Discrete Density Comonads and Graph Parameters

Game comonads have brought forth a new approach to studying finite model theory categorically. By representing comparison games semantically as comonads, they allow important logical and combinatorial properties be exressed in terms of their Eilenberg-Moore coalgebras. As result, number results from theory, such preservation theorems homomorphism counting theorems, been formalised parameterised by giving rise simply varying the comonad. In this paper we study limits comonadic homomorphism-counting aspect regardless whether any are involved. We show that standard graph parameter has corresponding comonad, classifying same class. This comonad is constructed via simple Kan extension formula, making it initial solution problem and, furthermore, automatically admitting theorem.