Abramsky, Dawar, and Wang (2017) introduced the pebbling comonad for k-variable counting logic and thereby initiated a line of work that imports category theoretic machinery to finite model theory. Such game comonads have been developed for various logics, yielding characterisations of logical equivalences in terms of isomorphisms in the associated co-Kleisli category. We show a first limitation of this approach by studying linear-algebraic logic, which is strictly more expressive than first-order counting logic and whose k-variable logical equivalence relations are known as invertible-map equivalences (IM). We show that there exists no finite-rank comonad on the category of graphs whose co-Kleisli isomorphisms characterise IM-equivalence, answering a question of \'O Conghaile and Dawar (CSL 2021). We obtain this result by ruling out a characterisation of IM-equivalence in terms of homomorphism indistinguishability and employing the Lov\'asz-type theorems for game comonads established by Dawar, Jakl, and Reggio (2021). Two graphs are homomorphism indistinguishable over a graph class if they admit the same number of homomorphisms from every graph in the class. The IM-equivalences cannot be characterised in this way, neither when counting homomorphisms in the natural numbers, nor in any finite prime field.

翻译：阿布拉姆斯基、达瓦尔和王(2017)引入了用于k变量计数逻辑的博弈共子，从而开启了一系列将范畴论工具引入有限模型理论的工作。此类游戏共子已被开发用于多种逻辑，在相应的余克莱斯利范畴中通过同构关系刻画了逻辑等价性。我们通过研究线性代数逻辑展示了该方法的第一个局限性——该逻辑严格强于一阶计数逻辑，其k变量逻辑等价关系被称为可逆映射等价(IM)。我们证明对于图范畴，不存在有限秩的共子使得其余克莱斯利范畴中的同构能够刻画IM等价，这回应了Ó康海勒和达瓦尔(CSL 2021)提出的问题。通过排除用同态不可区分性刻画IM等价的可能，并运用达瓦尔、雅克尔和雷吉奥(2021)建立的游戏共子洛夫式定理，我们得到了这一结论。若两个图在某个图类上从该类中每个图接收到的同态数量相同，则称它们在该图类上同态不可区分。IM等价无法通过这种方式刻画——无论是通过自然数还是任何有限素域中的同态计数。