Lov\'asz-Type Theorems and Game Comonads

TL;DR

This paper introduces a new categorical framework that generalizes Lovász's theorem, linking homomorphism counts to structural equivalences across various mathematical contexts, including graphs, logic, and modal logic.

Contribution

It provides a unifying categorical formulation that encompasses existing Lovász-type theorems and connects them through game comonads, extending their applicability.

Findings

Unified categorical framework for Lovász-type theorems

Characterization of graph equivalences via homomorphism counts

Application to modal logic homomorphism counts

Abstract

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.