A categorical characterization of a ${\scriptsize1}\!\!\!\!\bigcirc$-iteratively defined state of common knowledge

We present here a novel approach to the analysis of common knowledge based on category theory. In particular, we model the global epistemic state for a given set of agents through a hierarchy of beliefs represented by a presheaf construction. Then, by employing the properties of a categorical monad, we prove the existence of a state, obtained in an iterative fashion, in which all agents acquire common knowledge of some underlying statement. In order to guarantee the existence of a fixed point under certain suitable conditions, we make use of the properties entailed by Sergeyev's numeral system called grossone, which allows a finer control on the relevant structure of the infinitely nested epistemic states.