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 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引入的卵石伴子（pebbling comonad）为有限模型论中的k-卵石博弈提供了范畴论解释。该卵石伴子的coKleisli范畴刻画了无穷变量k-逻辑在不同片段和扩展下的等价关系。此外，该卵石伴子上的余代数刻画树宽（treewidth）并与树分解相对应。本文引入卵石-关系伴子（pebble-relation comonad），该伴子刻画路径宽（pathwidth）且其上的余代数对应于路径分解。我们进一步证明，该伴子中coKleisli态射的存在性等价于k-变量无穷逻辑受限合取片段中的真值保持性。为此，我们借助Dalmau的卵石-关系博弈及其等价的整合型卵石博弈。随后，我们通过带有隐藏卵石放置的双射版整合型卵石博弈，对相应的coKleisli同构给出类似处理。最后，我们推导出一个新的Lovász型定理，该定理将路径宽与带计数量词的k-变量无穷逻辑的受限合取片段相关联。