Classical sofic entropy theory, a la Lewis Bowen, counts the number of good colorings (models) on a sofic approximation to define the basic invariant of interest. In our approach with Benjy Weiss (also independently investigated by Tim Austin) one models the infinite process by a random coloring and uses classical Shannon's entropy to define sofic entropy.
Using this language we present a weak Yuzvinsky formula for factor maps, giving a lower bound on the entropy of the kernel. This in particular implies a strengthening of Gromov’s surjunctivity theorem. This is that prompted him to introduce sofic groups. We show that if a cellular map is not surjective, then typical fibers have continuum cardinality.
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