A Pebbling Comonad for Finite Rank and Variable Logic, and an Application to the Equirank-variable Homomorphism Preservation Theorem

In this paper we recast the proof of Rossman's equirank homomorphism preservation theorem using comonadic formulations of bounded quantifier rank and variable count (and dually tree width and tree-depth), and work towards generalisation of it that simultaneously preserves quantifier rank and variable count. Along the way, we give an exposition of the required comonads, showing how their properties arise.