Lov\'asz (1967) showed that two finite relational structures A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B for any finite structure C. Soon after, Pultr (1973) proved a categorical generalisation of this fact. We propose a new categorical formulation, which applies to any locally finite category with pushouts and a proper factorisation system. As special cases of this general theorem, we obtain two variants of Lov\'asz' theorem: the result by Dvo\v{r}\'ak (2010) that characterises equivalence of graphs in the k-dimensional Weisfeiler-Leman equivalence by homomorphism counts from graphs of tree-width at most k, and the result of Grohe (2020) characterising equivalence with respect to first-order logic with counting and quantifier depth k in terms of homomorphism counts from graphs of tree-depth at most k. The connection of our categorical formulation with these results is obtained by means of the game comonads of Abramsky et al. We also present a novel application to homomorphism counts in modal logic.

中文翻译：

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Lovász型定理和博弈公式

Lov'asz（1967）表明，当且仅当从C到A的同态数与对于任何有限结构C的从C到B的同态数相同时，两个有限关系结构A和B是同构的。不久之后，Pultr（1973）证明了这一事实的分类概括。我们提出了一种新的分类公式，该公式适用于任何带有出入点和适当分解系统的局部有限类别。作为该一般定理的特例，我们获得了Lov \'asz'定理的两个变体：Dvo \ v {r} \'ak（2010）的结果表征了k维Weisfeiler-Leman等价图的等价性。从最多k的树宽图计算同态，格罗（Grohe（2020））得出的结果是，在一阶逻辑上具有等价性的计数和量化器深度为k的等价性来自树深度图k。我们的分类表述与这些结果的联系是通过Abramsky等人的博弈论得出的。我们还提出了模态逻辑中同态计数的一种新颖应用。