The Garden of Eden theorem has the dual role of linking surjectivity of cellular automata to pre-injectivity and balance, and to characterize the class of amenable groups. We give an overview of several properties which, for cellular automata , are linked to surjectivity, and the role of several group properties in establishing such links.

Cellular automata are an archetypical comonadic notion of computation in that computation happens in the coKleisli category of a comonad. In this paper, we show that they can also be viewed as graded comonadic—a perspective that turns out to be both more informative and also more basic. We also discuss additive cellular automata to show that they admit both a graded comonadic and a graded monadic view. That these two perspectives are simultaneously available in this special case arises from a graded version of an observation by Kleiner about adjoint comonad-monad pairs.