Discrete density comonads and graph parameters

TL;DR

This paper explores the limits of game comonads in finite model theory, showing that any graph parameter can be represented by a corresponding comonad, which also supports homomorphism counting.

Contribution

It introduces a universal construction for graph parameters via comonads, extending the categorical approach beyond model comparison games.

Findings

Any standard graph parameter has a corresponding comonad.

The constructed comonad admits a homomorphism-counting theorem.

The approach is based on a simple Kan extension formula.

Abstract

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