戏及一个等价的全合一卵石游戏得以证明。随后，我们通过全合一卵石游戏的双射版本，对相应的余Kleisli同构给出了类似处理。最后，作为推论，我们证明了一个新的Lovász型定理，该定理将路径宽度与带计数量词的k-变量无穷逻辑的受限合取片段联系起来。